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      "rationale": "E is a subset of R^r, so its complement is taken in R^r. Confirmed by Fremlin's official Volume 1 errata and corrected current official chap12.tex.",
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      "rationale": "If c belongs to E the interval is right-closed; otherwise it is open. The derivative swaps two adjacent formula atoms while preserving their topology; both formula records link this one correction.",
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      "target_text": "D\\cap\\ocint{-\\infty,c} atau D\\cap\\ooint{-\\infty,c}, masing-masing menurut apakah c\\in E atau tidak",
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      "rationale": "The clause defines and discusses the limit inferior as n tends to infinity; normalize the lone inconsistent concluding subscript.",
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    {
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      "authority_text": "\\sum_{k=1}^{\\infty}|a_k|+|b_k|=\\infty",
      "classification": "mathematical-summation-scope-clarification",
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      "math_ordinal": "435",
      "rationale": "Group both absolute-value terms under the summation so k is bound throughout the intended non-absolute-summability condition.",
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      "authority_text": "\\phi(x)=(\\phi_i(x),\\ldots,\\phi_r(x))",
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      "correction_id": "O007-CORR-0012",
      "math_ordinal": "910",
      "rationale": "The full coordinate tuple begins with the first coordinate; i is only the running index in the preceding phrase.",
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      "target_text": "\\phi(x)=(\\phi_1(x),\\ldots,\\phi_r(x))",
      "unit_id": "O007-FREMLIN-V1-S121"
    },
    {
      "authority_line": "111",
      "authority_path": "authority/fremlin/source/mt1.2011/mt122.tex",
      "authority_text": "\\sum_{i=0}^n b_i\\chi G_i(x)",
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      "correction_id": "O007-CORR-0013",
      "math_ordinal": "95",
      "rationale": "The disjoint family and its coefficients are indexed from 0 through m in 122C(b); the immediately preceding construction confirms that n is the wrong upper bound here.",
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      "target_text": "\\sum_{i=0}^m b_i\\chi G_i(x)",
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    },
    {
      "authority_line": "155-156",
      "authority_path": "authority/fremlin/source/mt1.2011/mt122.tex",
      "authority_text": "pick on for the calculation",
      "classification": "source-grammatical-error",
      "correction_id": "O007-CORR-0014",
      "math_ordinal": "",
      "rationale": "Translate the intended statement after removing the stray word 'on'.",
      "source_normalized_sha256": "",
      "target_line": "156-157",
      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt122.tex",
      "target_text": "representasi $f$ yang kita pilih untuk perhitungan",
      "unit_id": "O007-FREMLIN-V1-S122"
    },
    {
      "authority_line": "283-284;334-335",
      "authority_path": "authority/fremlin/source/mt1.2011/mt122.tex",
      "authority_text": "an non-decreasing sequence",
      "classification": "source-grammatical-error",
      "correction_id": "O007-CORR-0015",
      "math_ordinal": "",
      "rationale": "Both occurrences use the wrong English indefinite article; the Indonesian derivative states the intended non-decreasing-sequence condition naturally.",
      "source_normalized_sha256": "",
      "target_line": "284-285;334-335",
      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt122.tex",
      "target_text": "barisan tak-menurun",
      "unit_id": "O007-FREMLIN-V1-S122"
    },
    {
      "authority_line": "340",
      "authority_path": "authority/fremlin/source/mt1.2011/mt122.tex",
      "authority_text": "\\sup\\{\\int g:g\\le f\\text{ is simple}\\}",
      "classification": "mathematical-predicate-order-and-qualifier-error",
      "correction_id": "O007-CORR-0016",
      "math_ordinal": "256",
      "rationale": "The source expression is syntactically malformed and omits the almost-everywhere qualifier. The corrected predicate exactly matches 122J(a)(gamma) and the repeated comparison at source line 359.",
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      "target_path": "source/id-ID/mt122.tex",
      "target_text": "\\sup\\{\\int g:g\\text{ adalah fungsi sederhana},\\,g\\leae f\\}",
      "unit_id": "O007-FREMLIN-V1-S122"
    },
    {
      "authority_line": "354",
      "authority_path": "authority/fremlin/source/mt1.2011/mt123.tex",
      "authority_text": "\\int\\limsup_{n\\to\\infty}f_n\\ge\\limsup_{n\\to\\infty}f_n",
      "classification": "mathematical-missing-integral-operator",
      "correction_id": "O007-CORR-0017",
      "math_ordinal": "262",
      "rationale": "Exercise 123Xd states the reverse-Fatou inequality for scalar integrals under an integrable dominator. The source right-hand side is a function and is dimensionally incompatible with the scalar left-hand side; inserting the missing integral yields the standard consequence of applying Fatou to -f_n.",
      "source_normalized_sha256": "c5102ef1ba28f1c0075c1fee9ce1cfd256cdeaf194fcfe0bcd772a78e0b29f71",
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      "target_path": "source/id-ID/mt123.tex",
      "target_text": "\\int\\limsup_{n\\to\\infty}f_n\\ge\\limsup_{n\\to\\infty}\\int f_n",
      "unit_id": "O007-FREMLIN-V1-S123"
    },
    {
      "authority_line": "214",
      "authority_path": "authority/fremlin/source/mt1.2011/mt131.tex",
      "authority_text": "fd\\mu=\\int_{\\coint{a,b}}fd\\mu=\\int_{\\ocint{a,b}}fd\\mu=\\int_{[a,b]}fd\\mu",
      "classification": "mathematical-missing-integral-operator",
      "correction_id": "O007-CORR-0018",
      "math_ordinal": "212",
      "rationale": "Exercise 131Xb(i) and its hint require four endpoint variants, but the authority display begins with bare f dmu and contains only three integral signs. The official PDF replay confirms the malformed first term; restoring the open-interval integral yields the required fourth variant.",
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      "target_line": "239-240",
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      "target_path": "source/id-ID/mt131.tex",
      "target_text": "\\int_{\\ooint{a,b}}fd\\mu=\\int_{\\coint{a,b}}fd\\mu=\\int_{\\ocint{a,b}}fd\\mu=\\int_{[a,b]}fd\\mu",
      "unit_id": "O007-FREMLIN-V1-S131"
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      "authority_line": "116",
      "authority_path": "authority/fremlin/source/mt1.2011/mt131.tex",
      "authority_text": "f\\restr E\\cap H",
      "classification": "source-notation-grouping-clarification",
      "correction_id": "O007-CORR-0019",
      "math_ordinal": "114",
      "rationale": "The preceding level-set equality and the claimed Sigma_H-measurability require restriction to the domain E intersection H. Parentheses make that intended scope explicit without changing the mathematical content.",
      "source_normalized_sha256": "c658fa5cf9f228f51dfe90790886bb293945fdd6673a94acfbca4510b1e25008",
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      "target_path": "source/id-ID/mt131.tex",
      "target_text": "f\\restr(E\\cap H)",
      "unit_id": "O007-FREMLIN-V1-S131"
    },
    {
      "authority_line": "325",
      "authority_path": "authority/fremlin/source/mt1.2011/mt136.tex",
      "authority_text": "\\mu(E\\cap F)\\le\\epsilon",
      "classification": "mathematical-wrong-set-operation",
      "correction_id": "O007-CORR-0020",
      "math_ordinal": "338",
      "rationale": "The proposition is the density-in-measure statement proved immediately below: its defining class and every estimate use symmetric difference. The printed intersection claim is trivial by choosing the empty algebra set and does not follow from the proof; replacing intersection with symmetric difference restores the theorem actually proved.",
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      "target_path": "source/id-ID/mt136.tex",
      "target_text": "\\mu(F\\symmdiff E)\\le\\epsilon",
      "unit_id": "O007-FREMLIN-V1-S136"
    },
    {
      "authority_line": "330",
      "authority_path": "authority/fremlin/source/mt1.2011/mt133.tex",
      "authority_text": "g_1=g-2^{-n}\\chi F",
      "classification": "mathematical-index-error",
      "correction_id": "O007-CORR-0021",
      "math_ordinal": "241",
      "rationale": "The preceding sentence chooses n with positive measure for F_n, and the next displayed integral subtracts 2^{-n} times mu(F_n). Subtracting the characteristic function of all F would not yield that integral and need not preserve the required domination; the indexed set F_n is the proof's intended set.",
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      "target_path": "source/id-ID/mt133.tex",
      "target_text": "g_1=g-2^{-n}\\chi F_n",
      "unit_id": "O007-FREMLIN-V1-S133"
    },
    {
      "authority_line": "464",
      "authority_path": "authority/fremlin/source/mt1.2011/mt133.tex",
      "authority_text": "g_n=\\inf_{m\\ge n}f_n",
      "classification": "mathematical-bound-variable-error",
      "correction_id": "O007-CORR-0022",
      "math_ordinal": "359",
      "rationale": "The infimum must range over the tail functions f_m. Using f_n makes the bound variable m vacuous, contradicts the following inequality against inf_{m>=n} of the upper integrals, and does not construct the monotone tail-infimum sequence needed for liminf.",
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      "target_path": "source/id-ID/mt133.tex",
      "target_text": "g_n=\\inf_{m\\ge n}f_m",
      "unit_id": "O007-FREMLIN-V1-S133"
    },
    {
      "authority_line": "158-161",
      "authority_path": "authority/fremlin/source/mt1.2011/mt136.tex",
      "authority_text": "If sequence A_n is disjoint in Cal A, then mu A = sum mu A_n = sum nu A_n = nu A",
      "classification": "mathematical-undefined-symbol",
      "correction_id": "O007-CORR-0023",
      "math_ordinal": "target-insertion-170",
      "rationale": "The authority proof uses A in the displayed equality without defining it. Declaring A as the union of the disjoint sequence is required by countable additivity and is the unique definition that makes the displayed calculation meaningful.",
      "source_normalized_sha256": "",
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      "target_path": "source/id-ID/mt136.tex",
      "target_text": "Jika barisan A_n saling lepas dalam Cal A dan A=union A_n, maka mu A=sum mu A_n=sum nu A_n=nu A",
      "unit_id": "O007-FREMLIN-V1-S136"
    },
    {
      "authority_line": "385-399",
      "authority_path": "authority/fremlin/source/mt1.2011/mt136.tex",
      "authority_text": "Exercise 136Xc invokes mu and nu without declaring either measure or their domains/agreement hypotheses",
      "classification": "mathematical-missing-hypotheses",
      "correction_id": "O007-CORR-0024",
      "math_ordinal": "target-insertions-416-421",
      "rationale": "The requested conclusion and hint use mu, nu and the restricted measures mu_J, nu_J, but the authority prompt never introduces them. Adding the finite agreement hypotheses from 136C makes the exercise well-typed and exactly supports the stated hint; all six inserted math atoms are hash-bound in qa/mt136-structural-qa.json.",
      "source_normalized_sha256": "",
      "target_line": "395-397",
      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt136.tex",
      "target_text": "Misalkan pula mu dan nu dua ukuran pada X, keduanya berdomain Sigma, sedemikian sehingga mu I=nu I<infinity untuk setiap I dalam Cal I.",
      "unit_id": "O007-FREMLIN-V1-S136"
    },
    {
      "authority_line": "405",
      "authority_path": "authority/fremlin/source/mt1.2011/mt134.tex",
      "authority_text": "H_2=\\bigcap_{n\\in\\Bbb N}G_n",
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      "math_ordinal": "235",
      "rationale": "The construction immediately above proves that every G'_n contains E and that mu(G'_n minus E) tends to zero. The unprimed G_n contain only E_n and need not contain E, so their intersection cannot support the stated conclusion; H_2 must be the intersection of the primed tail unions.",
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      "target_path": "source/id-ID/mt134.tex",
      "target_text": "H_2=\\bigcap_{n\\in\\Bbb N}G'_n",
      "unit_id": "O007-FREMLIN-V1-S134"
    },
    {
      "authority_line": "849",
      "authority_path": "authority/fremlin/source/mt1.2011/mt134.tex",
      "authority_text": "\\underline{\\int}fd\\nu",
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      "math_ordinal": "570",
      "rationale": "The entire proposition and both Darboux estimates use Lebesgue measure mu; nu is never introduced in this argument. Replacing the undefined nu with mu restores the equality between the lower Lebesgue integral and the Riemann integral established by the surrounding inequalities.",
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      "target_path": "source/id-ID/mt134.tex",
      "target_text": "\\underline{\\int}fd\\mu",
      "unit_id": "O007-FREMLIN-V1-S134"
    },
    {
      "authority_line": "385-386",
      "authority_path": "authority/fremlin/source/mt1.2011/mt1a1.tex",
      "authority_text": "is a partition partition of X",
      "classification": "source-duplicated-word",
      "correction_id": "O007-CORR-0027",
      "math_ordinal": "",
      "rationale": "The authority repeats 'partition' across a line break. The Indonesian derivative states the intended definition once.",
      "source_normalized_sha256": "",
      "target_line": "386",
      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt1a1.tex",
      "target_text": "adalah partisi dari X",
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    {
      "authority_line": "86-87",
      "authority_path": "authority/fremlin/source/mt1.2011/mt1a2.tex",
      "authority_text": "\\sum_{j=1}^r\\alpha\\beta\\xi_j^2+\\alpha\\beta\\eta_j^2+\\alpha^2\\eta_j^2+\\beta^2\\xi_j^2",
      "classification": "mathematical-summation-scope-error",
      "correction_id": "O007-CORR-0028",
      "math_ordinal": "70",
      "rationale": "The next equality sums all four j-dependent terms. Without grouping, only the first term lies under the summation sign and the displayed identity is false.",
      "source_normalized_sha256": "e7ca951f4fe69b1e7ba8f1c71d29183e9a35159c12595c900f79908500f134c9",
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      "target_path": "source/id-ID/mt1a2.tex",
      "target_text": "\\sum_{j=1}^r(\\alpha\\beta\\xi_j^2+\\alpha\\beta\\eta_j^2+\\alpha^2\\eta_j^2+\\beta^2\\xi_j^2)",
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    },
    {
      "authority_line": "116-117",
      "authority_path": "authority/fremlin/source/mt1.2011/mt1a3.tex",
      "authority_text": "\\limsup a_n=\\lim\\sup a_m\\ge\\lim\\inf a_m=\\limsup a_n",
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      "rationale": "This is the proof of liminf <= limsup. The final tail-infimum limit is liminf, as stated in Proposition 1A3B(a), not a second limsup.",
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      "target_text": "\\limsup a_n=\\lim\\sup a_m\\ge\\lim\\inf a_m=\\liminf a_n",
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    {
      "authority_line": "156",
      "authority_path": "authority/fremlin/source/mt1.2011/mt1a3.tex",
      "authority_text": "\\sup_{m\\ge n_1}a_n\\le v_1",
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      "rationale": "The supremum ranges over m, so its operand must be a_m. The source's a_n leaves m vacuous and cannot supply the following tail bound.",
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    {
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      "authority_path": "authority/fremlin/source/mt1.2011/mt1a3.tex",
      "authority_text": "\\sup_{m\\ge n_2}b_n\\le v_2",
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      "rationale": "The supremum ranges over m, so its operand must be b_m. The source's b_n leaves m vacuous and cannot supply the following tail bound.",
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    },
    {
      "authority_line": "159",
      "authority_path": "authority/fremlin/source/mt1.2011/mt1a3.tex",
      "authority_text": "\\sup_{m\\ge\\max(n_1,n_2)}a_m+b_m",
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      "rationale": "The proof needs the supremum of the pointwise sum. Parentheses make both summands part of the supremum operand and match Proposition 1A3B(d).",
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      "target_path": "source/id-ID/mt1a3.tex",
      "target_text": "\\sup_{m\\ge\\max(n_1,n_2)}(a_m+b_m)",
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    {
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      "authority_path": "authority/fremlin/source/mt1.2011/mt1a3.tex",
      "authority_text": "\\limsup a_n+b_n=\\inf_n\\sup_{m\\ge n}a_m+b_m",
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      "rationale": "The conclusion of part (d) concerns the limsup of the sum sequence. Grouping removes two free-standing addends and makes the equality mathematically well scoped.",
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      "target_text": "\\limsup(a_n+b_n)=\\inf_n\\sup_{m\\ge n}(a_m+b_m)",
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    {
      "authority_line": "174-175",
      "authority_path": "authority/fremlin/source/mt1.2011/mt1a3.tex",
      "authority_text": "\\liminf a_n+b_n=-\\limsup(-a_n)+(-b_n)",
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      "rationale": "Part (e) applies part (d) to the pointwise sum of the negated sequences. Parentheses are required on both sides; otherwise b_n lies outside the limit operator.",
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      "target_text": "\\liminf(a_n+b_n)=-\\limsup((-a_n)+(-b_n))",
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    },
    {
      "authority_line": "241",
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      "authority_text": "$g$",
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      "authority_line": "75",
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      "authority_text": "$\\mu(A\\setminus\\bigcup\\Cal I_0)=0$",
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      "math_ordinal": "56",
      "rationale": "Vitali's theorem was applied to A_m and the following bound requires A_m to be covered up to a null set; the authority unexpectedly falls back to A in this one sentence.",
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      "authority_line": "290",
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      "authority_text": "$|F(x+h)-F(x))-hF'(x)|$",
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      "correction_id": "O007-CORR-0045",
      "math_ordinal": "231",
      "rationale": "The authority has an unmatched closing parenthesis after F(x); removing it restores the difference-quotient error term used in the next displayed estimate.",
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      "target_text": "$|F(x+h)-F(x)-hF'(x)|$",
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    },
    {
      "authority_line": "619",
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      "authority_text": "$(\\DiniD^+f)(x)=(\\Dinid^+f)(x)=(\\DiniD^-f)(x)=(\\Dinid^+f)(x)=c$",
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      "rationale": "The characterization must equate all four Dini derivatives; the authority repeats the lower right derivative and omits the lower left derivative.",
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      "target_text": "$(\\DiniD^+f)(x)=(\\Dinid^+f)(x)=(\\DiniD^-f)(x)=(\\Dinid^-f)(x)=c$",
      "unit_id": "O007-FREMLIN-V2-S222"
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      "authority_line": "648",
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      "authority_text": "$(\\Dinid^+f)(x)=-\\infty$ and $(\\DiniD^+f)(x)=\\infty$",
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      "unit_id": "O007-FREMLIN-V2-S222"
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      "authority_line": "676",
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      "authority_text": "$g$",
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      "authority_text": "$h$",
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    {
      "authority_line": "112",
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      "authority_text": "$\\sum_{j=1}^n|f(b_i)-f(b_{i-1})|$",
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      "rationale": "The second sum is indexed by j and ranges over the b-sequence; the authority's b_i leaves j unused and conflicts with the concatenation in the following sentence.",
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      "authority_line": "114",
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      "authority_text": "$\\Var_{[a,b]}(f)$",
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      "math_ordinal": "73",
      "rationale": "This proposition has arbitrary D and no interval [a,b]; the concatenated sequence lies in D, so the defining bound is Var_D(f), exactly as the conclusion below requires.",
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      "authority_line": "307",
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      "authority_text": "$2^{-n}\\le\\delta$",
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      "authority_text": "$f_2=h_1-\\gamma_2+\\beta_2$",
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      "authority_text": "$A=\\bigcup_{n\\in\\Bbb N}A_n$",
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      "authority_text": "$x_{nk}\\in\\dom f\\cap\\ocint{a_{n,k-1},a_{nk}}$",
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      "authority_text": "$f=f_1-f_2+if_3-if_4$ on $D$",
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      "authority_text": "$\\int_F|f|\\le\\epsilon$",
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      "authority_text": "$a\\le a_1\\le b_1\\ldots\\le a_n\\le b_n\\le b$",
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      "authority_text": "$\\bigcup_{i\\le m}f[\\,[c_i,d_i],]$",
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      "rationale": "The decomposition of F_m defines intervals [c_i,d_i] only for i through n, with n at most m; the image union must therefore run through n, not through the unrelated upper bound m.",
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      "target_path": "source/id-ID/mt225.tex",
      "target_text": "$\\bigcup_{i\\le n}f[\\,[c_i,d_i],]$",
      "unit_id": "O007-FREMLIN-V2-S225"
    },
    {
      "authority_line": "509-510",
      "authority_path": "authority/fremlin/source/mt2.2016/mt225.tex",
      "authority_text": "$\\mu_D(D\\cap G_n\\setminus F_n)$ and $\\mu(G_n\\setminus F_n)$",
      "classification": "mathematical-undefined-set-variable",
      "correction_id": "O007-CORR-0060",
      "math_ordinal": "367",
      "rationale": "Only E_n and G_n are defined in this construction; h_2 and the preceding set F also use G_n setminus E_n, so both occurrences of the undefined F_n must be E_n.",
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      "target_text": "$\\mu_D(D\\cap G_n\\setminus E_n)$ dan $\\mu(G_n\\setminus E_n)$",
      "unit_id": "O007-FREMLIN-V2-S225"
    },
    {
      "authority_line": "611",
      "authority_path": "authority/fremlin/source/mt2.2016/mt225.tex",
      "authority_text": "$\\le 2^{-k}\\lim_{n\\to\\infty}2^{-k}|y-x_n|$",
      "classification": "mathematical-duplicated-factor",
      "correction_id": "O007-CORR-0061",
      "math_ordinal": "427",
      "rationale": "The defining estimate contributes one factor 2^{-k}; the authority repeats it inside the limit but immediately concludes with a single factor, so removing the duplicate restores the stated bound.",
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      "target_path": "source/id-ID/mt225.tex",
      "target_text": "$\\le 2^{-k}\\lim_{n\\to\\infty}|y-x_n|$",
      "unit_id": "O007-FREMLIN-V2-S225"
    },
    {
      "authority_line": "674",
      "authority_path": "authority/fremlin/source/mt2.2016/mt225.tex",
      "authority_text": "$|f(x)-f_k(x)|\\le 2^{-k}$",
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      "correction_id": "O007-CORR-0062",
      "math_ordinal": "492",
      "rationale": "The construction uses p_i to approximate f'(x), and the next sentence concludes f' is the limit of f_k; the bound must therefore compare f_k with f', not with f.",
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      "target_text": "$|f'(x)-f_k(x)|\\le 2^{-k}$",
      "unit_id": "O007-FREMLIN-V2-S225"
    },
    {
      "authority_line": "728",
      "authority_path": "authority/fremlin/source/mt2.2016/mt225.tex",
      "authority_text": "$|x-c|\\le\\delta$",
      "classification": "mathematical-missing-punctured-neighborhood",
      "correction_id": "O007-CORR-0063",
      "math_ordinal": "548",
      "rationale": "The displayed difference quotient has denominator x-c and is undefined at x=c; adding the strict positive lower bound states the punctured neighborhood required by differentiability while leaving the preceding condition on g unchanged.",
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      "target_text": "$0<|x-c|\\le\\delta$",
      "unit_id": "O007-FREMLIN-V2-S225"
    },
    {
      "authority_line": "192",
      "authority_path": "authority/fremlin/source/mt2.2016/mt226.tex",
      "authority_text": "$J\\subseteq\\{i:a_i\\ne 0\\}$",
      "classification": "mathematical-false-converse-hypothesis",
      "correction_id": "O007-CORR-0064",
      "math_ordinal": "100",
      "rationale": "The converse identifies the arbitrary-index sum with the series over J; this is false if nonzero terms lie outside J. Taking J to be the complete nonzero support is exactly the condition constructed in the preceding paragraph and makes the converse valid.",
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      "target_line": "192",
      "target_normalized_sha256": "5bb090a2063cf82e49acef5e571be5d763fbca2301dc0867ebccd6a0055c8634",
      "target_path": "source/id-ID/mt226.tex",
      "target_text": "$J=\\{i:a_i\\ne 0\\}$",
      "unit_id": "O007-FREMLIN-V2-S226"
    },
    {
      "authority_line": "210",
      "authority_path": "authority/fremlin/source/mt2.2016/mt226.tex",
      "authority_text": "$K=\\{i:(i,j)\\in M\\}$",
      "classification": "mathematical-unbound-projection-variable",
      "correction_id": "O007-CORR-0065",
      "math_ordinal": "113",
      "rationale": "K is intended to be the first-coordinate projection of the finite set M, but j is free in the authority set-builder. The existential quantifier closes the definition and gives the finite projection used below.",
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      "target_line": "210",
      "target_normalized_sha256": "05b9ea1ed03c619967935f79f8a7a5d68b48714dac72ad5dd7a8d1297d8c44e9",
      "target_path": "source/id-ID/mt226.tex",
      "target_text": "$K=\\{i:\\exists j\\ (i,j)\\in M\\}$",
      "unit_id": "O007-FREMLIN-V2-S226"
    },
    {
      "authority_line": "211",
      "authority_path": "authority/fremlin/source/mt2.2016/mt226.tex",
      "authority_text": "$L=\\{j:(i,j)\\in M\\}$",
      "classification": "mathematical-unbound-projection-variable",
      "correction_id": "O007-CORR-0066",
      "math_ordinal": "114",
      "rationale": "L is intended to be the second-coordinate projection of the finite set M, but i is free in the authority set-builder. The existential quantifier closes the definition and gives the finite projection used below.",
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      "target_line": "211",
      "target_normalized_sha256": "a3035ffdc8e231a132ba6a82bed8908abc3ff2008b0642ff439952b492b264ef",
      "target_path": "source/id-ID/mt226.tex",
      "target_text": "$L=\\{j:\\exists i\\ (i,j)\\in M\\}$",
      "unit_id": "O007-FREMLIN-V2-S226"
    },
    {
      "authority_line": "281",
      "authority_path": "authority/fremlin/source/mt2.2016/mt226.tex",
      "authority_text": "as defined in (b)",
      "classification": "cross-reference-wrong-subsection",
      "correction_id": "O007-CORR-0067",
      "math_ordinal": "",
      "rationale": "Subsection 226B(a) defines saltus functions and their associated families; 226B(b) only defines one-sided-limit notation. The reference must therefore point to (a).",
      "source_normalized_sha256": "",
      "target_line": "282",
      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt226.tex",
      "target_text": "sebagaimana didefinisikan dalam (a)",
      "unit_id": "O007-FREMLIN-V2-S226"
    },
    {
      "authority_line": "403-407",
      "authority_path": "authority/fremlin/source/mt2.2016/mt226.tex",
      "authority_text": "four-family list lacks a conjunction between the u-double-prime and v-double-prime families",
      "classification": "source-editorial-missing-conjunction",
      "correction_id": "O007-CORR-0068",
      "math_ordinal": "",
      "rationale": "The intended four-family list is determined by the immediately preceding definitions; supplying the missing final conjunction repairs the syntax without changing any formula or assertion.",
      "source_normalized_sha256": "",
      "target_line": "411-415",
      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt226.tex",
      "target_text": "empat keluarga dipisahkan dengan koma dan 'dan' sebelum keluarga v-double-prime",
      "unit_id": "O007-FREMLIN-V2-S226"
    },
    {
      "authority_line": "249",
      "authority_path": "authority/fremlin/source/mt2.2016/mt211.tex",
      "authority_text": "$B\\subseteq\\Bbb R$",
      "classification": "mathematical-wrong-ambient-dimension",
      "correction_id": "O007-CORR-0069",
      "math_ordinal": "193",
      "rationale": "Section 211M defines Lebesgue measure on R^r.  The completeness test must therefore take an arbitrary subset B of that same ambient space; R is valid only in the special case r=1.",
      "source_normalized_sha256": "282a1dd4cf4b21034d7086d7605ea16151d7bb70722ed7c0ead3ccde983c1b22",
      "target_line": "267",
      "target_normalized_sha256": "5f3106e1ddbecda96e88d235a978b68760b104483a734daece907b4bb0997529",
      "target_path": "source/id-ID/mt211.tex",
      "target_text": "$B\\subseteq\\BbbR^r$",
      "unit_id": "O007-FREMLIN-V2-S211"
    },
    {
      "authority_line": "266",
      "authority_path": "authority/fremlin/source/mt2.2016/mt211.tex",
      "authority_text": "$E\\in\\Sigma$",
      "classification": "mathematical-missing-positive-measure-hypothesis",
      "correction_id": "O007-CORR-0070",
      "math_ordinal": "207",
      "rationale": "The proof immediately uses lim f(n)=mu E>0 and the intermediate value theorem to find a subset of measure strictly between 0 and mu E.  That conclusion requires E to have positive measure.",
      "source_normalized_sha256": "afe4bbaaedba5158924d3a0bd77f0304472650e71de5aed22515cc3a0a8e1bd2",
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      "target_normalized_sha256": "0a52244951b5859f7d5c463794b9d25f2de3dd97a545734588d6644dcb00eccd",
      "target_path": "source/id-ID/mt211.tex",
      "target_text": "$E\\in\\Sigma,\\ \\mu E>0$",
      "unit_id": "O007-FREMLIN-V2-S211"
    },
    {
      "authority_line": "511",
      "authority_path": "authority/fremlin/source/mt2.2016/mt211.tex",
      "authority_text": "$a\\in\\frak A$",
      "classification": "mathematical-wrong-quantifier-domain",
      "correction_id": "O007-CORR-0071",
      "math_ordinal": "462",
      "rationale": "The least-upper-bound condition in 211Ya must quantify over members of the selected subfamily A.  Quantifying over every member of the whole measure algebra would demand a top element rather than the supremum of A.",
      "source_normalized_sha256": "b538a05fd20d55430c234fcba4e12ff08942be0ea2947040cfc1c9a155d66366",
      "target_line": "563",
      "target_normalized_sha256": "1e105b215fd2d4e7c99395aa9447b20db49916f38921e66c942de0ab5c834282",
      "target_path": "source/id-ID/mt211.tex",
      "target_text": "$a\\in A$",
      "unit_id": "O007-FREMLIN-V2-S211"
    },
    {
      "authority_line": "526",
      "authority_path": "authority/fremlin/source/mt2.2016/mt211.tex",
      "authority_text": "$a\\in\\frak A$",
      "classification": "mathematical-wrong-quantifier-domain",
      "correction_id": "O007-CORR-0072",
      "math_ordinal": "481",
      "rationale": "The countable-supremum condition in 211Yb must quantify over members of the given countable subfamily A, exactly as in 211Ya, not over every member of the measure algebra.",
      "source_normalized_sha256": "b538a05fd20d55430c234fcba4e12ff08942be0ea2947040cfc1c9a155d66366",
      "target_line": "580",
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      "target_path": "source/id-ID/mt211.tex",
      "target_text": "$a\\in A$",
      "unit_id": "O007-FREMLIN-V2-S211"
    },
    {
      "authority_line": "526",
      "authority_path": "authority/fremlin/source/mt2.2016/mt211.tex",
      "authority_text": "$h\\subseteq g$",
      "classification": "mathematical-wrong-order-symbol",
      "correction_id": "O007-CORR-0073",
      "math_ordinal": "482",
      "rationale": "The objects h and g are equivalence classes in the measure algebra, ordered by the relation Bsubseteq introduced in 211Ya; ordinary set inclusion is not the defined order on these objects.",
      "source_normalized_sha256": "0b31539bcb64bc14349cc7f749a590c104b9b0a930fbc497739f60e84c70b8f0",
      "target_line": "580",
      "target_normalized_sha256": "ed942be50bd9d288bf34398559489502e6076cf05261595dce32dd1ce7e7b14d",
      "target_path": "source/id-ID/mt211.tex",
      "target_text": "$h\\Bsubseteq g$",
      "unit_id": "O007-FREMLIN-V2-S211"
    },
    {
      "authority_line": "136-137",
      "authority_path": "authority/fremlin/source/mt2.2016/mt212.tex",
      "authority_text": "$\\mu(E''\\setminus E)$",
      "classification": "mathematical-wrong-set-difference",
      "correction_id": "O007-CORR-0074",
      "math_ordinal": "98",
      "rationale": "The displayed additivity step is mu E'' = mu E' + mu(E'' setminus E').  Replacing E' by the intermediate set E makes the asserted equality false in general.",
      "source_normalized_sha256": "50870048ca8d8da9225d906ac1a61cb47741fc35c20484906f536e9a2729e073",
      "target_line": "142-143",
      "target_normalized_sha256": "a708d028901f9f819715a3d88cec826edcb7b199697b0348f3fcda4f9bcce493",
      "target_path": "source/id-ID/mt212.tex",
      "target_text": "$\\mu(E''\\setminus E')$",
      "unit_id": "O007-FREMLIN-V2-S212"
    },
    {
      "authority_line": "449",
      "authority_path": "authority/fremlin/source/mt2.2016/mt212.tex",
      "authority_text": "$(X,\\hat\\Sigma,\\mu)$",
      "classification": "mathematical-wrong-measure-component",
      "correction_id": "O007-CORR-0075",
      "math_ordinal": "420",
      "rationale": "This half of 212G proves that the completed space is semi-finite.  Its measure is hat-mu; the original mu is not defined on every member of hat-Sigma.",
      "source_normalized_sha256": "c6c9e568cc53c65d55ced5708f5a7af9c8bc08ed1dd0f5fbb3b288fae6b306a3",
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      "target_path": "source/id-ID/mt212.tex",
      "target_text": "$(X,\\hat\\Sigma,\\hat\\mu)$",
      "unit_id": "O007-FREMLIN-V2-S212"
    },
    {
      "authority_line": "257",
      "authority_path": "authority/fremlin/source/mt2.2016/mt213.tex",
      "authority_text": "$(X,\\Sigma,\\mu)$",
      "classification": "mathematical-wrong-space-tuple",
      "correction_id": "O007-CORR-0076",
      "math_ordinal": "190",
      "rationale": "Part 213D(d) has just proved local determination for the c.l.d. version.  The concluding tuple must therefore be the tilde space, not the original measure space.",
      "source_normalized_sha256": "05e9b769403b4c7926a944d5aaa03f00bb216412bc3f850e41b3c28de657aa67",
      "target_line": "260",
      "target_normalized_sha256": "3e134a815dc912ab5b295db4411e9f08536230e6a0834f40378e0190aec12436",
      "target_path": "source/id-ID/mt213.tex",
      "target_text": "$(X,\\tilde\\Sigma,\\tilde\\mu)$",
      "unit_id": "O007-FREMLIN-V2-S213"
    },
    {
      "authority_line": "298",
      "authority_path": "authority/fremlin/source/mt2.2016/mt213.tex",
      "authority_text": "$\\tilde\\mu H=\\sup\\{\\mu F:E\\in\\Sigma$",
      "classification": "mathematical-wrong-bound-variable",
      "correction_id": "O007-CORR-0077",
      "math_ordinal": "229",
      "rationale": "The supremum ranges over measurable subsets F of H.  E is otherwise free in this set-builder, while the following proof consistently chooses and varies F.",
      "source_normalized_sha256": "79eef544bd0a0b95ad2c91a0bdf39faeb5a53ff98633eb7a2736a59d5dee5c29",
      "target_line": "301",
      "target_normalized_sha256": "3c939ad24373944bef1be5a9c63af8dd571afac3a55755438099adbed172f7a7",
      "target_path": "source/id-ID/mt213.tex",
      "target_text": "$\\tilde\\mu H=\\sup\\{\\mu F:F\\in\\Sigma$",
      "unit_id": "O007-FREMLIN-V2-S213"
    },
    {
      "authority_line": "575",
      "authority_path": "authority/fremlin/source/mt2.2016/mt213.tex",
      "authority_text": "$\\mu(H\\symmdiff F)=0$",
      "classification": "mathematical-wrong-measure-component",
      "correction_id": "O007-CORR-0078",
      "math_ordinal": "466",
      "rationale": "Here H is only assumed to belong to tilde-Sigma, so H symmetric-difference F need not lie in the domain of mu.  It does lie in the domain of tilde-mu, which is the measure used throughout 213H.",
      "source_normalized_sha256": "77a233e0e0ddab89db7bf686afd46faf16dd7a6d5087ed3ece3382eaf56d2f2e",
      "target_line": "588",
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      "target_path": "source/id-ID/mt213.tex",
      "target_text": "$\\tilde\\mu(H\\symmdiff F)=0$",
      "unit_id": "O007-FREMLIN-V2-S213"
    },
    {
      "authority_line": "823",
      "authority_path": "authority/fremlin/source/mt2.2016/mt213.tex",
      "authority_text": "$h(x)\\in\\Bbb R$",
      "classification": "mathematical-circular-piecewise-condition",
      "correction_id": "O007-CORR-0079",
      "math_ordinal": "687",
      "rationale": "The display is defining h from the extended-real-valued function h*.  Its first branch must test whether h*(x) is real; testing h(x) is circular because h has not yet been defined.",
      "source_normalized_sha256": "48d2c7f1fbd4f6d52115d8f78cb3dcebc955b57a51f7222b8675f3ea610e19d2",
      "target_line": "840",
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      "target_path": "source/id-ID/mt213.tex",
      "target_text": "$h^*(x)\\in\\Bbb R$",
      "unit_id": "O007-FREMLIN-V2-S213"
    },
    {
      "authority_line": "1068-1069",
      "authority_path": "authority/fremlin/source/mt2.2016/mt213.tex",
      "authority_text": "locally determined measurable sets",
      "classification": "mathematical-wrong-property-name",
      "correction_id": "O007-CORR-0080",
      "math_ordinal": "",
      "rationale": "Exercise 213Xh points back to 213I, whose defined property is locally determined negligible sets.  The measurable-set wording names a different property and does not state the intended implication to semi-finiteness.",
      "source_normalized_sha256": "",
      "target_line": "1087-1088",
      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt213.tex",
      "target_text": "himpunan terabaikan yang ditentukan secara lokal",
      "unit_id": "O007-FREMLIN-V2-S213"
    },
    {
      "authority_line": "753",
      "authority_path": "authority/fremlin/source/mt2.2016/mt214.tex",
      "authority_text": "$\\{x:(x,i)\\in F_a\\}=E_i\\cap\\dom f$",
      "classification": "mathematical-wrong-fiber-domain",
      "correction_id": "O007-CORR-0081",
      "math_ordinal": "712",
      "rationale": "The left side is the a-superlevel set of the component function f_i.  Its domain restriction is dom f_i, not the domain of the disjoint-sum function f.",
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      "target_line": "775",
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      "target_path": "source/id-ID/mt214.tex",
      "target_text": "$\\{x:(x,i)\\in F_a\\}=E_i\\cap\\dom f_i$",
      "unit_id": "O007-FREMLIN-V2-S214"
    },
    {
      "authority_line": "769-770",
      "authority_path": "authority/fremlin/source/mt2.2016/mt214.tex",
      "authority_text": "two occurrences of $\\mu F_{nki}$",
      "classification": "mathematical-wrong-component-measure",
      "correction_id": "O007-CORR-0082",
      "math_ordinal": "732",
      "rationale": "Each F_nki is a subset of the component space X_i.  Both iterated-sum terms must use the component measure mu_i, consistent with the immediately following integrals against mu_i.",
      "source_normalized_sha256": "22139831db5491f36aa89838dd8bd73807442c93715889ad7eea95756f153193",
      "target_line": "793-794",
      "target_normalized_sha256": "587e21b48404b00a4f969b9903573937c3c2cd2401f2cfe763e37a1faa1522d8",
      "target_path": "source/id-ID/mt214.tex",
      "target_text": "two occurrences of $\\mu_i F_{nki}$",
      "unit_id": "O007-FREMLIN-V2-S214"
    },
    {
      "authority_line": "779",
      "authority_path": "authority/fremlin/source/mt2.2016/mt214.tex",
      "authority_text": "$E_i=\\{x:(x,i)\\in X_i\\}$",
      "classification": "mathematical-wrong-set-in-fiber-definition",
      "correction_id": "O007-CORR-0083",
      "math_ordinal": "738",
      "rationale": "The preceding sentence chooses a conegligible set E in the domain of f.  E_i must be its i-th fiber; taking a fiber of X_i is ill-typed and would not support the subsequent conegligibility identity.",
      "source_normalized_sha256": "b16084689422bcbc7379752d8ce9f8c2da73ffb50f92ac3553eb2c8989903dc5",
      "target_line": "803",
      "target_normalized_sha256": "d9f360db5025b5200aae925bf8be6cba9d1b751a3f94c330aca6ed0cfd4b956d",
      "target_path": "source/id-ID/mt214.tex",
      "target_text": "$E_i=\\{x:(x,i)\\in E\\}$",
      "unit_id": "O007-FREMLIN-V2-S214"
    },
    {
      "authority_line": "859",
      "authority_path": "authority/fremlin/source/mt2.2016/mt214.tex",
      "authority_text": "$F\\in\\Sigma$",
      "classification": "mathematical-wrong-algebra-membership",
      "correction_id": "O007-CORR-0084",
      "math_ordinal": "828",
      "rationale": "This paragraph proves that Lambda is closed under countable unions.  The union F need only belong to Lambda; claiming it belongs to the original sigma-algebra Sigma is not established and is false in general.",
      "source_normalized_sha256": "65b127545966720002875b7921f02a35d214624fcf61cbaceb81857933de3616",
      "target_line": "890",
      "target_normalized_sha256": "c70aaa4404847c6c8d1579a37096de6fab1d7a778a5b489f3ca34c2d9c30fe7d",
      "target_path": "source/id-ID/mt214.tex",
      "target_text": "$F\\in\\Lambda$",
      "unit_id": "O007-FREMLIN-V2-S214"
    },
    {
      "authority_line": "105",
      "authority_path": "authority/fremlin/source/mt2.2016/mt215.tex",
      "authority_text": "$E\\setminus F$",
      "classification": "mathematical-wrong-set-variable",
      "correction_id": "O007-CORR-0085",
      "math_ordinal": "81",
      "rationale": "The sentence has just introduced G as an arbitrary measurable upper bound modulo negligible sets.  Its hypothesis must be E setminus G negligible; the conclusion then proves F setminus G negligible and establishes that F is the essential supremum.",
      "source_normalized_sha256": "c9e8c5493b74286611c1946b518754e7285cf53768fb26280d7d784865dffaaf",
      "target_line": "103",
      "target_normalized_sha256": "b12382caa7a50bdfd75b9ededf43a6aa8a549520855e54315f11821d2d4a07ac",
      "target_path": "source/id-ID/mt215.tex",
      "target_text": "$E\\setminus G$",
      "unit_id": "O007-FREMLIN-V2-S215"
    },
    {
      "authority_line": "113",
      "authority_path": "authority/fremlin/source/mt2.2016/mt215.tex",
      "authority_text": "$F=\\bigcup_{n\\in\\Bbb N}E_n$",
      "classification": "mathematical-undefined-sequence-variable",
      "correction_id": "O007-CORR-0086",
      "math_ordinal": "89",
      "rationale": "Part (c) explicitly reuses the sequence F_n from part (a); no sequence E_n has been defined in this proof.",
      "source_normalized_sha256": "0a60d992035013325aca470625c70ea6c00d538f4ce31e5d3c656e35be2f1a64",
      "target_line": "111",
      "target_normalized_sha256": "47df2bc651c7bd2c8297a701f54db1154e7a710a121d4e60928c10844f51034d",
      "target_path": "source/id-ID/mt215.tex",
      "target_text": "$F=\\bigcup_{n\\in\\Bbb N}F_n$",
      "unit_id": "O007-FREMLIN-V2-S215"
    },
    {
      "authority_line": "376",
      "authority_path": "authority/fremlin/source/mt2.2016/mt215.tex",
      "authority_text": "$\\mu(E\\cap F)=\\emptyset$",
      "classification": "mathematical-type-error",
      "correction_id": "O007-CORR-0087",
      "math_ordinal": "363",
      "rationale": "A measure takes a numerical extended-real value, not the empty set.  The defining disjointness-modulo-null condition is that each pairwise intersection has measure zero.",
      "source_normalized_sha256": "285f02b0b717cef550c33b0ef010362e5570c6df9c9e580aab8cbccb710c8a07",
      "target_line": "398",
      "target_normalized_sha256": "70e7aa8d7153a26a8a3e1419c449b53fbcbde8c93c18b19a043d25de4d9032b3",
      "target_path": "source/id-ID/mt215.tex",
      "target_text": "$\\mu(E\\cap F)=0$",
      "unit_id": "O007-FREMLIN-V2-S215"
    },
    {
      "authority_line": "129-130",
      "authority_path": "authority/fremlin/source/mt2.2016/mt216.tex",
      "authority_text": "$E[\\{0\\}]\\symmdiff E[\\{1\\}]\\subseteq\\bigcup_{n\\in\\Bbb N}E_n[\\{0\\}]\\symmdiff E_n[\\{1\\}]$",
      "classification": "mathematical-missing-union-scope",
      "correction_id": "O007-CORR-0088",
      "math_ordinal": "37",
      "rationale": "To prove the symmetric difference for the union is countable, the right side must be the union of the countable symmetric differences.  Parentheses make the intended scope of the union explicit.",
      "source_normalized_sha256": "6a391fcf933046f39da6a342c6b26714e50edfab95d6f6e4d27354a8397621c7",
      "target_line": "127-128",
      "target_normalized_sha256": "0c9d9ee0694c9cd0a33db7c2933e9a1a44059b19dc45f788cbc6e80769c14c83",
      "target_path": "source/id-ID/mt216.tex",
      "target_text": "$E[\\{0\\}]\\symmdiff E[\\{1\\}]\\subseteq\\bigcup_{n\\in\\Bbb N}(E_n[\\{0\\}]\\symmdiff E_n[\\{1\\}])$",
      "unit_id": "O007-FREMLIN-V2-S216"
    },
    {
      "authority_line": "422-423",
      "authority_path": "authority/fremlin/source/mt2.2016/mt216.tex",
      "authority_text": "$A\\subseteq E\\subseteq\\Sigma$",
      "classification": "mathematical-type-error",
      "correction_id": "O007-CORR-0089",
      "math_ordinal": "299",
      "rationale": "E is a measurable subset of X and therefore an element of the sigma-algebra Sigma, not a subset of that family.  The following sentence also uses E in every Sigma_gamma.",
      "source_normalized_sha256": "ed6a4766660afffd90ee02908bf36f7b838e7834b2e654cb2d7cf5e7ea696049",
      "target_line": "436-437",
      "target_normalized_sha256": "9843fd581b22f8651080272188e36a7afa47e1ace788269143ec301c69a0b5bb",
      "target_path": "source/id-ID/mt216.tex",
      "target_text": "$A\\subseteq E\\in\\Sigma$",
      "unit_id": "O007-FREMLIN-V2-S216"
    },
    {
      "authority_line": "484-486",
      "authority_path": "authority/fremlin/source/mt2.2016/mt216.tex",
      "authority_text": "$C\\setminus(K_j\\cup K_j)$",
      "classification": "mathematical-wrong-ambient-and-duplicate-index",
      "correction_id": "O007-CORR-0090",
      "math_ordinal": "362",
      "rationale": "The function x belongs to {0,1}^I, so its remaining coordinates range over I rather than the index set C.  The two prescribed pieces are K_j and K_k; repeating K_j leaves the K_k branch inconsistent with the intended piecewise definition.",
      "source_normalized_sha256": "6473c507490fe536b4ff9b66d57b9455eb13c91f65edac26f2e72f41f8f39e7b",
      "target_line": "503-505",
      "target_normalized_sha256": "677fce5a77f706e2a94b04028c78a307cb727b30c7db64dc708b94f215472fbb",
      "target_path": "source/id-ID/mt216.tex",
      "target_text": "$I\\setminus(K_j\\cup K_k)$",
      "unit_id": "O007-FREMLIN-V2-S216"
    },
    {
      "authority_line": "168",
      "authority_path": "authority/fremlin/source/mt2.2016/mt02.tex",
      "authority_text": "positive bilinear operators and the ordering 48 $L^1(\\mu\\times\\nu)$;",
      "classification": "source-typographical-error",
      "correction_id": "O007-CORR-0091",
      "math_ordinal": "",
      "rationale": "The isolated token 48 has no syntactic or mathematical role.  Removing it and restoring the preposition 'pada' yields the coherent contents description without changing the referenced topic.",
      "source_normalized_sha256": "",
      "target_line": "169",
      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt02.tex",
      "target_text": "operator bilinear positif dan pengurutan pada $L^1(\\mu\\times\\nu)$;",
      "unit_id": "O007-FREMLIN-V2-FRONT-MT02"
    },
    {
      "authority_line": "339",
      "authority_path": "authority/fremlin/source/mt2.2016/mt02.tex",
      "authority_text": "\\section{2A3}{General topology}{25.7.07}{503}{}",
      "classification": "source-wrong-page-reference",
      "correction_id": "O007-CORR-0092",
      "math_ordinal": "",
      "rationale": "A clean deterministic build of the frozen official Volume II places the appendix headings at 2A1 p.518, 2A2 p.524, 2A3 p.527, 2A4 p.536, 2A5 p.539, and 2A6 p.543.  The neighboring mt02 references agree with that build; only 2A3's 503 is wrong.  See qa/MT02_SOURCE_REFERENCE_PROOF.json.",
      "source_normalized_sha256": "",
      "target_line": "341",
      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt02.tex",
      "target_text": "\\section{2A3}{Topologi umum}{25.7.07}{527}{}",
      "unit_id": "O007-FREMLIN-V2-FRONT-MT02"
    },
    {
      "authority_line": "333-334",
      "authority_path": "authority/fremlin/source/mt2.2016/mt233.tex",
      "authority_text": "By (iii) and (iv)",
      "classification": "mathematical-wrong-part-reference",
      "correction_id": "O007-CORR-0093",
      "math_ordinal": "",
      "rationale": "The proposition labels its relevant order-preservation and monotone-limit results as parts (c) and (d); there are no parts (iii) and (iv).",
      "source_normalized_sha256": "",
      "target_line": "340",
      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt233.tex",
      "target_text": "Menurut (c) dan (d)",
      "unit_id": "O007-FREMLIN-V2-S233"
    },
    {
      "authority_line": "358-359",
      "authority_path": "authority/fremlin/source/mt2.2016/mt233.tex",
      "authority_text": "\\int_Ff\\times\\chi F_i ... \\int_Ff\\times h",
      "classification": "mathematical-undefined-variable",
      "correction_id": "O007-CORR-0094",
      "math_ordinal": "312",
      "rationale": "Part (g) is about the fixed pair f_0 and g_0.  No un-subscripted f is defined in this proposition; both integrals must use f_0 to establish the stated conditional-expectation identity.",
      "source_normalized_sha256": "a239ddf869969afde75c4060d714f31c7adc739fb005c961edea68cb97d4a36b",
      "target_line": "367-368",
      "target_normalized_sha256": "a9b0e7eba99602a2cc3d5372832c5a110d15d65cde8a0209737f1bec2c4cca4b",
      "target_path": "source/id-ID/mt233.tex",
      "target_text": "\\int_Ff_0\\times\\chi F_i ... \\int_Ff_0\\times h",
      "unit_id": "O007-FREMLIN-V2-S233"
    },
    {
      "authority_line": "458-459",
      "authority_path": "authority/fremlin/source/mt2.2016/mt233.tex",
      "authority_text": "\\phi(x)\\ge\\sup_{q\\in\\Bbb Q}\\phi_q(x)",
      "classification": "mathematical-undefined-index-range",
      "correction_id": "O007-CORR-0095",
      "math_ordinal": "381",
      "rationale": "The preceding statement defines b_q and hence phi_q only for q in I intersect Q.  The supremum must use that domain; the rest of the proof already does so.",
      "source_normalized_sha256": "589c21155d6bd754199b0731aba21a4564e8989f1ec56571397b59511ab7a873",
      "target_line": "472-473",
      "target_normalized_sha256": "8add8e97dca11be57e2d3856d0aef743e4145770e2a157b84ed1a389f69195b0",
      "target_path": "source/id-ID/mt233.tex",
      "target_text": "\\phi(x)\\ge\\sup_{q\\in I\\cap\\Bbb Q}\\phi_q(x)",
      "unit_id": "O007-FREMLIN-V2-S233"
    },
    {
      "authority_line": "590",
      "authority_path": "authority/fremlin/source/mt2.2016/mt233.tex",
      "authority_text": "\\tilde h(x)\\ge 2^{-n}",
      "classification": "mathematical-wrong-threshold",
      "correction_id": "O007-CORR-0096",
      "math_ordinal": "497",
      "rationale": "The first branch has 0<=k<4^n and therefore covers values below 2^{-n}4^n=2^n.  The constant cap branch must begin at 2^n; 2^{-n} overlaps almost the entire first branch and does not define a function.",
      "source_normalized_sha256": "606c0a4604fb468d6dc656ed463552c3b4c6f377c2804b7aec59ead21d6ad588",
      "target_line": "608",
      "target_normalized_sha256": "0ea870d036063a07a26606e87a9c40cfc3d55722c2f9b1ff4867731708e3a4e2",
      "target_path": "source/id-ID/mt233.tex",
      "target_text": "\\tilde h(x)\\ge 2^n",
      "unit_id": "O007-FREMLIN-V2-S233"
    },
    {
      "authority_line": "812",
      "authority_path": "authority/fremlin/source/mt2.2016/mt233.tex",
      "authority_text": "$\\phi:C\\to\\BbbR^r$",
      "classification": "mathematical-type-error",
      "correction_id": "O007-CORR-0097",
      "math_ordinal": "737",
      "rationale": "The exercise calls phi convex and compares phi of the vector of integrals with the scalar integral of phi.  The immediately preceding scalar-valued convex-function results and the inequality require codomain R, not R^r.",
      "source_normalized_sha256": "c3b979b675e2d679eca275aee52f219f5c47d3b775ecb23f37a7b53e5247dd3d",
      "target_line": "849",
      "target_normalized_sha256": "3a11aafba515bbbd2cc791402efe78d9f2ff7f06f2563a7c92417e5066c10bde",
      "target_path": "source/id-ID/mt233.tex",
      "target_text": "$\\phi:C\\to\\Bbb R$",
      "unit_id": "O007-FREMLIN-V2-S233"
    },
    {
      "authority_line": "235",
      "authority_path": "authority/fremlin/source/mt2.2016/mt232.tex",
      "authority_text": "$\\mu(F\\cap E)\\le\\delta'$",
      "classification": "mathematical-wrong-continuity-witness",
      "correction_id": "O007-CORR-0098",
      "math_ordinal": "207",
      "rationale": "The primed continuity premise is for nu-prime and was introduced with E-prime and delta-prime on the preceding lines.  Using E again leaves the E-prime witness unused and does not justify the subsequent estimate.",
      "source_normalized_sha256": "8aa17ce0c730c077f32599c4587aab1ec72a10c60b56189c3bb4bc477ea9e4f0",
      "target_line": "232",
      "target_normalized_sha256": "af334d84bc6074fde7fd55e27798345ce39f4f173fbde8eeeb0c3c7f6fe684f2",
      "target_path": "source/id-ID/mt232.tex",
      "target_text": "$\\mu(F\\cap E')\\le\\delta'$",
      "unit_id": "O007-FREMLIN-V2-S232"
    },
    {
      "authority_line": "385",
      "authority_path": "authority/fremlin/source/mt2.2016/mt232.tex",
      "authority_text": "$\\int_Ef=\\lim_{n\\to\\infty}\\int_Ef_n\\le\\nu E$",
      "classification": "mathematical-wrong-convergent-sequence",
      "correction_id": "O007-CORR-0099",
      "math_ordinal": "334",
      "rationale": "The proof defines g_n as the increasing finite suprema and then defines f as their pointwise limit.  B. Levi applies to g_n; the original f_n sequence need not be monotone or converge pointwise.",
      "source_normalized_sha256": "73806288125df251460e83767b114b0b0af3071534cb87a08c1bc0c863c64915",
      "target_line": "378",
      "target_normalized_sha256": "353cb337bf6f4fcea1114b175e6254efaac764a5a2770acc6b48c4ce01f6b6ef",
      "target_path": "source/id-ID/mt232.tex",
      "target_text": "$\\int_Ef=\\lim_{n\\to\\infty}\\int_Eg_n\\le\\nu E$",
      "unit_id": "O007-FREMLIN-V2-S232"
    },
    {
      "authority_line": "682-683",
      "authority_path": "authority/fremlin/source/mt2.2016/mt232.tex",
      "authority_text": "$\\mu F=\\mu F'=\\nu_s'(X\\setminus F)=\\nu_s''(X\\setminus F)=0$",
      "classification": "mathematical-undefined-and-mismatched-witnesses",
      "correction_id": "O007-CORR-0100",
      "math_ordinal": "593",
      "rationale": "The sentence introduces separate singularity witnesses F-prime and F-double-prime.  Each must be mu-null and support its corresponding singular component; unprimed F is not introduced in this argument.",
      "source_normalized_sha256": "e286197dbf648ad6e53c67aec801e51a33aac6506e9f3b1e5dbfa17a7b84f202",
      "target_line": "662-663",
      "target_normalized_sha256": "96aa1e6ec2aa15e78e11842d690ff19f28f5b2d9ed2edbbdbeff16822e08cd44",
      "target_path": "source/id-ID/mt232.tex",
      "target_text": "$\\mu F'=\\mu F''=\\nu_s'(X\\setminus F')=\\nu_s''(X\\setminus F'')=0$",
      "unit_id": "O007-FREMLIN-V2-S232"
    },
    {
      "authority_line": "107",
      "authority_path": "authority/fremlin/source/mt2.2016/mt234.tex",
      "authority_text": "0<\\mu(E\\cap\\phi^{-1}[F_{ni})",
      "classification": "source-mismatched-delimiter",
      "correction_id": "O007-CORR-0101",
      "math_ordinal": "85",
      "rationale": "The inverse-image bracket opened before F_{ni} is closed by a parenthesis in the authority, leaving the formula unbalanced.  Closing the bracket before the outer measure parenthesis restores the intended set expression.",
      "source_normalized_sha256": "fbb71eb274beed55cc9447f8f71171b12ead42f8f0d7b9e04dbae3f317a11aa6",
      "target_line": "111",
      "target_normalized_sha256": "5dd3704c7a7e74da0db17dd72c7e374001cbdcc476ca855db73724257c8b5649",
      "target_path": "source/id-ID/mt234.tex",
      "target_text": "0<\\mu(E\\cap\\phi^{-1}[F_{ni}])",
      "unit_id": "O007-FREMLIN-V2-S234"
    },
    {
      "authority_line": "257",
      "authority_path": "authority/fremlin/source/mt2.2016/mt234.tex",
      "authority_text": "(\\mu\\phi^{-1})(\\psi^{-1}[G]))",
      "classification": "source-extra-delimiter",
      "correction_id": "O007-CORR-0102",
      "math_ordinal": "195",
      "rationale": "The displayed composition identity has one unmatched closing parenthesis after the argument of the image measure.  Removing it restores balanced application syntax and agrees with the preceding and following lines.",
      "source_normalized_sha256": "edfae7c7da8150743726a488d706a46a0070c611810f803a784896a70756cf61",
      "target_line": "265",
      "target_normalized_sha256": "2b6a10d9eba1a2fb14d29050fd78c7d39796a6c96b9af224e6f5b8c4dafe34fd",
      "target_path": "source/id-ID/mt234.tex",
      "target_text": "(\\mu\\phi^{-1})(\\psi^{-1}[G])",
      "unit_id": "O007-FREMLIN-V2-S234"
    },
    {
      "authority_line": "362",
      "authority_path": "authority/fremlin/source/mt2.2016/mt234.tex",
      "authority_text": "\\sum_{i\\in I}f^+d\\mu_i",
      "classification": "mathematical-missing-integral-operator",
      "correction_id": "O007-CORR-0103",
      "math_ordinal": "291",
      "rationale": "The proposition requires the sum of positive-part integrals; without the integral sign the expression is not a scalar and conflicts with the proof in 234H(c)(iii).",
      "source_normalized_sha256": "7ce2aec77ceb058acd71831ccc09b99a1bc72ee674459f3f78760f39bb927356",
      "target_line": "377",
      "target_normalized_sha256": "7c5dc9429a630a645d384eed2e4288339e956cdb785bd9e26df0f4d3a2a70747",
      "target_path": "source/id-ID/mt234.tex",
      "target_text": "\\sum_{i\\in I}\\int f^+d\\mu_i",
      "unit_id": "O007-FREMLIN-V2-S234"
    },
    {
      "authority_line": "362",
      "authority_path": "authority/fremlin/source/mt2.2016/mt234.tex",
      "authority_text": "\\sum_{i\\in I}f^-d\\mu_i",
      "classification": "mathematical-missing-integral-operator",
      "correction_id": "O007-CORR-0104",
      "math_ordinal": "292",
      "rationale": "The proposition requires the sum of negative-part integrals; without the integral sign the expression is not a scalar and conflicts with the proof in 234H(c)(iii).",
      "source_normalized_sha256": "fc8b78a5f7088e9654aa1d121587beac63794b824f6efd545d76fe9c34519563",
      "target_line": "377",
      "target_normalized_sha256": "86a04d59b89539d78e2eddccdbe315727a9d8525c29c5823ede31c245cb369ef",
      "target_path": "source/id-ID/mt234.tex",
      "target_text": "\\sum_{i\\in I}\\int f^-d\\mu_i",
      "unit_id": "O007-FREMLIN-V2-S234"
    },
    {
      "authority_line": "371",
      "authority_path": "authority/fremlin/source/mt2.2016/mt234.tex",
      "authority_text": "E_i\\in\\Sigma_i",
      "classification": "mathematical-undefined-variable",
      "correction_id": "O007-CORR-0105",
      "math_ordinal": "302",
      "rationale": "Only E and F are introduced in this proof.  Completeness of each mu_i yields E in Sigma_i from E subseteq F and mu_i F=0; no indexed E_i exists.",
      "source_normalized_sha256": "b251cbed3bc7e1fb2954006af3974141472b4f8a8b3adf075cd98d8711f3d5ea",
      "target_line": "386",
      "target_normalized_sha256": "eb175f9df9ccdcbf76627baec7ec0cb7b79460c26ea7165e0ddce22bb0a0fa58",
      "target_path": "source/id-ID/mt234.tex",
      "target_text": "E\\in\\Sigma_i",
      "unit_id": "O007-FREMLIN-V2-S234"
    },
    {
      "authority_line": "421",
      "authority_path": "authority/fremlin/source/mt2.2016/mt234.tex",
      "authority_text": "[\\infty,\\infty]",
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      "math_ordinal": "331",
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      "target_text": "[-\\infty,\\infty]",
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    },
    {
      "authority_line": "741",
      "authority_path": "authority/fremlin/source/mt2.2016/mt234.tex",
      "authority_text": "\\hat\\mu(H\\setminus((H'\\cup(X\\setminus G)))=0",
      "classification": "source-mismatched-delimiter",
      "correction_id": "O007-CORR-0107",
      "math_ordinal": "620",
      "rationale": "The essential-supremum equivalence measures H minus the union of H' and X minus G.  The authority has an unmatched extra opening parenthesis and fails to close the measure application; the corrected grouping is balanced and yields the next equivalence.",
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      "target_text": "\\hat\\mu(H\\setminus(H'\\cup(X\\setminus G)))=0",
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    },
    {
      "authority_line": "821",
      "authority_path": "authority/fremlin/source/mt2.2016/mt234.tex",
      "authority_text": "indefinite-integral measure over \\nu",
      "classification": "mathematical-wrong-reference-measure",
      "correction_id": "O007-CORR-0108",
      "math_ordinal": "710",
      "rationale": "The theorem assumes that nu is an indefinite-integral measure over mu.  Referring to nu as its own base measure contradicts the theorem statement and the cited consequences, all of which use mu.",
      "source_normalized_sha256": "84eb2b123d1102b8f7c00b34116df32107cffa82d46efbac7fd9543fa7459c38",
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      "target_text": "ukuran integral tak tentu terhadap \\mu",
      "unit_id": "O007-FREMLIN-V2-S234"
    },
    {
      "authority_line": "943-946",
      "authority_path": "authority/fremlin/source/mt2.2016/mt234.tex",
      "authority_text": "\\sum_{n=0}^{\\infty}\\nu(F\\cap G_n)-\\mu(F\\cap G_n)",
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      "correction_id": "O007-CORR-0109",
      "math_ordinal": "839",
      "rationale": "The equality expands nu F minus mu F term by term.  Without grouping, only the nu terms are summed and a single free n remains in the mu term; parentheses make both terms part of every summand.",
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      "target_path": "source/id-ID/mt234.tex",
      "target_text": "\\sum_{n=0}^{\\infty}(\\nu(F\\cap G_n)-\\mu(F\\cap G_n))",
      "unit_id": "O007-FREMLIN-V2-S234"
    },
    {
      "authority_line": "953",
      "authority_path": "authority/fremlin/source/mt2.2016/mt234.tex",
      "authority_text": "\\sum_{n=0}^m\\nu F_n-\\mu F_n\\ge\\gamma",
      "classification": "mathematical-missing-summand-grouping",
      "correction_id": "O007-CORR-0110",
      "math_ordinal": "850",
      "rationale": "The finite choice is made from the sum of the differences defining lambda G_n.  Without parentheses only nu F_n is summed and the final mu F_n has a free index.",
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      "target_path": "source/id-ID/mt234.tex",
      "target_text": "\\sum_{n=0}^m(\\nu F_n-\\mu F_n)\\ge\\gamma",
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    },
    {
      "authority_line": "957-958",
      "authority_path": "authority/fremlin/source/mt2.2016/mt234.tex",
      "authority_text": "\\sum_{n=0}^m\\nu F_n-\\mu F_n",
      "classification": "mathematical-missing-summand-grouping",
      "correction_id": "O007-CORR-0111",
      "math_ordinal": "855",
      "rationale": "The equality uses finite additivity for nu F minus mu F.  Both terms must be grouped inside the sum; otherwise the displayed equality leaves the last mu term unsummed.",
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      "target_path": "source/id-ID/mt234.tex",
      "target_text": "\\sum_{n=0}^m(\\nu F_n-\\mu F_n)",
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    },
    {
      "authority_line": "1160",
      "authority_path": "authority/fremlin/source/mt2.2016/mt234.tex",
      "authority_text": "\\mu E=\\nu\\phi[E] for E\\in\\Tau",
      "classification": "mathematical-wrong-domain",
      "correction_id": "O007-CORR-0112",
      "math_ordinal": "1062",
      "rationale": "The exercise defines Sigma as a sigma-algebra on X and then defines mu on X.  Tau is the sigma-algebra on Y, so E belongs to Sigma; using Tau is a type error.",
      "source_normalized_sha256": "2c51c23a716fda0bf481e7a5326d3f3bfbae2b118b054427c24bebecf5853122",
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      "target_path": "source/id-ID/mt234.tex",
      "target_text": "\\mu E=\\nu\\phi[E] untuk E\\in\\Sigma",
      "unit_id": "O007-FREMLIN-V2-S234"
    },
    {
      "authority_line": "114",
      "authority_path": "authority/fremlin/source/mt2.2016/mt235.tex",
      "authority_text": "g\\text{ is }\\nu-\\text{integrable}",
      "classification": "source-tex-hyphenation-error",
      "correction_id": "O007-CORR-0113",
      "math_ordinal": "",
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      "source_normalized_sha256": "",
      "target_line": "120",
      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt235.tex",
      "target_text": "g\\text{ terintegralkan terhadap }\\nu",
      "unit_id": "O007-FREMLIN-V2-S235"
    },
    {
      "authority_line": "113-119",
      "authority_path": "authority/fremlin/source/mt2.2016/mt235.tex",
      "authority_text": "\\overline{\\int}fd\\nu ... \\le\\inf\\{\\int h\\,d\\mu:...\\}=\\overline{\\intop}J\\times f\\phi\\,d\\mu",
      "classification": "mathematical-reversed-inequality",
      "correction_id": "O007-CORR-0114",
      "math_ordinal": "72",
      "rationale": "Mapped nu-integrable majorants form a subset of all mu-integrable majorants of the pulled-back function.  The infimum over that restricted class is therefore greater than or equal to the unrestricted infimum, as required by the theorem's upper-integral chain.  The full math-atom hashes also include the localized repair recorded in O007-CORR-0113.",
      "source_normalized_sha256": "ea47dbdef709903ab57a6dd75d18532ad524f0d18cff2179b2c70db1b9122ee4",
      "target_line": "119-126",
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      "target_path": "source/id-ID/mt235.tex",
      "target_text": "\\overline{\\int}fd\\nu ... \\ge\\inf\\{\\int h\\,d\\mu:...\\}=\\overline{\\intop}J\\times f\\phi\\,d\\mu",
      "unit_id": "O007-FREMLIN-V2-S235"
    },
    {
      "authority_line": "122",
      "authority_path": "authority/fremlin/source/mt2.2016/mt235.tex",
      "authority_text": "\\underline{\\int}J\\times f\\phi\\,d\\mu\\le\\underline{\\int}f\\,d\\nu",
      "classification": "mathematical-reversed-inequality",
      "correction_id": "O007-CORR-0115",
      "math_ordinal": "74",
      "rationale": "Applying the corrected upper-integral inequality to -f and negating reverses the relation.  This yields lower integral of the pullback greater than or equal to lower integral of f, exactly as the theorem states.",
      "source_normalized_sha256": "0ab0bad14e4e0e82b261e2384441d4425e42a909349c06c366247803fa1e8f0d",
      "target_line": "130",
      "target_normalized_sha256": "6504c0c94e662bc36a39981b61e8848d6151b88ed58db7abc697d3c2b044bb31",
      "target_path": "source/id-ID/mt235.tex",
      "target_text": "\\underline{\\int}J\\times f\\phi\\,d\\mu\\ge\\underline{\\int}f\\,d\\nu",
      "unit_id": "O007-FREMLIN-V2-S235"
    },
    {
      "authority_line": "383",
      "authority_path": "authority/fremlin/source/mt2.2016/mt235.tex",
      "authority_text": "\\int J\\times \\phi\\,d\\hat\\mu=\\int g\\,d\\hat\\nu",
      "classification": "mathematical-missing-factor",
      "correction_id": "O007-CORR-0116",
      "math_ordinal": "286",
      "rationale": "This branch concerns J times the composition g after phi, and the preceding display proves precisely that integral.  Omitting g changes the integrand and leaves phi itself in a scalar product.",
      "source_normalized_sha256": "1bfca017ba17097acaaecc2d92bb1ee5ba975597e6434d617e2b3a9c6e342505",
      "target_line": "403",
      "target_normalized_sha256": "32a39a08c590856a83aed1c672e14f8b419af7ec836531e453954709f309155c",
      "target_path": "source/id-ID/mt235.tex",
      "target_text": "\\int J\\times g\\phi\\,d\\hat\\mu=\\int g\\,d\\hat\\nu",
      "unit_id": "O007-FREMLIN-V2-S235"
    },
    {
      "authority_line": "1171",
      "authority_path": "authority/fremlin/source/mt2.2016/mt235.tex",
      "authority_text": "intepreting",
      "classification": "source-spelling-error",
      "correction_id": "O007-CORR-0117",
      "math_ordinal": "",
      "rationale": "The English participle is misspelled; the Indonesian target already renders the intended word correctly.",
      "source_normalized_sha256": "",
      "target_line": "1223",
      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt235.tex",
      "target_text": "menafsirkan",
      "unit_id": "O007-FREMLIN-V2-S235"
    },
    {
      "authority_line": "301",
      "authority_path": "authority/fremlin/source/mt2.2016/mt241.tex",
      "authority_text": "stable header 241Gg",
      "classification": "stable-id-wrong-section-prefix",
      "correction_id": "O007-CORR-0118",
      "math_ordinal": "",
      "rationale": "The seventh subitem of 241E follows 241Ea through 241Ef, and mt242 explicitly cites 241Eg. The authority label 241Gg is a prefix typo; the source spelling remains a lookup alias.",
      "source_normalized_sha256": "",
      "target_line": "305",
      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt241.tex",
      "target_text": "stable header 241Eg",
      "unit_id": "O007-FREMLIN-V2-S241"
    },
    {
      "authority_line": "574",
      "authority_path": "authority/fremlin/source/mt2.2016/mt241.tex",
      "authority_text": "f_q(x)=\\Real(e^{iqx}f(x))",
      "classification": "mathematical-ill-typed-phase",
      "correction_id": "O007-CORR-0119",
      "math_ordinal": "589",
      "rationale": "The underlying x is an arbitrary point of X, so qx is undefined. Rational scalar phases e^{iq} give the dense family needed for the stated equality and bound.",
      "source_normalized_sha256": "c83eec622e0abd47a314809b99ff3385f91bc938b2d620f6d1955a420d5a6ef3",
      "target_line": "589",
      "target_normalized_sha256": "bea6e0e2b16eb3379b993f95cc64fdf0ff4b1c114b2736ba5c836439d82ca16d",
      "target_path": "source/id-ID/mt241.tex",
      "target_text": "f_q(x)=\\Real(e^{iq}f(x))",
      "unit_id": "O007-FREMLIN-V2-S241"
    },
    {
      "authority_line": "904-908",
      "authority_path": "authority/fremlin/source/mt2.2016/mt242.tex",
      "authority_text": "+\\int(\\sum \\Imag(f_n))",
      "classification": "mathematical-missing-imaginary-factor",
      "correction_id": "O007-CORR-0120",
      "math_ordinal": "764",
      "rationale": "For complex-valued functions f=Real(f)+i Imag(f). The printed intermediate equality omits i and would be real-valued despite its complex left side.",
      "source_normalized_sha256": "cdf0d5a936187669fb2d5bc3a311607b22bb1947323ffbf3f4273f284a63a895",
      "target_line": "845-849",
      "target_normalized_sha256": "f3a1c74952137d8d8a2765283b02b67219139b1ea579010c8dc3a51c288ecc66",
      "target_path": "source/id-ID/mt242.tex",
      "target_text": "+i\\int(\\sum \\Imag(f_n))",
      "unit_id": "O007-FREMLIN-V2-S242"
    },
    {
      "authority_line": "999",
      "authority_path": "authority/fremlin/source/mt2.2016/mt242.tex",
      "authority_text": "P_2:L_1(\\mu\\restrp\\Tau)\\to L^1(\\mu\\restrp\\Upsilon)",
      "classification": "mathematical-wrong-function-space-notation",
      "correction_id": "O007-CORR-0121",
      "math_ordinal": "834",
      "rationale": "P_1 maps into L^1(mu restricted to Tau) and the requested identity is P=P_2P_1, so this must be the domain of P_2. L_1 is an isolated typo.",
      "source_normalized_sha256": "5c1bc51e7d424ff4a87e037c287a280685a7e701db92f4d9aa5a58a4998800c3",
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      "target_path": "source/id-ID/mt242.tex",
      "target_text": "P_2:L^1(\\mu\\restrp\\Tau)\\to L^1(\\mu\\restrp\\Upsilon)",
      "unit_id": "O007-FREMLIN-V2-S242"
    },
    {
      "authority_line": "1012-1014",
      "authority_path": "authority/fremlin/source/mt2.2016/mt242.tex",
      "authority_text": "phi_delta defined, then phi declared smooth",
      "classification": "mathematical-undefined-referent",
      "correction_id": "O007-CORR-0122",
      "math_ordinal": "845",
      "rationale": "The exercise defines the parameterized function phi_delta and uses it in the next part. Bare phi is undefined, so the smoothness assertion must refer to phi_delta.",
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      "target_line": "950-952",
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      "target_path": "source/id-ID/mt242.tex",
      "target_text": "phi_delta defined and declared smooth",
      "unit_id": "O007-FREMLIN-V2-S242"
    },
    {
      "authority_line": "1123-1125",
      "authority_path": "authority/fremlin/source/mt2.2016/mt242.tex",
      "authority_text": "if U is a Banach lattice, it is U is Dedekind complete",
      "classification": "source-prose-duplication",
      "correction_id": "O007-CORR-0123",
      "math_ordinal": "",
      "rationale": "The authority duplicates the words 'it is' and U; the Indonesian sentence retains the single intended implication.",
      "source_normalized_sha256": "",
      "target_line": "1055-1057",
      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt242.tex",
      "target_text": "jika U suatu kisi Banach, maka U lengkap Dedekind",
      "unit_id": "O007-FREMLIN-V2-S242"
    },
    {
      "authority_line": "1210",
      "authority_path": "authority/fremlin/source/mt2.2016/mt242.tex",
      "authority_text": "conditional expections",
      "classification": "source-spelling-error",
      "correction_id": "O007-CORR-0124",
      "math_ordinal": "",
      "rationale": "The authority misspells expectations; the Indonesian text renders the intended mathematical noun.",
      "source_normalized_sha256": "",
      "target_line": "1135-1136",
      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt242.tex",
      "target_text": "ekspektasi bersyarat",
      "unit_id": "O007-FREMLIN-V2-S242"
    },
    {
      "authority_line": "52",
      "authority_path": "authority/fremlin/source/mt2.2016/mt243.tex",
      "authority_text": "|cf|\\le|c||M_1|",
      "classification": "mathematical-redundant-absolute-value",
      "correction_id": "O007-CORR-0125",
      "math_ordinal": "56",
      "rationale": "M_1 was chosen nonnegative, so absolute-value bars around it are redundant and typographically misleading.",
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      "target_path": "source/id-ID/mt243.tex",
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      "unit_id": "O007-FREMLIN-V2-S243"
    },
    {
      "authority_line": "579",
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      "authority_text": "E_k={x:k epsilon <= f_tilde(x)<k+1)epsilon",
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      "correction_id": "O007-CORR-0126",
      "math_ordinal": "553",
      "rationale": "The printed upper endpoint has an unmatched closing parenthesis and the set lacks its closing brace. The corrected interval is the partition used in the following approximation.",
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      "target_text": "E_k={x:k epsilon <= f_tilde(x)<(k+1)epsilon}",
      "unit_id": "O007-FREMLIN-V2-S243"
    },
    {
      "authority_line": "420-421",
      "authority_path": "authority/fremlin/source/mt2.2016/mt244.tex",
      "authority_text": "two bounds both use integral_X |f-g|^p <= epsilon^p",
      "classification": "mathematical-duplicated-approximant",
      "correction_id": "O007-CORR-0127",
      "math_ordinal": "363",
      "rationale": "The analogous construction and the immediately following use of both approximants require separate bounds for g and h; repeating g leaves h uncontrolled.",
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      "target_text": "second bound uses integral_X |f-h|^p <= epsilon^p",
      "unit_id": "O007-FREMLIN-V2-S244"
    },
    {
      "authority_line": "576-578",
      "authority_path": "authority/fremlin/source/mt2.2016/mt244.tex",
      "authority_text": "f belongs to \\eusm L^q",
      "classification": "mathematical-wrong-function-space",
      "correction_id": "O007-CORR-0128",
      "math_ordinal": "499",
      "rationale": "The argument begins with f in L^p, h acts on L^p, and the vanishing property is required for L^p. L^q is the wrong space.",
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    },
    {
      "authority_line": "784-785",
      "authority_path": "authority/fremlin/source/mt2.2016/mt244.tex",
      "authority_text": "w=\\sup A'=\\sup A",
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      "correction_id": "O007-CORR-0129",
      "math_ordinal": "661",
      "rationale": "w is an arbitrary upper bound; the constructed w_1 has just been proved to be the least upper bound.",
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      "target_text": "w_1=\\sup A'=\\sup A",
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    },
    {
      "authority_line": "927-930",
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      "authority_text": "\\bar\\tau_F(u)",
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      "authority_line": "242",
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      "authority_text": "v^{\\ssbullet}=v",
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      "authority_text": "\\rho_F(\\bar h(v),\\bar h(u))",
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      "authority_text": "\\rho_F(v,u)",
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      "authority_line": "321",
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      "authority_text": "\\min(1,|h(g_n(x))-h(f(x))|\\,\\mu(dx)",
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      "authority_line": "474",
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      "authority_text": "\\bigcap_{k\\le n}A_n(F)",
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      "authority_line": "658",
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      "authority_text": "\\min(f(x),1)",
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      "math_ordinal": "563",
      "rationale": "The defining functional tau_F uses the modulus of f, so the equivalent formula must also use |f(x)|.",
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      "authority_line": "659",
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      "authority_text": "f(x)>\\epsilon",
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      "authority_line": "1074",
      "authority_path": "authority/fremlin/source/mt2.2016/mt245.tex",
      "authority_text": "h(f_{1n},...,f_{kn})",
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      "authority_text": "h(f_1,...,f_k)",
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      "authority_line": "1182",
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      "authority_text": "|f(x)|\\ge a",
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      "rationale": "With >=, a non-null space prevents the stated minimum at rho(0,0). The strict distribution tail gives the intended metric and matches 245F.",
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      "authority_text": "rho formula with unclosed \\min(",
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      "rationale": "The minimum expression defining rho is missing its closing parenthesis.",
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      "authority_line": "124",
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      "authority_text": "|a|\\int_E(|u|-M\\chi E)^+",
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      "rationale": "An integral restricted to E cannot control the tail outside E; homogeneity and uniform integrability require the whole-space integral.",
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      "authority_line": "125",
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      "authority_text": "\\int_E(|v|-|a|M\\chi E)^+",
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      "rationale": "The conclusion likewise needs the whole-space integral from Definition 246Ab.",
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      "authority_text": "\\int_{X\\setminus E}|u|\\le\\delta",
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      "rationale": "Applying 246F to u chi(X minus E) and the preceding epsilon bound gives 2 epsilon; delta has no such role.",
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      "authority_line": "636",
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      "authority_text": "\\int_E(v-M\\chi E)^+",
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      "authority_line": "643-644",
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      "authority_text": "concluding display uses two \\int_E terms",
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      "correction_id": "O007-CORR-0149",
      "math_ordinal": "528",
      "rationale": "Both displayed integrals must be whole-space integrals to prove Definition 246Ab.",
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      "authority_line": "848",
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      "authority_text": "\\mu=\\sum a_n\\nu_n",
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      "authority_text": "\\sum a_i\\int_{F_i}u",
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      "rationale": "This display decomposes the integral of u_0 and must use the just-proved identity integral over F_i of u_0 = nu F_i. Bare u is the unrelated filter variable.",
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      "authority_text": "shows that 243Gb it is nearly valid",
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      "rationale": "The stray 'it' makes the authority sentence ungrammatical; the Indonesian text retains the intended clause.",
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      "rationale": "The preceding lines define the second projection onto R^s and establish measurability of its inverse image of F; applying pi_1 to F is generally ill-typed.",
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      "authority_line": "843",
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      "authority_text": "F\\subseteq\\bigcup_{n\\in\\Bbb N}F_n",
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      "authority_line": "941",
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      "authority_text": "\\mu F=\\sum_{j\\in J}\\mu(F\\cap Y_j)",
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      "rationale": "F and Y_j are subsets of the second factor Y governed by nu; the following estimate already uses nu F and nu(F intersect Y_j).",
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      "target_text": "\\nu F=\\sum_{j\\in J}\\nu(F\\cap Y_j)",
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      "authority_line": "284",
      "authority_path": "authority/fremlin/source/mt2.2016/mt251.tex",
      "authority_text": "\\nu E_n",
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      "math_ordinal": "221",
      "rationale": "The two rectangle factors are E_n and F_n; nu is the measure on the second factor and every adjacent formula correctly uses nu F_n.",
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      "target_text": "\\nu F_n",
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      "authority_line": "904",
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      "authority_text": "Putting (c) and (d) together",
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      "rationale": "Part (b) proves theta is at most the outer measure and part (d) proves the reverse inequality; part (c) concerns rectangles and does not supply the stated equality.",
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      "target_text": "Dengan menggabungkan (b) dan (d)",
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      "authority_text": "writing \\mu for Lebesgue measure",
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      "math_ordinal": "744",
      "rationale": "Section 252N puts Lebesgue measure on the second factor R and every adjacent iterated integral denotes that measure by nu; mu remains the measure on X.",
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      "authority_text": "\\mu W[\\{x\\}]=\\sup_n\\nu(Y_n\\cap W[\\{x\\}])",
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      "authority_text": "(X_2,\\Sigma_3,\\mu_3)",
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      "rationale": "The exercise declares three spaces and immediately uses X_1 times X_2 times X_3; the third triple must begin with X_3.",
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      "authority_text": "\\Omega^*\\symmdiff\\Omega^*_g",
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      "authority_text": "premature math delimiter after \\dom g and \\mu(\\{...\\}da",
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      "authority_path": "authority/fremlin/source/mt2.2016/mt252.tex",
      "authority_text": "\\Bover12((g(x^-)+g(x^+))",
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      "authority_text": "x\\in X\\times Y",
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      "rationale": "The piecewise definition is on X times Y and the preceding branch uses both coordinates; the zero branch must quantify the other product points (x,y), not a single x alleged to lie in the product.",
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      "authority_text": "premature $$ before +\\epsilon\\sum_{i=0}^na_i",
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      "authority_line": "565-566",
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      "authority_text": "integral exists $=\\int\\tilde f_nd\\lambda$",
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      "rationale": "The authority places a bare equality sign after the prose predicate 'exists', yielding no left-hand operand in its second math atom. The derivative states the intended existence-and-equality claim without the orphaned sign.",
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      "authority_text": "for other x, y\\in\\BbbR^2",
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    {
      "authority_line": "180",
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      "authority_text": "\\sequencen{f_n\\times g_n}",
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      "math_ordinal": "151",
      "rationale": "The functions live on different factor spaces and 253B defines their tensor product; ordinary pointwise multiplication is undefined here. The following convergence and integral identities also use the tensor product.",
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    {
      "authority_line": "353",
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      "authority_line": "755-756",
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      "authority_text": "\\nu F",
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      "authority_line": "966-967",
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      "authority_text": "\\sup\\{\\phi(u_1,\\ldots,u_n):\\|u_i\\|_1\\le1\\Forall i\\le n\\}",
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      "authority_line": "1151",
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      "authority_text": "%253Yg, 253Yi, 253+",
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      "authority_text": "Tu\\ge0 whenever u\\in L^1(\\lambda)",
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      "authority_text": "\\nu(E_{\\gamma}\\setminus H_{\\gamma})=0",
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    },
    {
      "authority_line": "1777",
      "authority_path": "authority/fremlin/source/mt2.2016/mt254.tex",
      "authority_text": "\\mu(E_{\\gamma}\\cap H_{\\gamma}\\cap E)=\\nu(E_{\\gamma}\\cap E)",
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      "target_path": "source/id-ID/mt254.tex",
      "target_text": "\\mu(E_{\\gamma}\\cap H_{\\gamma}\\cap E)=\\mu(E_{\\gamma}\\cap E)",
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    },
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      "authority_line": "1812",
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      "authority_text": "distinct elements of X",
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      "target_text": "unsur-unsur I yang berbeda",
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      "authority_text": "\\{x:x\\in X, x(i)=x(j)\\}",
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      "authority_text": "\\{x:x\\in X, x(i)=x(j) for some distinct i,j\\in I\\}",
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      "target_text": "\\{x:x\\in X^I, x(i)=x(j) untuk unsur-unsur berbeda i,j\\in I\\}",
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      "authority_text": "product (X,\\Lambda,\\mu)",
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      "rationale": "The exercise declares and subsequently uses lambda as the product measure; unindexed mu is unavailable.",
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      "authority_text": "\\int_Wf\\le\\int_Wf",
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      "authority_line": "1913",
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      "authority_text": "F\\in\\Tensorhat_{i\\in J}X_j",
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      "math_ordinal": "1870",
      "rationale": "F must belong to the measurable subproduct sigma-algebra; X_j is the wrong object type and j is unbound.",
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      "authority_line": "1969",
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      "authority_text": "L^p(\\mu)",
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      "target_text": "L^p(\\lambda)",
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    },
    {
      "authority_line": "2149-2154",
      "authority_path": "authority/fremlin/source/mt2.2016/mt254.tex",
      "authority_text": "Lemma 254G ... It is therefore a universal mapping theorem",
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      "math_ordinal": "",
      "rationale": "Section 254G is explicitly headed Theorem, so calling it a lemma misstates its source structure.",
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      "target_path": "source/id-ID/mt254.tex",
      "target_text": "Teorema 254G ... teorema ini merupakan suatu teorema pemetaan universal",
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      "authority_line": "2251",
      "authority_path": "authority/fremlin/source/mt2.2016/mt254.tex",
      "authority_text": "see part (b) of the proof of 254R, and 254Xq(iii)",
      "classification": "source-broken-cross-reference-unresolved",
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      "math_ordinal": "",
      "rationale": "254Xq contains only parts (i) and (ii). Part (ii) is an L1 intersection result while this note states an L0 intersection, so the intended replacement is uncertain; the reader preserves and explicitly flags the source citation rather than silently inventing a target.",
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      "target_line": "2250-2251",
      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt254.tex",
      "target_text": "lihat bagian (b) dari bukti 254R; sumber juga mengacu ke 254Xq(iii), tetapi bagian tersebut tidak ada dalam seksi yang dirujuk",
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      "authority_text": "\\cmmnt{... \\header{254Ta}}{\\bf (a)}",
      "classification": "source-hidden-section-boundary",
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      "target_line": "1666-1672",
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      "authority_line": "99",
      "authority_path": "authority/fremlin/source/mt2.2016/mt255.tex",
      "authority_text": "\\mu^*A\\le\\mu^*(-(-A))=\\mu^*A",
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      "correction_id": "O007-CORR-0214",
      "math_ordinal": "54",
      "rationale": "The preceding argument proves mu-star(-A) <= mu-star(A); applying it to -A supplies the reverse inequality. The printed self-inequality is tautological.",
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      "target_text": "\\mu^*A=\\mu^*(-(-A))\\le\\mu^*(-A)",
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    },
    {
      "authority_line": "147-148",
      "authority_path": "authority/fremlin/source/mt2.2016/mt255.tex",
      "authority_text": "\\mu(\\phi[H_2]\\setminus\\phi[H_1])=\\mu\\phi[H_2\\setminus H_1]=\\mu(H_2\\setminus H_1)=0",
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      "target_text": "\\mu_2(\\phi[H_2]\\setminus\\phi[H_1])=\\mu_2\\phi[H_2\\setminus H_1]=\\mu_2(H_2\\setminus H_1)=0",
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      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt255.tex",
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      "authority_line": "319",
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      "authority_text": "g^{\\ssbullet}=u",
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      "rationale": "The function theta(u,v) is defined from representatives f of u and g of v; assigning both representatives to u drops the second argument.",
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      "authority_line": "574",
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      "authority_text": "+|h_x-h_{x'}\\|_p",
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      "authority_text": "\\|f_x-f_{x'}|_p\\|g\\|_q",
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      "authority_line": "603-608",
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      "authority_text": "if I_n=coint{a_n,b_n} ... writing a_n=(alpha_n1,...,alpha_nr)",
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      "authority_line": "751",
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      "authority_text": "\\mu(-(E\\cap\\ooint{-\\pi,\\pi})",
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      "math_ordinal": "473",
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      "authority_line": "937",
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      "authority_text": "defined, then then u*(v*w)",
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    {
      "authority_line": "992-993",
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      "rationale": "This is the second occurrence of the same approximate-identity hypothesis in part (ii); it requires the same decreasing right-hand tail.",
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    {
      "authority_line": "1003",
      "authority_path": "authority/fremlin/source/mt2.2016/mt255.tex",
      "authority_text": "\\Bover1a\\int_x^{\\infty}f(y)e^{-a(y-x)}dy",
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      "target_text": "a\\int_x^{\\infty}f(y)e^{-a(y-x)}dy",
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      "authority_text": "255Ac-255Ad",
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      "rationale": "Parts 255A(a)-(c) are one-dimensional, while the two-dimensional assertions are exactly 255A(d)-(e). Protected-reference ordinals 146-147 are ledgered by the unit validator.",
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      "target_text": "255Ad-255Ae",
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      "authority_line": "82",
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      "authority_text": "sets of finite measurea covering $\\BbbR^r$",
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      "rationale": "The final 'a' in 'measurea' is a stray character; removing it restores ordinary grammatical prose without changing the mathematical claim.",
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      "target_text": "himpunan dengan ukuran berhingga yang menutupi $\\BbbR^r$",
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      "authority_text": "$\\nu_0E_1=\\nu E$",
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      "math_ordinal": "117",
      "rationale": "Part 256C(a)(iii) defines the Borel set H and proves H subset E with negligible difference; E_1 belongs to the preceding subpart and is unavailable here.",
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      "authority_line": "181;254-274",
      "authority_path": "authority/fremlin/source/mt2.2016/mt256.tex",
      "authority_text": "reuse of $\\Sigma$ for the auxiliary family in 256C(c)-(e)",
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      "math_ordinal": "",
      "rationale": "Sigma is already bound to the domain of nu. A distinct auxiliary family is necessary to avoid circularity while proving closure and then returning to the original measure domain. Grouped math ordinals 135, 203, 207, 208, 212, 214, 218, 219 and 220 are sealed in the unit QA.",
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      "target_normalized_sha256": "",
      "target_path": "source/id-ID/mt256.tex",
      "target_text": "fresh auxiliary family $\\Cal C$ used through 256C(d) and early 256C(e); original domain $\\Sigma$ retained at line 274",
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      "authority_line": "199;201;254",
      "authority_path": "authority/fremlin/source/mt2.2016/mt256.tex",
      "authority_text": "three occurrences of $\\Bbb R$ inside the proof for arbitrary $\\BbbR^r$",
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      "rationale": "The theorem fixes arbitrary dimension r; the closed-set exhaustion and auxiliary sigma-algebra argument must remain in the ambient Euclidean space R^r. Grouped math ordinals 156, 159 and 204 are sealed in the unit QA.",
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      "target_text": "$\\Bbb R^r$ restored at all three occurrences",
      "unit_id": "O007-FREMLIN-V2-S256"
    },
    {
      "authority_line": "247",
      "authority_path": "authority/fremlin/source/mt2.2016/mt256.tex",
      "authority_text": "$\\nu(F\\cap B(\\tbf{0},k)\\setminus H_k)$",
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      "math_ordinal": "199",
      "rationale": "The proof defines compact approximants K_m and their union H but no indexed family H_k; the preceding estimate applies to K_k.",
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      "target_path": "source/id-ID/mt256.tex",
      "target_text": "$\\nu(F\\cap B(\\tbf{0},k)\\setminus K_k)$",
      "unit_id": "O007-FREMLIN-V2-S256"
    },
    {
      "authority_line": "260",
      "authority_path": "authority/fremlin/source/mt2.2016/mt256.tex",
      "authority_text": "By (a-iii) and (a-iv)",
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      "rationale": "The union and intersection closure results invoked here are established in 256C(c)(iii)-(iv), not in part (a).",
      "source_normalized_sha256": "",
      "target_line": "260",
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      "target_path": "source/id-ID/mt256.tex",
      "target_text": "Berdasarkan (c-iii) dan (c-iv)",
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    },
    {
      "authority_line": "732",
      "authority_path": "authority/fremlin/source/mt2.2016/mt256.tex",
      "authority_text": "$\\hat f(y)\\ge a$ whenever $y\\in D\\cap[q,q']$",
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      "math_ordinal": "636",
      "rationale": "Membership in Phi_{aqq'} is defined by a strict inequality throughout the rational box; continuity and hat-f(x)>a justify the strict bound.",
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      "target_text": "$\\hat f(y)>a$ kapan pun $y\\in D\\cap[q,q']$",
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      "authority_line": "789-790",
      "authority_path": "authority/fremlin/source/mt2.2016/mt256.tex",
      "authority_text": "a linear combination of / of Radon-Nikod\\'ym derivatives",
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      "math_ordinal": "",
      "rationale": "The second line-opening 'of' duplicates the preposition at the end of the preceding line; the derivative removes the duplicate while translating the sentence.",
      "source_normalized_sha256": "",
      "target_line": "797-798",
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      "target_path": "source/id-ID/mt256.tex",
      "target_text": "suatu kombinasi linear dari turunan-turunan Radon-Nikod\\'ym",
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    {
      "authority_line": "824",
      "authority_path": "authority/fremlin/source/mt2.2016/mt256.tex",
      "authority_text": "$\\nu A=\\inf\\{\\nu G:G\\supseteq A$ is open$\\}$",
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      "math_ordinal": "742",
      "rationale": "The exercise introduces the corresponding outer measure and quantifies over every subset A; nu(A) need not be defined for a nonmeasurable A, whereas nu-star(A) is.",
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      "target_text": "$\\nu^*A=\\inf\\{\\nu G:G\\supseteq A$ terbuka$\\}$",
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      "authority_line": "894",
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      "authority_text": "$\\psi(x)(i)=x(i+1)$ for $x\\in\\{0,1\\}^{\\Bbb N}$ and $j\\in\\Bbb N$",
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      "rationale": "The defining formula binds coordinate i; j occurs nowhere in the formula and leaves i unquantified.",
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      "target_text": "$\\psi(x)(i)=x(i+1)$ untuk $x\\in\\{0,1\\}^{\\Bbb N}$ dan $i\\in\\Bbb N$",
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      "authority_line": "51-53",
      "authority_path": "authority/fremlin/source/mt2.2016/mt257.tex",
      "authority_text": "domain $\\Lambda$ of product measure $\\lambda$ used to qualify $h$ on $\\BbbR^r$",
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      "rationale": "The product measure lives on R^r x R^r while h lives on R^r; the conclusion integrates h against the convolution measure nu, so its domain and null sets are the required structures.",
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      "target_path": "source/id-ID/mt257.tex",
      "target_text": "domain $\\Sigma$ of convolution measure $\\nu$ used for measurability and almost-everywhere qualification",
      "unit_id": "O007-FREMLIN-V2-S257"
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    {
      "authority_line": "117",
      "authority_path": "authority/fremlin/source/mt2.2016/mt257.tex",
      "authority_text": "$\\int f_1*f_2d\\mu=1$",
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      "math_ordinal": "89",
      "rationale": "The measures are only totally finite, not probability measures; the convolution density integrates to the product of their total masses, which is finite but need not equal one.",
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      "target_text": "$\\int f_1*f_2d\\mu=\\nu_1(\\BbbR^r)\\nu_2(\\BbbR^r)<\\infty$",
      "unit_id": "O007-FREMLIN-V2-S257"
    },
    {
      "authority_line": "131",
      "authority_path": "authority/fremlin/source/mt2.2016/mt257.tex",
      "authority_text": "Radon-Nikod\\'ym derivative of $\\nu$",
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      "math_ordinal": "94",
      "rationale": "No unindexed nu is bound in 257F; the theorem and preceding calculation concern the convolution nu_1*nu_2.",
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      "target_path": "source/id-ID/mt257.tex",
      "target_text": "turunan Radon--Nikod\\'ym dari $\\nu_1*\\nu_2$",
      "unit_id": "O007-FREMLIN-V2-S257"
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      "authority_line": "141",
      "authority_path": "authority/fremlin/source/mt2.2016/mt257.tex",
      "authority_text": "for every totally finite Radon measure on $\\BbbR^r$",
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      "math_ordinal": "104",
      "rationale": "The identity uses nu but the source noun phrase omits the measure name; inserting nu binds the quantified symbol without changing the claim.",
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      "target_line": "141-142",
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      "target_path": "source/id-ID/mt257.tex",
      "target_text": "untuk setiap $\\nu$ yang merupakan ukuran Radon berhingga total pada $\\BbbR^r$",
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      "authority_line": "192",
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      "authority_text": "Lebesgue measure on $\\BbbR^r$",
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      "rationale": "Exercise 257Ya defines M over Borel subsets of the real line and binds no r; its L1 subalgebra must use Lebesgue measure on the same real line.",
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      "authority_line": "69",
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      "authority_text": "$C\\subseteq\\Bbb R$",
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      "rationale": "D is a subset of R^r and the asserted Minkowski sum C+D and proof both add vectors x+y; C must lie in the same r-dimensional ambient space.",
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      "authority_text": "$\\prod_{i=0}^r(\\alpha'_i-\\alpha_i)^{1/r}+\\prod_{i=0}^r(\\beta'_i-\\beta_i)^{1/r}$",
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      "authority_text": "union of at most $m-1$ disjoint half-open intervals",
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      "rationale": "A is presented as m+1 rectangles; when m=1 each nonempty side already contains one rectangle, so 'at most m-1' is impossible. The induction needs at most m rectangles, whose terminal index is at most m-1.",
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      "authority_text": "$B''=B\\setminus B$",
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      "rationale": "B setminus B is empty, contradicting the immediately following nonempty split and every use of B-double-prime; it is the complement of the just-defined B-prime within B.",
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      "authority_text": "unions of at most $n$ half-open intervals",
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      "rationale": "B is presented as n+1 rectangles and a hyperplane cut can leave a piece of every rectangle on either side; when n=0 each nonempty piece is one rectangle, not zero. The induction still decreases through the A-side count.",
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      "authority_line": "296",
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      "authority_text": "$(\\mu B)^{1/r}$",
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      "rationale": "Part (f) treats arbitrary nonempty B, for which mu(B) need not be defined; translation invariance gives outer measure mu-star(x+B)=mu-star(B), exactly as in the theorem statement.",
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      "authority_line": "499",
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      "authority_text": "applying (iii) to $\\phi\\restr H_k$",
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      "authority_text": "$r\\in\\coint{1,\\infty}$ and $\\rho(x,y)=\\|x-y\\|^{s/r}$",
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      "math_ordinal": "780",
      "rationale": "The snowflake exponent s/r must be at most one for the triangle inequality; r>=s is necessary, while the printed range admits counterexamples such as s=2 and r=1.",
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      "authority_line": "425",
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      "authority_text": "$|\\phi_i(y)-\\phi_i(x)-T_iy|$",
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      "math_ordinal": "325",
      "rationale": "The derivative definition and the immediately following squared estimate both apply the row T_i to y-x; T_i y would not approximate the increment at a general base point x.",
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      "authority_line": "702",
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      "authority_text": "$T(x)(x-y)$",
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      "rationale": "The derivative condition used to choose delta and the definition of H_n both use y-x; reversing it changes the linear term's sign.",
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      "target_text": "$T(x)(y-x)$",
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      "authority_line": "707",
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      "authority_text": "$S_k(x-y)$",
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      "target_text": "$S_k(y-x)$",
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      "authority_line": "850",
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      "authority_text": "$\\|T(x)-T_n\\|\\le 2^{-k}$",
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      "math_ordinal": "651",
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      "unit_id": "O007-FREMLIN-V2-S262"
    },
    {
      "authority_line": "1181",
      "authority_path": "authority/fremlin/source/mt2.2016/mt262.tex",
      "authority_text": "an $r\\times s$ matrix $T$",
      "classification": "mathematical-transposed-matrix-dimensions",
      "correction_id": "O007-CORR-0280",
      "math_ordinal": "892",
      "rationale": "A derivative of a map from R^r to R^s multiplies r-component displacements and returns s components, so its matrix has s rows and r columns.",
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      "target_path": "source/id-ID/mt262.tex",
      "target_text": "matriks $s\\times r$, yaitu $T$",
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      "authority_line": "359",
      "authority_path": "authority/fremlin/source/mt2.2016/mt265.tex",
      "authority_text": "$\\sum_{n=0}^{\\infty}J_n\\mu_r^*D_n+\\epsilon\\mu_r^*D_n$",
      "classification": "mathematical-missing-summand-grouping",
      "correction_id": "O007-CORR-0281",
      "math_ordinal": "286",
      "rationale": "Without parentheses the epsilon term lies outside the n-sum and leaves n free; the next bound uses the sum of both n-indexed terms over the disjoint partition.",
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      "target_text": "$\\sum_{n=0}^{\\infty}(J_n\\mu_r^*D_n+\\epsilon\\mu_r^*D_n)$",
      "unit_id": "O007-FREMLIN-V2-S265"
    },
    {
      "authority_line": "363",
      "authority_path": "authority/fremlin/source/mt2.2016/mt265.tex",
      "authority_text": "$\\int_D\\sum_nJ_n\\chi D_n\\,d\\mu$",
      "classification": "mathematical-undefined-measure-symbol",
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      "math_ordinal": "286",
      "rationale": "The section defines mu_r as Lebesgue measure and uses it throughout; no unindexed measure mu is bound in this theorem.",
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      "target_text": "$\\int_D\\sum_nJ_n\\chi D_n\\,d\\mu_r$",
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    },
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      "authority_line": "371",
      "authority_path": "authority/fremlin/source/mt2.2016/mt265.tex",
      "authority_text": "$\\int_DJ\\,d\\mu$",
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      "correction_id": "O007-CORR-0283",
      "math_ordinal": "293",
      "rationale": "The theorem's integration measure is mu_r and no unindexed mu is defined; the following terms and conclusion all use mu_r.",
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      "target_text": "$\\int_DJ\\,d\\mu_r$",
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    {
      "authority_line": "119",
      "authority_path": "authority/fremlin/source/mt2.2016/mt262.tex",
      "authority_text": "$\\sum_{i=1}^nb_i-a_i$",
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      "math_ordinal": "85",
      "rationale": "Absolute continuity bounds the sum of the interval lengths; without parentheses only b_i is summed and a_i remains free.",
      "source_normalized_sha256": "3439301b34703a48af92639ee73edb5df63ae41da5f9d70d8345cd050878240a",
      "target_line": "117",
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      "target_path": "source/id-ID/mt262.tex",
      "target_text": "$\\sum_{i=1}^n(b_i-a_i)$",
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    {
      "authority_line": "1265",
      "authority_path": "authority/fremlin/source/mt2.2016/mt262.tex",
      "authority_text": "$e^{1/(\\delta^2-\\|x\\|^2)}$",
      "classification": "mathematical-wrong-bump-function-sign",
      "correction_id": "O007-CORR-0285",
      "math_ordinal": "962",
      "rationale": "The positive exponent diverges at the boundary and cannot define the stated smooth compactly supported mollifier; the standard negative exponent tends to zero there.",
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      "target_line": "1254",
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      "target_text": "$e^{-1/(\\delta^2-\\|x\\|^2)}$",
      "unit_id": "O007-FREMLIN-V2-S262"
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      "authority_line": "348-350",
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      "authority_text": "writing $\\mu_{Hr}$ for r-dimensional Hausdorff outer measure",
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      "correction_id": "O007-CORR-0286",
      "math_ordinal": "238",
      "rationale": "The proposition and displayed inequality concern Hausdorff outer measure, whose notation here is mu_Hr-star; unstarred mu_Hr denotes the completed measure.",
      "source_normalized_sha256": "e743fe8316fcff8f2d9538efb00d23bbd8328edc5fe4cd44d96c16d433de340e",
      "target_line": "344-346",
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      "target_path": "source/id-ID/mt264.tex",
      "target_text": "menuliskan $\\mu_{Hr}^*$ untuk ukuran luar Hausdorff berdimensi $r$",
      "unit_id": "O007-FREMLIN-V2-S264"
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    {
      "authority_line": "923-925",
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      "authority_text": "if X is complete and $\\mu_{Hr}E$ is defined and finite, then $\\mu_{Hr}(\\phi[E])$ is defined",
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      "correction_id": "O007-CORR-0287",
      "math_ordinal": "target-insertion-707",
      "rationale": "Because phi is defined only on D, phi[E] is not defined for arbitrary E; the conclusion requires E to be a subset of D.",
      "source_normalized_sha256": "",
      "target_line": "920-922",
      "target_normalized_sha256": "809f6da7c90f034888299fe2f4046fe685f1865181f2236e7b0aee02758f9c67",
      "target_path": "source/id-ID/mt264.tex",
      "target_text": "jika $X$ lengkap dan $E\\subseteq D$ dan $\\mu_{Hr}E$ terdefinisi dan berhingga, maka $\\mu_{Hr}(\\phi[E])$ terdefinisi",
      "unit_id": "O007-FREMLIN-V2-S264"
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      "authority_text": "$h:\\coint{0,\\infty}\\to\\coint{0,\\infty}$ a non-decreasing function",
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      "correction_id": "O007-CORR-0288",
      "math_ordinal": "753",
      "rationale": "Without h(0)=0 the construction need not send the empty set to zero; for example h identically one gives theta_h(emptyset)=infinity, so it is not an outer measure.",
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      "target_line": "959-961",
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      "target_text": "$h:\\coint{0,\\infty}\\to\\coint{0,\\infty},\\ h(0)=0$ suatu fungsi tak menurun",
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    {
      "authority_line": "291-295",
      "authority_path": "authority/fremlin/source/mt2.2016/mt276.tex",
      "authority_text": "$\\Expn(Y_n+X'_n)^2$",
      "classification": "mathematical-missing-square-grouping",
      "correction_id": "O007-CORR-0289",
      "math_ordinal": "194",
      "rationale": "The next equality expands the expectation of the square of Y_n+X'_n; squaring its expectation would not yield the displayed second-moment identity.",
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      "target_line": "290-294",
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      "target_path": "source/id-ID/mt276.tex",
      "target_text": "$\\Expn((Y_n+X'_n)^2)$",
      "unit_id": "O007-FREMLIN-V2-S276"
    },
    {
      "authority_line": "377-378",
      "authority_path": "authority/fremlin/source/mt2.2016/mt276.tex",
      "authority_text": "$|\\int_{X\\setminus E}W_k|\\le\\lim_{n\\to\\Cal F}\\int_{X\\setminus E}|F_k(X_n)-F_{k-1}(X_n)|$",
      "classification": "mathematical-undefined-underlying-set",
      "correction_id": "O007-CORR-0290",
      "math_ordinal": "249",
      "rationale": "The lemma fixes Omega as the probability-space domain and defines E as a subset of Omega; no underlying set X is bound here.",
      "source_normalized_sha256": "15f368e3d22100ed73f2f79f3c67766c94b665c7a466dd23d8db0488e33918b2",
      "target_line": "379-380",
      "target_normalized_sha256": "e1245ebc3792ed49e2e20292679a077a15f14f79eec435c5f62d267223c69254",
      "target_path": "source/id-ID/mt276.tex",
      "target_text": "$|\\int_{\\Omega\\setminus E}W_k|\\le\\lim_{n\\to\\Cal F}\\int_{\\Omega\\setminus E}|F_k(X_n)-F_{k-1}(X_n)|$",
      "unit_id": "O007-FREMLIN-V2-S276"
    },
    {
      "authority_line": "382-384",
      "authority_path": "authority/fremlin/source/mt2.2016/mt276.tex",
      "authority_text": "$\\Expn(|W_k|)=\\int_EW_k-\\int_{X\\setminus E}W_k\\le\\cdots$",
      "classification": "mathematical-undefined-underlying-set",
      "correction_id": "O007-CORR-0291",
      "math_ordinal": "250",
      "rationale": "This is the same positive/negative-part decomposition as the preceding bound and its complementary domain must again be Omega minus E.",
      "source_normalized_sha256": "0accc69423e42cc7205b308e0680f17490c63f2a9979d2a5c5b4a894e6e30578",
      "target_line": "384-386",
      "target_normalized_sha256": "b0c865c458d80059b76aa201bd318a16251c3d6db15b507f536bdc7ebf9ad7b5",
      "target_path": "source/id-ID/mt276.tex",
      "target_text": "$\\Expn(|W_k|)=\\int_EW_k-\\int_{\\Omega\\setminus E}W_k\\le\\cdots$",
      "unit_id": "O007-FREMLIN-V2-S276"
    },
    {
      "authority_line": "426",
      "authority_path": "authority/fremlin/source/mt2.2016/mt276.tex",
      "authority_text": "$\\mu X=1$",
      "classification": "mathematical-wrong-underlying-set-symbol",
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      "math_ordinal": "272",
      "rationale": "The theorem and this proof use the measure space (Omega,Sigma,mu); the probability-space reduction requires mu(Omega)=1 and no set X is bound.",
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      "target_line": "429",
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      "target_path": "source/id-ID/mt276.tex",
      "target_text": "$\\mu\\Omega=1$",
      "unit_id": "O007-FREMLIN-V2-S276"
    },
    {
      "authority_line": "512-515",
      "authority_path": "authority/fremlin/source/mt2.2016/mt276.tex",
      "authority_text": "$\\Expn_m(X)(\\omega)=\\cdots$ if $x\\in A\\in\\Cal A_m$",
      "classification": "mathematical-wrong-bound-variable",
      "correction_id": "O007-CORR-0293",
      "math_ordinal": "337",
      "rationale": "The conditional expectation is being evaluated at omega and the atom containing that same point must be selected; x is unbound in the definition.",
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      "target_text": "$\\Expn_m(X)(\\omega)=\\cdots$ jika $\\omega\\in A\\in\\Cal A_m$",
      "unit_id": "O007-FREMLIN-V2-S276"
    },
    {
      "authority_line": "567",
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      "authority_text": "$\\Bover1{n+1}\\sum_{i=0}^nX_{r(i)}-F_i(X_{r(i)})\\to0$",
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      "correction_id": "O007-CORR-0294",
      "math_ordinal": "361",
      "rationale": "The argument averages the first component of the four-term decomposition for every i; without parentheses only X is summed and the index i remains free in F_i.",
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      "target_path": "source/id-ID/mt276.tex",
      "target_text": "$\\Bover1{n+1}\\sum_{i=0}^n(X_{r(i)}-F_i(X_{r(i)}))\\to0$",
      "unit_id": "O007-FREMLIN-V2-S276"
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    {
      "authority_line": "577",
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      "authority_text": "$\\Bover1{n+1}\\sum_{i=0}^nF_i(X_{r(i)})-Y_i-Z_{i,r(i)}\\to0$",
      "classification": "mathematical-missing-summand-grouping",
      "correction_id": "O007-CORR-0295",
      "math_ordinal": "365",
      "rationale": "The second component is the average of the complete approximation error for each i; ungrouped Y_i and Z_i would lie outside the sum with free i.",
      "source_normalized_sha256": "d7a4da552ebe81595f54365a6f2eb098db866ef899a2ff657ae7b55365448cc1",
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      "target_path": "source/id-ID/mt276.tex",
      "target_text": "$\\Bover1{n+1}\\sum_{i=0}^n(F_i(X_{r(i)})-Y_i-Z_{i,r(i)})\\to0$",
      "unit_id": "O007-FREMLIN-V2-S276"
    },
    {
      "authority_line": "606-614",
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      "authority_text": "$\\int F_n(X_{r(n)}-Y_n)^2$",
      "classification": "mathematical-misapplied-truncation",
      "correction_id": "O007-CORR-0296",
      "math_ordinal": "377",
      "rationale": "Membership in J_n supplies the squared L2 error between the truncated variable F_n(X_r(n)) and Y_n; applying F_n to their difference is a different unsupported quantity.",
      "source_normalized_sha256": "3b29719d4f527062b8d8701d7d7c7e3061c1a69be130553f1c7055bf8c97880d",
      "target_line": "616-628",
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      "target_path": "source/id-ID/mt276.tex",
      "target_text": "$\\int(F_n(X_{r(n)})-Y_n)^2$",
      "unit_id": "O007-FREMLIN-V2-S276"
    },
    {
      "authority_line": "627",
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      "authority_text": "$\\Bover1{n+1}\\sum_{i=0}^nX_{r(i)}\\to0$",
      "classification": "mathematical-wrong-limit",
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      "math_ordinal": "382",
      "rationale": "Components (i), (ii), and (iv) tend to zero while component (iii) tends to Y; the assertion at the start of part (d) and its later use both require the total to tend to Y.",
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      "target_path": "source/id-ID/mt276.tex",
      "target_text": "$\\Bover1{n+1}\\sum_{i=0}^nX_{r(i)}\\to Y$",
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      "authority_line": "661",
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      "authority_text": "$\\sup_{n\\in\\Bbb N}\\int\\Bover1f\\times X_nd\\nu<\\infty$",
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      "correction_id": "O007-CORR-0298",
      "math_ordinal": "417",
      "rationale": "The probability-space case requires a uniform L1 bound. Under dnu=f dmu this equals the stated hypothesis only with the absolute value around f^{-1}X_n.",
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      "target_path": "source/id-ID/mt276.tex",
      "target_text": "$\\sup_{n\\in\\Bbb N}\\int|\\Bover1f\\times X_n|d\\nu<\\infty$",
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    {
      "authority_line": "745",
      "authority_path": "authority/fremlin/source/mt2.2016/mt276.tex",
      "authority_text": "$\\sequencen{\\Bover1{n+1}\\sum_{i=0}^{\\infty}X_i}$",
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      "correction_id": "O007-CORR-0299",
      "math_ordinal": "488",
      "rationale": "The exercise asks for convergence of the sequence of finite Cesaro means; an infinite numerator is independent of n and need not even be defined under the stated finite-expectation hypothesis.",
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      "target_path": "source/id-ID/mt276.tex",
      "target_text": "$\\sequencen{\\Bover1{n+1}\\sum_{i=0}^{n}X_i}$",
      "unit_id": "O007-FREMLIN-V2-S276"
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      "authority_line": "271",
      "authority_path": "authority/fremlin/source/mt2.2016/mt271.tex",
      "authority_text": "for all $\\alpha_0,\\ldots,\\alpha_n\\in\\Bbb R$",
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      "math_ordinal": "183",
      "rationale": "The family consists of X_1 through X_n and the displayed linear combination has coefficients alpha_1 through alpha_n; alpha_0 is extraneous while alpha_1 is required.",
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      "target_path": "source/id-ID/mt271.tex",
      "target_text": "untuk semua $\\alpha_1,\\ldots,\\alpha_n\\in\\Bbb R$",
      "unit_id": "O007-FREMLIN-V2-S271"
    },
    {
      "authority_line": "491-492",
      "authority_path": "authority/fremlin/source/mt2.2016/mt271.tex",
      "authority_text": "the indefinite-integral measure over $\\mu$ associated with $f$",
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      "correction_id": "O007-CORR-0301",
      "math_ordinal": "328",
      "rationale": "The surrounding result declares Lebesgue measure mu_L and the next displayed identity integrates f against mu_L; no measure named bare mu is bound in this argument.",
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      "target_path": "source/id-ID/mt271.tex",
      "target_text": "ukuran integral tak tentu di atas $\\mu_L$ yang berkaitan dengan $f$",
      "unit_id": "O007-FREMLIN-V2-S271"
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      "authority_line": "545-546",
      "authority_path": "authority/fremlin/source/mt2.2016/mt271.tex",
      "authority_text": "$g$ is zero off $\\phi[D_k]$",
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      "correction_id": "O007-CORR-0302",
      "math_ordinal": "374",
      "rationale": "The parenthetical argument is proving integrability of the fixed summand g_k, whose definition makes it zero outside phi[D_k]; the total density g generally has support on the other phi[D_j].",
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      "target_text": "$g_k$ nol di luar $\\phi[D_k]$",
      "unit_id": "O007-FREMLIN-V2-S271"
    },
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      "authority_line": "760-761",
      "authority_path": "authority/fremlin/source/mt2.2016/mt271.tex",
      "authority_text": "$F(\\alpha_1,\\alpha_2)+F(\\beta_1,\\beta_2)\\ge F(\\alpha_1,\\beta_2)+F(\\alpha_2,\\beta_1)$",
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      "correction_id": "O007-CORR-0303",
      "math_ordinal": "566",
      "rationale": "The mixed lower-left corner of the rectangle is (beta_1,alpha_2), not (alpha_2,beta_1); the latter reverses the two coordinate roles.",
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      "target_path": "source/id-ID/mt271.tex",
      "target_text": "$F(\\alpha_1,\\alpha_2)+F(\\beta_1,\\beta_2)\\ge F(\\alpha_1,\\beta_2)+F(\\beta_1,\\alpha_2)$",
      "unit_id": "O007-FREMLIN-V2-S271"
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      "authority_line": "770-771",
      "authority_path": "authority/fremlin/source/mt2.2016/mt271.tex",
      "authority_text": "$\\lambda\\ocint{a,b}=F(\\alpha_1,\\alpha_2)+F(\\beta_1,\\beta_2)-F(\\alpha_1,\\beta_2)-F(\\alpha_2,\\beta_1)$",
      "classification": "mathematical-swapped-coordinate-arguments",
      "correction_id": "O007-CORR-0304",
      "math_ordinal": "579",
      "rationale": "The hint must use the same four rectangle corners as the corrected condition; its second mixed corner is (beta_1,alpha_2).",
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      "target_normalized_sha256": "05cb6065f34a4d089476fe94feaca64f016d996150d2c09446f7687bbe4ea897",
      "target_path": "source/id-ID/mt271.tex",
      "target_text": "$\\lambda\\ocint{a,b}=F(\\alpha_1,\\alpha_2)+F(\\beta_1,\\beta_2)-F(\\alpha_1,\\beta_2)-F(\\beta_1,\\alpha_2)$",
      "unit_id": "O007-FREMLIN-V2-S271"
    },
    {
      "authority_line": "79-82",
      "authority_path": "authority/fremlin/source/mt2.2016/mt274.tex",
      "authority_text": "\\int_{\\infty}^ce^{-(x-a)^2/2\\sigma^2}dx",
      "classification": "mathematical-wrong-integral-bound",
      "correction_id": "O007-CORR-0305",
      "math_ordinal": "31",
      "rationale": "The change of variables covers the normal left tail up to c and the next integral explicitly starts at negative infinity; a positive-infinity lower bound would reverse or empty the intended integral.",
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      "target_line": "77-80",
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      "target_path": "source/id-ID/mt274.tex",
      "target_text": "\\int_{-\\infty}^ce^{-(x-a)^2/2\\sigma^2}dx",
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    },
    {
      "authority_line": "128-129",
      "authority_path": "authority/fremlin/source/mt2.2016/mt274.tex",
      "authority_text": "\\Bover1{2\\pi\\sigma_1\\sigma_2}e^{-(\\zeta_1^2+\\zeta_2^2)/2}",
      "classification": "mathematical-spurious-scale-denominator",
      "correction_id": "O007-CORR-0306",
      "math_ordinal": "60",
      "rationale": "Z_1 and Z_2 were just standardized, so their joint density is the product of two standard-normal densities and contains no sigma_1 sigma_2 scale factor.",
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      "target_line": "125-126",
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      "target_path": "source/id-ID/mt274.tex",
      "target_text": "\\Bover1{2\\pi}e^{-(\\zeta_1^2+\\zeta_2^2)/2}",
      "unit_id": "O007-FREMLIN-V2-S274"
    },
    {
      "authority_line": "184",
      "authority_path": "authority/fremlin/source/mt2.2016/mt274.tex",
      "authority_text": "$W_j=\\sum_{i=0}^{j-1}U_j+\\sum_{i=j+1}^nV_j$",
      "classification": "mathematical-wrong-summand-indices",
      "correction_id": "O007-CORR-0307",
      "math_ordinal": "91",
      "rationale": "W_j must contain all the other indexed summands so that Z_j=W_j+V_j and Z_{j+1}=W_j+U_j; repeating U_j and V_j does not yield either identity.",
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      "target_path": "source/id-ID/mt274.tex",
      "target_text": "$W_j=\\sum_{i=0}^{j-1}U_i+\\sum_{i=j+1}^nV_i$",
      "unit_id": "O007-FREMLIN-V2-S274"
    },
    {
      "authority_line": "195-197",
      "authority_path": "authority/fremlin/source/mt2.2016/mt274.tex",
      "authority_text": "$(w,u)\\mapsto h(w,u)$",
      "classification": "mathematical-wrong-function-argument",
      "correction_id": "O007-CORR-0308",
      "math_ordinal": "102",
      "rationale": "The lemma declares h as a unary real function, and both the preceding addition map and the following three-variable integrand require composition h(w+u), not an undefined binary h(w,u).",
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      "target_path": "source/id-ID/mt274.tex",
      "target_text": "$(w,u)\\mapsto h(w+u)$",
      "unit_id": "O007-FREMLIN-V2-S274"
    },
    {
      "authority_line": "309-311",
      "authority_path": "authority/fremlin/source/mt2.2016/mt274.tex",
      "authority_text": "for $|x|\\le\\epsilon$.",
      "classification": "mathematical-missing-domain-extension",
      "correction_id": "O007-CORR-0309",
      "math_ordinal": "target-insertion-151",
      "rationale": "The lemma requires a function on all of R with the stated constant tails, but the proof only defined it on the closed transition interval; the constant extensions complete the construction and match the endpoint derivatives.",
      "source_normalized_sha256": "",
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      "target_path": "source/id-ID/mt274.tex",
      "target_text": "$h(x)=1\\text{ jika }x\\le-\\epsilon,\\quad\\text{dan}\\quad h(x)=0\\text{ jika }x\\ge\\epsilon$",
      "unit_id": "O007-FREMLIN-V2-S274"
    },
    {
      "authority_line": "445-447",
      "authority_path": "authority/fremlin/source/mt2.2016/mt274.tex",
      "authority_text": "$\\sup_{x\\in\\Bbb R}|h'''(x)|=M_3$",
      "classification": "mathematical-missing-translated-function-subscript",
      "correction_id": "O007-CORR-0310",
      "math_ordinal": "239",
      "rationale": "This paragraph states derivative bounds for the translate h_t; the second bound must name h_t just as the preceding h_t double-prime bound does.",
      "source_normalized_sha256": "dd011b751a2083b09a6017bf8dc1e64f60ceace057f73b996d3fcde302907205",
      "target_line": "440-442",
      "target_normalized_sha256": "d3757114ea4fc0736c11999dbedecaeb77ae21784d05e59a2d6c0883360e02a4",
      "target_path": "source/id-ID/mt274.tex",
      "target_text": "$\\sup_{x\\in\\Bbb R}|h_t'''(x)|=M_3$",
      "unit_id": "O007-FREMLIN-V2-S274"
    },
    {
      "authority_line": "470-472",
      "authority_path": "authority/fremlin/source/mt2.2016/mt274.tex",
      "authority_text": "$\\chi\\ocint{-\\infty,a-2\\epsilon}\\le h_{a-\\epsilon}\\le\\chi\\ocint{\\infty,a}\\le h_{a+\\epsilon}\\le\\chi\\ocint{-\\infty,a+\\epsilon}$",
      "classification": "mathematical-wrong-indicator-endpoint",
      "correction_id": "O007-CORR-0311",
      "math_ordinal": "244",
      "rationale": "The middle indicator is the same left-tail distribution event as the two surrounding indicators; an interval beginning at positive infinity is empty and breaks the sandwich.",
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      "target_path": "source/id-ID/mt274.tex",
      "target_text": "$\\chi\\ocint{-\\infty,a-2\\epsilon}\\le h_{a-\\epsilon}\\le\\chi\\ocint{-\\infty,a}\\le h_{a+\\epsilon}\\le\\chi\\ocint{-\\infty,a+\\epsilon}$",
      "unit_id": "O007-FREMLIN-V2-S274"
    },
    {
      "authority_line": "646-647",
      "authority_path": "authority/fremlin/source/mt2.2016/mt274.tex",
      "authority_text": "\\int_{\\{x:|x|>\\epsilon\\sigma\\sqrt{n}\\}}x^2\\nu(dx)",
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      "correction_id": "O007-CORR-0312",
      "math_ordinal": "324",
      "rationale": "The proof defines s_n=sigma sqrt(n+1) immediately above, so expanding psi_{epsilon s_n} requires the same n+1 threshold.",
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      "target_path": "source/id-ID/mt274.tex",
      "target_text": "\\int_{\\{x:|x|>\\epsilon\\sigma\\sqrt{n+1}\\}}x^2\\nu(dx)",
      "unit_id": "O007-FREMLIN-V2-S274"
    },
    {
      "authority_line": "880-885",
      "authority_path": "authority/fremlin/source/mt2.2016/mt274.tex",
      "authority_text": "$S_n=\\Bover1{\\sqrt{n+1}}(X_0+\\ldots+X_n)$",
      "classification": "mathematical-missing-centering",
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      "math_ordinal": "471",
      "rationale": "For common expectation c, the uncentered normalized sum has expectation c sqrt(n+1) and cannot converge to N(c,sigma^2). Centering each summand and adding c yields exactly the asserted limiting mean and variance.",
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      "target_path": "source/id-ID/mt274.tex",
      "target_text": "$S_n=c+\\Bover1{\\sqrt{n+1}}\\sum_{i=0}^n(X_i-c)$",
      "unit_id": "O007-FREMLIN-V2-S274"
    },
    {
      "authority_line": "932-935",
      "authority_path": "authority/fremlin/source/mt2.2016/mt274.tex",
      "authority_text": "$\\lim_{m\\to\\infty}\\Pr(|\\sum_{i=n}^mt_iX_i|\\ge 1)\\ge\\bover12)$",
      "classification": "mathematical-stray-delimiter",
      "correction_id": "O007-CORR-0314",
      "math_ordinal": "505",
      "rationale": "The final closing parenthesis has no opening mate and lies outside the probability expression; removing it restores the intended one-half lower bound.",
      "source_normalized_sha256": "cc2758cce32dca5279a2d4f39b4d8d076d29bc5463cab8140e3c11bf87c8cfd7",
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      "target_path": "source/id-ID/mt274.tex",
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    {
      "authority_line": "963-965",
      "authority_path": "authority/fremlin/source/mt2.2016/mt274.tex",
      "authority_text": "$\\chi\\ocint{-\\infty,-\\epsilon}\\le h\\le\\chi\\coint{\\epsilon,\\infty}$",
      "classification": "mathematical-impossible-indicator-sandwich",
      "correction_id": "O007-CORR-0315",
      "math_ordinal": "531",
      "rationale": "At x below -epsilon the lower indicator is one while the printed upper right-tail indicator is zero, making the inequalities impossible. The upper indicator must be the left tail through epsilon, as in Lemma 274E.",
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      "target_line": "953-955",
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      "target_text": "$\\chi\\ocint{-\\infty,-\\epsilon}\\le h\\le\\chi\\ocint{-\\infty,\\epsilon}$",
      "unit_id": "O007-FREMLIN-V2-S274"
    },
    {
      "authority_line": "86-90",
      "authority_path": "authority/fremlin/source/mt2.2016/mt275.tex",
      "authority_text": "$\\mu$-negligible subsets of $X$ belong to $\\Sigma_0$",
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      "correction_id": "O007-CORR-0316",
      "math_ordinal": "76",
      "rationale": "The surrounding martingale examples are defined on the probability space Omega, while X denotes a random variable rather than an underlying set whose negligible subsets could be measured.",
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      "target_line": "80-83",
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      "target_path": "source/id-ID/mt275.tex",
      "target_text": "himpunan yang terabaikan terhadap $\\mu$ di dalam $\\Omega$ termasuk dalam $\\Sigma_0$",
      "unit_id": "O007-FREMLIN-V2-S275"
    },
    {
      "authority_line": "781-785",
      "authority_path": "authority/fremlin/source/mt2.2016/mt275.tex",
      "authority_text": "$X\\setminus E\\in\\tilde\\Sigma_{\\tau}$",
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      "correction_id": "O007-CORR-0317",
      "math_ordinal": "641",
      "rationale": "The displayed complement immediately above is Omega minus E and the proof is verifying closure of a sigma-subalgebra of Sigma on Omega; X is not an underlying set here.",
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      "target_line": "771-775",
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      "target_text": "$\\Omega\\setminus E\\in\\tilde\\Sigma_{\\tau}$",
      "unit_id": "O007-FREMLIN-V2-S275"
    },
    {
      "authority_line": "841-844",
      "authority_path": "authority/fremlin/source/mt2.2016/mt275.tex",
      "authority_text": "partitions of conegligible subsets of $X$",
      "classification": "mathematical-wrong-underlying-set-symbol",
      "correction_id": "O007-CORR-0318",
      "math_ordinal": "675",
      "rationale": "The H_k and H'_k are level sets of stopping times on Omega and form partitions of conegligible subsets of that probability-space domain; no set X is bound.",
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      "authority_text": "$\\Pr(\\sup_{n\\in\\Bbb N}|X_n|)\\ge\\epsilon)\\le\\bover1\\epsilon\\sup_{n\\in\\Bbb N}\\Expn(|X_n|)$",
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      "rationale": "The maximal inequality bounds the probability of the event that the supremum is at least epsilon; the printed parenthesis closes Pr before the event comparison and leaves an unmatched final parenthesis.",
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      "target_text": "$\\Pr(\\sup_{n\\in\\Bbb N}|X_n|\\ge\\epsilon)\\le\\bover1\\epsilon\\sup_{n\\in\\Bbb N}\\Expn(|X_n|)$",
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      "authority_text": "for each $n\\in\\Bbb N\\,\\,X_n$ is a conditional expectation",
      "classification": "mathematical-missing-math-delimiter-and-clause-punctuation",
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      "math_ordinal": "802",
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    {
      "authority_line": "1024-1027",
      "authority_path": "authority/fremlin/source/mt2.2016/mt275.tex",
      "authority_text": "intervals of the type $[0,2^{-n}r]$ for $r\\le 2^{-n}$",
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      "authority_text": "$\\sequencen{X_n}$ be a uniformly integrable martingale adapted to $\\Sigma_n$",
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      "target_text": "$\\sequencen{X_n}$ adalah martingal terintegralkan secara seragam yang teradaptasi terhadap $\\sequencen{\\Sigma_n}$",
      "unit_id": "O007-FREMLIN-V2-S275"
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    {
      "authority_line": "93",
      "authority_path": "authority/fremlin/source/mt2.2016/mt272.tex",
      "authority_text": "\\prod_{i\\in J}\\alpha_j",
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      "rationale": "The product is indexed by i over J, so its factor must be alpha_i; alpha_j leaves j free and does not describe the stated family product.",
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      "target_text": "\\prod_{i\\in J}\\alpha_i",
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    },
    {
      "authority_line": "232",
      "authority_path": "authority/fremlin/source/mt2.2016/mt272.tex",
      "authority_text": "\\prod_{i=1}^n\\hat\\mu E_j",
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      "rationale": "The preceding factors and intersection are indexed by j; the printed product binds i while leaving j free.",
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      "target_path": "source/id-ID/mt272.tex",
      "target_text": "\\prod_{j=1}^n\\hat\\mu E_j",
      "unit_id": "O007-FREMLIN-V2-S272"
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    {
      "authority_line": "234",
      "authority_path": "authority/fremlin/source/mt2.2016/mt272.tex",
      "authority_text": "E_1,\\ldots,E_k",
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      "correction_id": "O007-CORR-0325",
      "math_ordinal": "135",
      "rationale": "The proof fixed E_1 through E_n and no k is bound in this implication; the arbitrariness claim must refer to those n sets.",
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      "target_text": "E_1,\\ldots,E_n",
      "unit_id": "O007-FREMLIN-V2-S272"
    },
    {
      "authority_line": "247",
      "authority_path": "authority/fremlin/source/mt2.2016/mt272.tex",
      "authority_text": "\\prod_{i=1}^n\\hat\\mu X_{i_j}^{-1}[F_j]",
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      "correction_id": "O007-CORR-0326",
      "math_ordinal": "145",
      "rationale": "The factor is parameterized by j through both i_j and F_j; binding i leaves j free.",
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      "target_text": "\\prod_{j=1}^n\\hat\\mu X_{i_j}^{-1}[F_j]",
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    },
    {
      "authority_line": "265",
      "authority_path": "authority/fremlin/source/mt2.2016/mt272.tex",
      "authority_text": "h(X_i)",
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      "correction_id": "O007-CORR-0327",
      "math_ordinal": "159;160",
      "rationale": "The proposition declares the family h_i, so the generated sigma-algebra and measurability argument must use h_i(X_i), not an unbound common h. Both occurrences carry the same repair.",
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      "target_path": "source/id-ID/mt272.tex",
      "target_text": "h_i(X_i)",
      "unit_id": "O007-FREMLIN-V2-S272"
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    {
      "authority_line": "446",
      "authority_path": "authority/fremlin/source/mt2.2016/mt272.tex",
      "authority_text": "\\bigcap_{j\\le n}E_{i_j}",
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      "correction_id": "O007-CORR-0328",
      "math_ordinal": "281",
      "rationale": "The proof declared E_j in Sigma_{i_j}; E_{i_j} is not declared and must be E_j in the diagonal-preimage identity.",
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      "target_text": "\\bigcap_{j\\le n}E_j",
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    {
      "authority_line": "449-452",
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      "authority_text": "\\mu_{i_j}E_{i_j}",
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      "math_ordinal": "283",
      "rationale": "The same declared events E_j must appear in the measure-product computation; E_{i_j} is undefined in this proof.",
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      "target_text": "\\mu_{i_j}E_j",
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    {
      "authority_line": "597",
      "authority_path": "authority/fremlin/source/mt2.2016/mt272.tex",
      "authority_text": "E_i=X_i",
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      "math_ordinal": "406",
      "rationale": "In the product-space cylinder argument E_i must equal the full factor space Omega_i outside finite J; X_i denotes a random variable and cannot serve as the factor set.",
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    {
      "authority_line": "620",
      "authority_path": "authority/fremlin/source/mt2.2016/mt272.tex",
      "authority_text": "\\Pr(X_{ij}\\in E_{ij})",
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      "math_ordinal": "421",
      "rationale": "F_{ij} are the declared Borel target sets; E_i is the product-coordinate event and E_{ij} is nowhere defined.",
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    {
      "authority_line": "669-675",
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      "authority_text": "\\mu\\phi^{-1}[\\bigcap_{i\\in I}H_i]",
      "classification": "mathematical-product-space-type-and-index-errors",
      "correction_id": "O007-CORR-0332",
      "math_ordinal": "453",
      "rationale": "The H_i are coordinate subsets whose product lies in the product space; J is undefined, and membership in Sigma_i concerns each coordinate map phi_i. The printed intersection and unindexed preimage are ill-typed.",
      "source_normalized_sha256": "7a02b7f71d04b1d82bc007811b3b207b5a9986771c9893c02483a69a22e042d2",
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      "target_text": "\\mu\\phi^{-1}[\\prod_{i\\in I}H_i]",
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    },
    {
      "authority_line": "771",
      "authority_path": "authority/fremlin/source/mt2.2016/mt272.tex",
      "authority_text": "E\\in\\Sigma_{\\infty}^*",
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      "correction_id": "O007-CORR-0333",
      "math_ordinal": "512",
      "rationale": "The proof defined the tail event E_a and immediately uses its measure; bare E is not defined in this argument.",
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    {
      "authority_line": "804",
      "authority_path": "authority/fremlin/source/mt2.2016/mt272.tex",
      "authority_text": "j_1,\\ldots,j_r",
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      "math_ordinal": "534",
      "rationale": "The theorem quantifies E_r for r up to n and its displayed product also runs to n; j_r leaves the family length inconsistent.",
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      "target_text": "j_1,\\ldots,j_n",
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    {
      "authority_line": "832",
      "authority_path": "authority/fremlin/source/mt2.2016/mt272.tex",
      "authority_text": "j_1,\\ldots,j_r",
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      "math_ordinal": "580",
      "rationale": "The proof repeats the theorem's n-member family and uses r only as the running index; the terminal subscript must be n.",
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      "authority_line": "926",
      "authority_path": "authority/fremlin/source/mt2.2016/mt272.tex",
      "authority_text": "\\int(x+a)^2\\mu_{X_1}(dx)",
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      "correction_id": "O007-CORR-0336",
      "math_ordinal": "646",
      "rationale": "The proof declared nu_{X_1} as the distribution of X_1; mu_{X_1} is not defined in this section.",
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    {
      "authority_line": "1092",
      "authority_path": "authority/fremlin/source/mt2.2016/mt272.tex",
      "authority_text": "h_i''(t)",
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      "rationale": "The Taylor argument defines the function h in the immediately preceding sentence; no h_i function is declared there.",
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    {
      "authority_line": "1104-1107",
      "authority_path": "authority/fremlin/source/mt2.2016/mt272.tex",
      "authority_text": "\\Expn(\\exp(b\\sum_{i=0}^nX_i-a_i-c))",
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      "math_ordinal": "741",
      "rationale": "The subsequent factorization subtracts b a_i and b c for every i. The printed expression leaves i free outside the sum and subtracts c only once syntactically.",
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    {
      "authority_line": "1192",
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      "authority_text": "\\Expn(Z)=\\gamma\\Expn(Y)",
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      "math_ordinal": "813",
      "rationale": "Here Z sums X_n inclusively from n=0 through Y, so E(Z)=gamma sum_{n>=0}P(Y>=n)=gamma(E(Y)+1). The unshifted Wald form applies to a one-indexed or exclusive convention.",
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      "target_text": "\\Expn(Z)=\\gamma(\\Expn(Y)+1)",
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    {
      "authority_line": "1388",
      "authority_path": "authority/fremlin/source/mt2.2016/mt272.tex",
      "authority_text": "\\exp(a_i+\\bover18b^2)",
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      "math_ordinal": "964",
      "rationale": "The endnote summarizes the Taylor estimate proved above, whose linear term is b a_i; omitting b contradicts both h'(0)=a_i and the preceding displayed bound.",
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    },
    {
      "authority_line": "63",
      "authority_path": "authority/fremlin/source/mt2.2016/mt273.tex",
      "authority_text": "\\langle X_i\\rangle_{i\\ge n_k}",
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      "correction_id": "O007-CORR-0341",
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      "rationale": "The partial-sum difference S_m-S_{n_k} is generated by increments with i strictly greater than n_k; including X_{n_k} would shift the Etemadi sequence by one term.",
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      "authority_text": "\\beta_n=\\sum_{i=0}^n\\mu E_n",
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      "authority_text": "\\|X\\times\\chi F)\\|_p",
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      "authority_text": "\\tilde Y_n+Y''_n+\\alpha_n",
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      "authority_text": "Of the four",
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      "rationale": "The immediately preceding sentence lists five applications: 273G, 273J, 273Xd, 273Xg, and 273Xi.",
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      "authority_text": "\\|\\alpha u-\\alpha u_n\\|=|\\alpha|\\|u-u_n\\|\\to 0",
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      "target_path": "source/id-ID/mt282.tex",
      "target_text": "sum (|c_k|^2+|d_k|^2)",
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      "authority_text": "varhat(g_3)y dy",
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      "authority_text": "\\Bbb R\\setminus K",
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        "heading": "Preferred terms and variants",
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            "cells": {
              "Attested/search variant(s)": "`teori ukur`",
              "Concept": "measure theory / visible edition title",
              "Decision and basis": "Refined to the noun compound attested by BAREKENG and explicitly paired with “measure theory” by UII. This does not change `ruang ukur`.",
              "Preferred O007 form": "`teori ukuran`; `Fondasi Teori Ukuran`"
            },
            "line": 29,
            "raw_markdown": "| measure theory / visible edition title | `teori ukuran`; `Fondasi Teori Ukuran` | `teori ukur` | Refined to the noun compound attested by BAREKENG and explicitly paired with “measure theory” by UII. This does not change `ruang ukur`. |"
          },
          {
            "cells": {
              "Attested/search variant(s)": "`ruang ukuran`",
              "Concept": "measure space",
              "Decision and basis": "Retain the concise UII-attested compound; record BAREKENG’s form as a variant.",
              "Preferred O007 form": "`ruang ukur`"
            },
            "line": 30,
            "raw_markdown": "| measure space | `ruang ukur` | `ruang ukuran` | Retain the concise UII-attested compound; record BAREKENG’s form as a variant. |"
          },
          {
            "cells": {
              "Attested/search variant(s)": "—",
              "Concept": "measurable space",
              "Decision and basis": "Retain; exact field agreement.",
              "Preferred O007 form": "`ruang terukur`"
            },
            "line": 31,
            "raw_markdown": "| measurable space | `ruang terukur` | — | Retain; exact field agreement. |"
          },
          {
            "cells": {
              "Attested/search variant(s)": "`aljabar-σ`",
              "Concept": "sigma-algebra",
              "Decision and basis": "Retain the text form; preserve the symbolic spelling as a display/search variant.",
              "Preferred O007 form": "`aljabar-sigma`"
            },
            "line": 32,
            "raw_markdown": "| sigma-algebra | `aljabar-sigma` | `aljabar-σ` | Retain the text form; preserve the symbolic spelling as a display/search variant. |"
          },
          {
            "cells": {
              "Attested/search variant(s)": "—",
              "Concept": "measure",
              "Decision and basis": "Retain; exact field agreement.",
              "Preferred O007 form": "`ukuran`"
            },
            "line": 33,
            "raw_markdown": "| measure | `ukuran` | — | Retain; exact field agreement. |"
          },
          {
            "cells": {
              "Attested/search variant(s)": "`terukur Lebesgue` where qualified",
              "Concept": "measurable set/function",
              "Decision and basis": "Retain; exact field agreement.",
              "Preferred O007 form": "`himpunan terukur`; `fungsi terukur`"
            },
            "line": 34,
            "raw_markdown": "| measurable set/function | `himpunan terukur`; `fungsi terukur` | `terukur Lebesgue` where qualified | Retain; exact field agreement. |"
          },
          {
            "cells": {
              "Attested/search variant(s)": "—",
              "Concept": "outer measure",
              "Decision and basis": "Retain; exact field agreement.",
              "Preferred O007 form": "`ukuran luar`"
            },
            "line": 35,
            "raw_markdown": "| outer measure | `ukuran luar` | — | Retain; exact field agreement. |"
          },
          {
            "cells": {
              "Attested/search variant(s)": "—",
              "Concept": "countable",
              "Decision and basis": "Retain; exact field agreement.",
              "Preferred O007 form": "`terhitung`"
            },
            "line": 36,
            "raw_markdown": "| countable | `terhitung` | — | Retain; exact field agreement. |"
          },
          {
            "cells": {
              "Attested/search variant(s)": "`saling asing`",
              "Concept": "disjoint",
              "Decision and basis": "Retain the established reader term; record the BAREKENG usage as a variant.",
              "Preferred O007 form": "`saling lepas`"
            },
            "line": 37,
            "raw_markdown": "| disjoint | `saling lepas` | `saling asing` | Retain the established reader term; record the BAREKENG usage as a variant. |"
          },
          {
            "cells": {
              "Attested/search variant(s)": "`h.d.`",
              "Concept": "almost everywhere",
              "Decision and basis": "Retain the unabbreviated form for readability; record the attested abbreviation for lookup.",
              "Preferred O007 form": "`hampir di mana-mana`"
            },
            "line": 38,
            "raw_markdown": "| almost everywhere | `hampir di mana-mana` | `h.d.` | Retain the unabbreviated form for readability; record the attested abbreviation for lookup. |"
          },
          {
            "cells": {
              "Attested/search variant(s)": "—",
              "Concept": "characteristic function",
              "Decision and basis": "Retain; exact field agreement.",
              "Preferred O007 form": "`fungsi karakteristik`"
            },
            "line": 39,
            "raw_markdown": "| characteristic function | `fungsi karakteristik` | — | Retain; exact field agreement. |"
          },
          {
            "cells": {
              "Attested/search variant(s)": "—",
              "Concept": "simple function",
              "Decision and basis": "Retain; exact field agreement.",
              "Preferred O007 form": "`fungsi sederhana`"
            },
            "line": 40,
            "raw_markdown": "| simple function | `fungsi sederhana` | — | Retain; exact field agreement. |"
          },
          {
            "cells": {
              "Attested/search variant(s)": "`fungsi langkah`",
              "Concept": "step function",
              "Decision and basis": "Prospective decision only; no current source propagation.",
              "Preferred O007 form": "`fungsi tangga` when first needed"
            },
            "line": 41,
            "raw_markdown": "| step function | `fungsi tangga` when first needed | `fungsi langkah` | Prospective decision only; no current source propagation. |"
          },
          {
            "cells": {
              "Attested/search variant(s)": "`terintegralkan Lebesgue`; `terintegral Lebesgue`",
              "Concept": "Lebesgue integrable",
              "Decision and basis": "Retain the explicit established form; record the attested shorter forms as variants.",
              "Preferred O007 form": "`terintegralkan secara Lebesgue`"
            },
            "line": 42,
            "raw_markdown": "| Lebesgue integrable | `terintegralkan secara Lebesgue` | `terintegralkan Lebesgue`; `terintegral Lebesgue` | Retain the explicit established form; record the attested shorter forms as variants. |"
          },
          {
            "cells": {
              "Attested/search variant(s)": "`tak-negatif`",
              "Concept": "nonnegative",
              "Decision and basis": "Refine to the repeatedly attested field form. Propagated through the 29 exact reader-facing matches in S112, S114, S122, S123, and S131.",
              "Preferred O007 form": "`nonnegatif`"
            },
            "line": 43,
            "raw_markdown": "| nonnegative | `nonnegatif` | `tak-negatif` | Refine to the repeatedly attested field form. Propagated through the 29 exact reader-facing matches in S112, S114, S122, S123, and S131. |"
          },
          {
            "cells": {
              "Attested/search variant(s)": "`ukuran bayangan`",
              "Concept": "image measure",
              "Decision and basis": "Normalize the same mathematical concept to the S123 wording for corpus consistency. This is an internal consistency decision, not an external-witness claim.",
              "Preferred O007 form": "`ukuran citra`"
            },
            "line": 44,
            "raw_markdown": "| image measure | `ukuran citra` | `ukuran bayangan` | Normalize the same mathematical concept to the S123 wording for corpus consistency. This is an internal consistency decision, not an external-witness claim. |"
          }
        ]
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        "heading": "Chapter 21 terminology decisions",
        "rows": [
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            "cells": {
              "Concept or construction": "sigma-algebra in reader prose",
              "Decision and boundary": "Use the spelled-out Indonesian compound in running prose. This is an intentional, lossless topology/localization exception: a source math atom `\\sigma` used only as the lexical prefix in “sigma-algebra” may become text, while symbolic variables such as `\\Sigma`, formulas, identifiers, and actual mathematical operators remain unchanged. Keep `aljabar-σ` as a search/display variant.",
              "Preferred O007 form": "`aljabar-sigma`"
            },
            "line": 62,
            "raw_markdown": "| sigma-algebra in reader prose | `aljabar-sigma` | Use the spelled-out Indonesian compound in running prose. This is an intentional, lossless topology/localization exception: a source math atom `\\sigma` used only as the lexical prefix in “sigma-algebra” may become text, while symbolic variables such as `\\Sigma`, formulas, identifiers, and actual mathematical operators remain unchanged. Keep `aljabar-σ` as a search/display variant. |"
          },
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            "cells": {
              "Concept or construction": "sigma-finite",
              "Decision and boundary": "Retain the conventional symbolic technical compound. Do not apply the `aljabar-sigma` prose exception to this established adjective.",
              "Preferred O007 form": "`$\\sigma$-hingga`"
            },
            "line": 63,
            "raw_markdown": "| sigma-finite | `$\\sigma$-hingga` | Retain the conventional symbolic technical compound. Do not apply the `aljabar-sigma` prose exception to this established adjective. |"
          },
          {
            "cells": {
              "Concept or construction": "semi-finite",
              "Decision and boundary": "Use one closed compound throughout Chapter 21; do not alternate with a literal “berhingga-sebagian” rendering.",
              "Preferred O007 form": "`semihingga`"
            },
            "line": 64,
            "raw_markdown": "| semi-finite | `semihingga` | Use one closed compound throughout Chapter 21; do not alternate with a literal “berhingga-sebagian” rendering. |"
          },
          {
            "cells": {
              "Concept or construction": "localizable",
              "Decision and boundary": "Use for Fremlin’s *localizable*.",
              "Preferred O007 form": "`dapat dilokalkan`"
            },
            "line": 65,
            "raw_markdown": "| localizable | `dapat dilokalkan` | Use for Fremlin’s *localizable*. |"
          },
          {
            "cells": {
              "Concept or construction": "strictly localizable",
              "Decision and boundary": "Preserve the strictness modifier explicitly and consistently.",
              "Preferred O007 form": "`dapat dilokalkan secara ketat`"
            },
            "line": 66,
            "raw_markdown": "| strictly localizable | `dapat dilokalkan secara ketat` | Preserve the strictness modifier explicitly and consistently. |"
          },
          {
            "cells": {
              "Concept or construction": "locally determined",
              "Decision and boundary": "Reserve for the property of a measure space or measure. For *locally determined negligible sets*, use `himpunan terabaikan yang ditentukan secara lokal`; do not collapse the two concepts.",
              "Preferred O007 form": "`ditentukan secara lokal`"
            },
            "line": 67,
            "raw_markdown": "| locally determined | `ditentukan secara lokal` | Reserve for the property of a measure space or measure. For *locally determined negligible sets*, use `himpunan terabaikan yang ditentukan secara lokal`; do not collapse the two concepts. |"
          },
          {
            "cells": {
              "Concept or construction": "measure-qualified predicates",
              "Decision and boundary": "Put the qualifier after the predicate in natural Indonesian and retain the governing measure explicitly where needed for contrast. Avoid English-order calques such as “μ-terukur”. Unqualified forms remain appropriate where the measure is unambiguous.",
              "Preferred O007 form": "`terukur terhadap μ`; `terabaikan terhadap μ`; `koterabaikan terhadap μ`; `dapat diintegralkan terhadap μ`"
            },
            "line": 68,
            "raw_markdown": "| measure-qualified predicates | `terukur terhadap μ`; `terabaikan terhadap μ`; `koterabaikan terhadap μ`; `dapat diintegralkan terhadap μ` | Put the qualifier after the predicate in natural Indonesian and retain the governing measure explicitly where needed for contrast. Avoid English-order calques such as “μ-terukur”. Unqualified forms remain appropriate where the measure is unambiguous. |"
          },
          {
            "cells": {
              "Concept or construction": "almost everywhere",
              "Decision and boundary": "Expand the abbreviation `h.d.` in reader-facing prose. Keep `h.d.` only as a lookup variant, not as the primary running form.",
              "Preferred O007 form": "`hampir di mana-mana`"
            },
            "line": 69,
            "raw_markdown": "| almost everywhere | `hampir di mana-mana` | Expand the abbreviation `h.d.` in reader-facing prose. Keep `h.d.` only as a lookup variant, not as the primary running form. |"
          },
          {
            "cells": {
              "Concept or construction": "purely atomic",
              "Decision and boundary": "Use consistently for *purely atomic*.",
              "Preferred O007 form": "`atomik murni`"
            },
            "line": 70,
            "raw_markdown": "| purely atomic | `atomik murni` | Use consistently for *purely atomic*. |"
          },
          {
            "cells": {
              "Concept or construction": "essential supremum",
              "Decision and boundary": "Preserve the standard mathematical noun phrase and its order-theoretic meaning.",
              "Preferred O007 form": "`supremum esensial`"
            },
            "line": 71,
            "raw_markdown": "| essential supremum | `supremum esensial` | Preserve the standard mathematical noun phrase and its order-theoretic meaning. |"
          },
          {
            "cells": {
              "Concept or construction": "principle of exhaustion",
              "Decision and boundary": "Use as the section title and Chapter 21 term for *principle of exhaustion*.",
              "Preferred O007 form": "`prinsip penghabisan`"
            },
            "line": 72,
            "raw_markdown": "| principle of exhaustion | `prinsip penghabisan` | Use as the section title and Chapter 21 term for *principle of exhaustion*. |"
          }
        ]
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          "Decision and boundary"
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        "heading": "Chapter 23 terminology decisions",
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              "Concept or construction": "finitely additive functional",
              "Decision and boundary": "Preserve *functional* as `fungsional`, not `fungsi`, because the object acts on a family of sets. Keep the finite-additivity qualifier after the noun.",
              "Preferred O007 form": "`fungsional aditif hingga`"
            },
            "line": 89,
            "raw_markdown": "| finitely additive functional | `fungsional aditif hingga` | Preserve *functional* as `fungsional`, not `fungsi`, because the object acts on a family of sets. Keep the finite-additivity qualifier after the noun. |"
          },
          {
            "cells": {
              "Concept or construction": "countably additive functional",
              "Decision and boundary": "Use in parallel with `fungsional aditif hingga`; do not substitute the measure-specific noun `ukuran` before the measure construction is actually introduced.",
              "Preferred O007 form": "`fungsional aditif terhitung`"
            },
            "line": 90,
            "raw_markdown": "| countably additive functional | `fungsional aditif terhitung` | Use in parallel with `fungsional aditif hingga`; do not substitute the measure-specific noun `ukuran` before the measure construction is actually introduced. |"
          },
          {
            "cells": {
              "Concept or construction": "signed measure / signed functional",
              "Decision and boundary": "Use `bertanda` for the signed scalar-valued objects. Do not use `ukuran berarah`, which suggests a different geometric concept.",
              "Preferred O007 form": "`ukuran bertanda`; `fungsional bertanda`"
            },
            "line": 91,
            "raw_markdown": "| signed measure / signed functional | `ukuran bertanda`; `fungsional bertanda` | Use `bertanda` for the signed scalar-valued objects. Do not use `ukuran berarah`, which suggests a different geometric concept. |"
          },
          {
            "cells": {
              "Concept or construction": "Hahn decomposition",
              "Decision and boundary": "Retain the proper name and use the established mathematical noun `dekomposisi`.",
              "Preferred O007 form": "`dekomposisi Hahn`"
            },
            "line": 92,
            "raw_markdown": "| Hahn decomposition | `dekomposisi Hahn` | Retain the proper name and use the established mathematical noun `dekomposisi`. |"
          },
          {
            "cells": {
              "Concept or construction": "Jordan decomposition",
              "Decision and boundary": "Retain the proper name and keep it distinct from the Hahn set decomposition.",
              "Preferred O007 form": "`dekomposisi Jordan`"
            },
            "line": 93,
            "raw_markdown": "| Jordan decomposition | `dekomposisi Jordan` | Retain the proper name and keep it distinct from the Hahn set decomposition. |"
          },
          {
            "cells": {
              "Concept or construction": "Radon--Nikodým theorem / derivative",
              "Decision and boundary": "Preserve the named theorem and use `turunan` for the density/derivative object. TeX source retains Fremlin’s accent encoding.",
              "Preferred O007 form": "`teorema Radon--Nikodým`; `turunan Radon--Nikodým`"
            },
            "line": 94,
            "raw_markdown": "| Radon--Nikodým theorem / derivative | `teorema Radon--Nikodým`; `turunan Radon--Nikodým` | Preserve the named theorem and use `turunan` for the density/derivative object. TeX source retains Fremlin’s accent encoding. |"
          },
          {
            "cells": {
              "Concept or construction": "absolute continuity",
              "Decision and boundary": "Always retain the reference measure after `terhadap`; do not collapse the relation into an unqualified adjective when two measures are in play.",
              "Preferred O007 form": "`kontinu mutlak terhadap`"
            },
            "line": 95,
            "raw_markdown": "| absolute continuity | `kontinu mutlak terhadap` | Always retain the reference measure after `terhadap`; do not collapse the relation into an unqualified adjective when two measures are in play. |"
          },
          {
            "cells": {
              "Concept or construction": "conditional expectation",
              "Decision and boundary": "Use the standard probability phrase and retain the conditioning algebra or subalgebra explicitly.",
              "Preferred O007 form": "`ekspektasi bersyarat`"
            },
            "line": 96,
            "raw_markdown": "| conditional expectation | `ekspektasi bersyarat` | Use the standard probability phrase and retain the conditioning algebra or subalgebra explicitly. |"
          },
          {
            "cells": {
              "Concept or construction": "convex function",
              "Decision and boundary": "Use `konveks`, with `konveks-tengah` reserved for *mid-convex*.",
              "Preferred O007 form": "`fungsi konveks`"
            },
            "line": 97,
            "raw_markdown": "| convex function | `fungsi konveks` | Use `konveks`, with `konveks-tengah` reserved for *mid-convex*. |"
          },
          {
            "cells": {
              "Concept or construction": "image measure",
              "Decision and boundary": "Reaffirm the corpus-wide decision above for every occurrence in 234; the inverse-image construction remains visible in formulas.",
              "Preferred O007 form": "`ukuran citra`"
            },
            "line": 98,
            "raw_markdown": "| image measure | `ukuran citra` | Reaffirm the corpus-wide decision above for every occurrence in 234; the inverse-image construction remains visible in formulas. |"
          },
          {
            "cells": {
              "Concept or construction": "inverse-measure-preserving function (`\\imp`)",
              "Decision and boundary": "The source macro is retained in every unit for structural replay. `source/id-ID/id-overrides.tex` supplies this Indonesian reader expansion; the English macro expansion must never leak into the Indonesian PDF.",
              "Preferred O007 form": "`fungsi pelestari ukuran melalui prapeta`"
            },
            "line": 99,
            "raw_markdown": "| inverse-measure-preserving function (`\\imp`) | `fungsi pelestari ukuran melalui prapeta` | The source macro is retained in every unit for structural replay. `source/id-ID/id-overrides.tex` supplies this Indonesian reader expansion; the English macro expansion must never leak into the Indonesian PDF. |"
          },
          {
            "cells": {
              "Concept or construction": "indefinite-integral measure",
              "Decision and boundary": "Preserve the relation to an underlying measure with `terhadap`; do not shorten it to `ukuran integral` because that loses Fremlin’s construction.",
              "Preferred O007 form": "`ukuran integral tak tentu`"
            },
            "line": 100,
            "raw_markdown": "| indefinite-integral measure | `ukuran integral tak tentu` | Preserve the relation to an underlying measure with `terhadap`; do not shorten it to `ukuran integral` because that loses Fremlin’s construction. |"
          },
          {
            "cells": {
              "Concept or construction": "upper / lower integral",
              "Decision and boundary": "Preserve the order relation and the overline/underline formulas exactly; the prose terms name those formula surfaces.",
              "Preferred O007 form": "`integral atas`; `integral bawah`"
            },
            "line": 101,
            "raw_markdown": "| upper / lower integral | `integral atas`; `integral bawah` | Preserve the order relation and the overline/underline formulas exactly; the prose terms name those formula surfaces. |"
          },
          {
            "cells": {
              "Concept or construction": "preimage / pullback measure",
              "Decision and boundary": "Use `prapeta` for inverse images of sets and `tarik-balik` for the induced measure construction.",
              "Preferred O007 form": "`prapeta`; `ukuran tarik-balik`"
            },
            "line": 102,
            "raw_markdown": "| preimage / pullback measure | `prapeta`; `ukuran tarik-balik` | Use `prapeta` for inverse images of sets and `tarik-balik` for the induced measure construction. |"
          }
        ]
      },
      {
        "headers": [
          "Concept or construction",
          "Preferred O007 form",
          "Decision and boundary"
        ],
        "heading": "Chapter 24 terminology decisions",
        "rows": [
          {
            "cells": {
              "Concept or construction": "function space",
              "Decision and boundary": "Use as the chapter title and generic name; retain the specific \\(L^p\\), \\(L^0\\), and quotient-space notation in formulas.",
              "Preferred O007 form": "ruang fungsi"
            },
            "line": 116,
            "raw_markdown": "| function space | ruang fungsi | Use as the chapter title and generic name; retain the specific \\(L^p\\), \\(L^0\\), and quotient-space notation in formulas. |"
          },
          {
            "cells": {
              "Concept or construction": "normed space",
              "Decision and boundary": "Prefer the established Indonesian mathematical phrase over a literal calque of normed linear space when linearity is already explicit.",
              "Preferred O007 form": "ruang bernorma"
            },
            "line": 117,
            "raw_markdown": "| normed space | ruang bernorma | Prefer the established Indonesian mathematical phrase over a literal calque of normed linear space when linearity is already explicit. |"
          },
          {
            "cells": {
              "Concept or construction": "convergence in measure",
              "Decision and boundary": "Use consistently for the topology and convergence relation generated by Fremlin's pseudometrics.",
              "Preferred O007 form": "konvergensi dalam ukuran"
            },
            "line": 118,
            "raw_markdown": "| convergence in measure | konvergensi dalam ukuran | Use consistently for the topology and convergence relation generated by Fremlin's pseudometrics. |"
          },
          {
            "cells": {
              "Concept or construction": "semi-finite / localizable",
              "Decision and boundary": "Keep the measure-space properties distinct and use the same forms as Chapter 21.",
              "Preferred O007 form": "semihingga; dapat dilokalkan"
            },
            "line": 119,
            "raw_markdown": "| semi-finite / localizable | semihingga; dapat dilokalkan | Keep the measure-space properties distinct and use the same forms as Chapter 21. |"
          },
          {
            "cells": {
              "Concept or construction": "non-negative",
              "Decision and boundary": "Use the closed form throughout running prose and definitions.",
              "Preferred O007 form": "nonnegatif"
            },
            "line": 120,
            "raw_markdown": "| non-negative | nonnegatif | Use the closed form throughout running prose and definitions. |"
          },
          {
            "cells": {
              "Concept or construction": "essentially bounded / essential supremum",
              "Decision and boundary": "Use the adverbial form for the predicate and the established noun phrase for the order bound.",
              "Preferred O007 form": "terbatas secara esensial; supremum esensial"
            },
            "line": 121,
            "raw_markdown": "| essentially bounded / essential supremum | terbatas secara esensial; supremum esensial | Use the adverbial form for the predicate and the established noun phrase for the order bound. |"
          },
          {
            "cells": {
              "Concept or construction": "order unit",
              "Decision and boundary": "Preserve the order-theoretic meaning; do not translate it as an algebraic identity element.",
              "Preferred O007 form": "satuan urutan"
            },
            "line": 122,
            "raw_markdown": "| order unit | satuan urutan | Preserve the order-theoretic meaning; do not translate it as an algebraic identity element. |"
          },
          {
            "cells": {
              "Concept or construction": "uniformly convex",
              "Decision and boundary": "Use for Banach-space geometry; keep it distinct from uniform integrability.",
              "Preferred O007 form": "konveks seragam"
            },
            "line": 123,
            "raw_markdown": "| uniformly convex | konveks seragam | Use for Banach-space geometry; keep it distinct from uniform integrability. |"
          },
          {
            "cells": {
              "Concept or construction": "square-integrable",
              "Decision and boundary": "The hyphen keeps the square qualifier attached to the integrability predicate without altering \\(L^2\\) notation.",
              "Preferred O007 form": "terintegralkan-kuadrat"
            },
            "line": 124,
            "raw_markdown": "| square-integrable | terintegralkan-kuadrat | The hyphen keeps the square qualifier attached to the integrability predicate without altering \\(L^2\\) notation. |"
          },
          {
            "cells": {
              "Concept or construction": "uniform integrability / uniformly integrable",
              "Decision and boundary": "Use a noun for the property and an adverbial predicate for families or functions.",
              "Preferred O007 form": "keterintegralan seragam; terintegralkan secara seragam"
            },
            "line": 125,
            "raw_markdown": "| uniform integrability / uniformly integrable | keterintegralan seragam; terintegralkan secara seragam | Use a noun for the property and an adverbial predicate for families or functions. |"
          },
          {
            "cells": {
              "Concept or construction": "weak compactness / weakly compact",
              "Decision and boundary": "Keep the noun and predicate forms parallel while retaining the topological sense of weak.",
              "Preferred O007 form": "kekompakan lemah; kompak secara lemah"
            },
            "line": 126,
            "raw_markdown": "| weak compactness / weakly compact | kekompakan lemah; kompak secara lemah | Keep the noun and predicate forms parallel while retaining the topological sense of weak. |"
          },
          {
            "cells": {
              "Concept or construction": "countably additive",
              "Decision and boundary": "Reaffirm the Chapter 23 form for functionals and measures; do not use a literal “tak hingga terhitung”.",
              "Preferred O007 form": "aditif terhitung"
            },
            "line": 127,
            "raw_markdown": "| countably additive | aditif terhitung | Reaffirm the Chapter 23 form for functionals and measures; do not use a literal “tak hingga terhitung”. |"
          },
          {
            "cells": {
              "Concept or construction": "order-continuous",
              "Decision and boundary": "Make the governing order relation explicit and avoid suggesting ordinary metric continuity.",
              "Preferred O007 form": "kontinu menurut urutan"
            },
            "line": 128,
            "raw_markdown": "| order-continuous | kontinu menurut urutan | Make the governing order relation explicit and avoid suggesting ordinary metric continuity. |"
          },
          {
            "cells": {
              "Concept or construction": "conditional-expectation operator",
              "Decision and boundary": "Preserve the standard probability term and name the operator explicitly where the source discusses its mapping properties.",
              "Preferred O007 form": "operator ekspektasi bersyarat"
            },
            "line": 129,
            "raw_markdown": "| conditional-expectation operator | operator ekspektasi bersyarat | Preserve the standard probability term and name the operator explicitly where the source discusses its mapping properties. |"
          },
          {
            "cells": {
              "Concept or construction": "Archimedean / Archimedean Riesz space",
              "Decision and boundary": "Use the adjectival property in prose and the compact named-space form as a noun phrase.",
              "Preferred O007 form": "bersifat Archimedes; ruang Riesz Archimedes"
            },
            "line": 130,
            "raw_markdown": "| Archimedean / Archimedean Riesz space | bersifat Archimedes; ruang Riesz Archimedes | Use the adjectival property in prose and the compact named-space form as a noun phrase. |"
          }
        ]
      },
      {
        "headers": [
          "Concept or construction",
          "Preferred O007 form",
          "Decision and boundary"
        ],
        "heading": "Chapter 25 — Product measures (2026-08-26)",
        "rows": [
          {
            "cells": {
              "Concept or construction": "Fubini's theorem / Tonelli's theorem",
              "Decision and boundary": "Retain the proper names and distinguish the integrability and nonnegative-function forms in surrounding prose.",
              "Preferred O007 form": "`teorema Fubini`; `teorema Tonelli`"
            },
            "line": 159,
            "raw_markdown": "| Fubini's theorem / Tonelli's theorem | `teorema Fubini`; `teorema Tonelli` | Retain the proper names and distinguish the integrability and nonnegative-function forms in surrounding prose. |"
          },
          {
            "cells": {
              "Concept or construction": "repeated integral",
              "Decision and boundary": "Use for the iterated integral as an object; retain `integrasi berulang` for the process or topic.",
              "Preferred O007 form": "`integral berulang`"
            },
            "line": 160,
            "raw_markdown": "| repeated integral | `integral berulang` | Use for the iterated integral as an object; retain `integrasi berulang` for the process or topic. |"
          },
          {
            "cells": {
              "Concept or construction": "ordinate set / section",
              "Decision and boundary": "Use `himpunan ordinat` for Fremlin's named construction in 252N and `penampang` for horizontal/vertical set sections elsewhere.",
              "Preferred O007 form": "`himpunan ordinat`; `penampang`"
            },
            "line": 161,
            "raw_markdown": "| ordinate set / section | `himpunan ordinat`; `penampang` | Use `himpunan ordinat` for Fremlin's named construction in 252N and `penampang` for horizontal/vertical set sections elsewhere. |"
          },
          {
            "cells": {
              "Concept or construction": "counting measure",
              "Decision and boundary": "Prefer the established Indonesian mathematical term; do not use the literal but less conventional `ukuran hitung`.",
              "Preferred O007 form": "`ukuran cacah`"
            },
            "line": 162,
            "raw_markdown": "| counting measure | `ukuran cacah` | Prefer the established Indonesian mathematical term; do not use the literal but less conventional `ukuran hitung`. |"
          },
          {
            "cells": {
              "Concept or construction": "volume of a ball",
              "Decision and boundary": "Use as the result title and preserve the dimension and normalization formulas exactly.",
              "Preferred O007 form": "`volume bola`"
            },
            "line": 163,
            "raw_markdown": "| volume of a ball | `volume bola` | Use as the result title and preserve the dimension and normalization formulas exactly. |"
          },
          {
            "cells": {
              "Concept or construction": "complex-valued function",
              "Decision and boundary": "Keep the codomain adjective attached to `bernilai`; do not suggest that the function itself is structurally complicated.",
              "Preferred O007 form": "`fungsi bernilai kompleks`"
            },
            "line": 164,
            "raw_markdown": "| complex-valued function | `fungsi bernilai kompleks` | Keep the codomain adjective attached to `bernilai`; do not suggest that the function itself is structurally complicated. |"
          },
          {
            "cells": {
              "Concept or construction": "measure-qualified almost-everywhere quantifier",
              "Decision and boundary": "Attach the governing measure directly to the almost-everywhere quantifier. This prevents ambiguous readings such as “relative to μ, for almost every x,” especially inside nested Fubini quantifiers.",
              "Preferred O007 form": "`untuk $\\mu$-hampir setiap $x$`"
            },
            "line": 165,
            "raw_markdown": "| measure-qualified almost-everywhere quantifier | `untuk $\\mu$-hampir setiap $x$` | Attach the governing measure directly to the almost-everywhere quantifier. This prevents ambiguous readings such as “relative to μ, for almost every x,” especially inside nested Fubini quantifiers. |"
          },
          {
            "cells": {
              "Concept or construction": "lambda-simple function",
              "Decision and boundary": "Keep the measure qualifier attached to the simple-function construction and place `nonnegatif` after it when required.",
              "Preferred O007 form": "`fungsi sederhana-$\\lambda$`"
            },
            "line": 166,
            "raw_markdown": "| lambda-simple function | `fungsi sederhana-$\\lambda$` | Keep the measure qualifier attached to the simple-function construction and place `nonnegatif` after it when required. |"
          }
        ]
      },
      {
        "headers": [
          "Concept or construction",
          "Preferred O007 form",
          "Decision and boundary"
        ],
        "heading": "Chapter 25 — Product measures (2026-08-26)",
        "rows": [
          {
            "cells": {
              "Concept or construction": "tensor product",
              "Decision and boundary": "Use for the section title, the universal construction, and the operation represented by `\\otimes`; never substitute ordinary pointwise multiplication across distinct factor spaces.",
              "Preferred O007 form": "`produk tensor`"
            },
            "line": 179,
            "raw_markdown": "| tensor product | `produk tensor` | Use for the section title, the universal construction, and the operation represented by `\\otimes`; never substitute ordinary pointwise multiplication across distinct factor spaces. |"
          },
          {
            "cells": {
              "Concept or construction": "bilinear / multilinear operator",
              "Decision and boundary": "Keep linearity in each argument explicit and attach `terbatas` or `kontinu` only when the source does.",
              "Preferred O007 form": "`operator bilinear`; `operator multilinear`"
            },
            "line": 180,
            "raw_markdown": "| bilinear / multilinear operator | `operator bilinear`; `operator multilinear` | Keep linearity in each argument explicit and attach `terbatas` or `kontinu` only when the source does. |"
          },
          {
            "cells": {
              "Concept or construction": "Banach lattice",
              "Decision and boundary": "Continue the established corpus form `kisi`; reject `kekisi`, which is not the selected technical noun in this edition.",
              "Preferred O007 form": "`kisi Banach`"
            },
            "line": 181,
            "raw_markdown": "| Banach lattice | `kisi Banach` | Continue the established corpus form `kisi`; reject `kekisi`, which is not the selected technical noun in this edition. |"
          },
          {
            "cells": {
              "Concept or construction": "Bochner-integrable / Bochner integral",
              "Decision and boundary": "Use the adverbial predicate for functions and the noun phrase for the resulting vector-valued integral. Preserve its codomain rather than conflating it with a scalar iterated integral.",
              "Preferred O007 form": "`terintegralkan menurut Bochner`; `integral Bochner`"
            },
            "line": 182,
            "raw_markdown": "| Bochner-integrable / Bochner integral | `terintegralkan menurut Bochner`; `integral Bochner` | Use the adverbial predicate for functions and the noun phrase for the resulting vector-valued integral. Preserve its codomain rather than conflating it with a scalar iterated integral. |"
          },
          {
            "cells": {
              "Concept or construction": "primitive product measure",
              "Decision and boundary": "Keep it distinct from the c.l.d. product measure and preserve the source measure symbol in every statement.",
              "Preferred O007 form": "`ukuran produk primitif`"
            },
            "line": 183,
            "raw_markdown": "| primitive product measure | `ukuran produk primitif` | Keep it distinct from the c.l.d. product measure and preserve the source measure symbol in every statement. |"
          },
          {
            "cells": {
              "Concept or construction": "universal mapping theorem / property",
              "Decision and boundary": "Use the theorem form for 253F and the property form in explanatory notes; retain the exact factorization and norm clauses.",
              "Preferred O007 form": "`teorema pemetaan universal`; `sifat pemetaan universal`"
            },
            "line": 184,
            "raw_markdown": "| universal mapping theorem / property | `teorema pemetaan universal`; `sifat pemetaan universal` | Use the theorem form for 253F and the property form in explanatory notes; retain the exact factorization and norm clauses. |"
          },
          {
            "cells": {
              "Concept or construction": "conditional expectation",
              "Decision and boundary": "Reaffirm the existing probability term and preserve the conditioning subalgebra and operator domains.",
              "Preferred O007 form": "`ekspektasi bersyarat`"
            },
            "line": 185,
            "raw_markdown": "| conditional expectation | `ekspektasi bersyarat` | Reaffirm the existing probability term and preserve the conditioning subalgebra and operator domains. |"
          }
        ]
      },
      {
        "headers": [
          "Concept or construction",
          "Preferred O007 form",
          "Decision and boundary"
        ],
        "heading": "Section 255 — Convolutions of functions (2026-08-28)",
        "rows": [
          {
            "cells": {
              "Concept or construction": "convolution / convolution of functions",
              "Decision and boundary": "Retain the standard Indonesian technical noun and use the second form for the section title. Preserve the source operation `f*g` and its domain clauses exactly.",
              "Preferred O007 form": "`konvolusi`; `konvolusi fungsi`"
            },
            "line": 200,
            "raw_markdown": "| convolution / convolution of functions | `konvolusi`; `konvolusi fungsi` | Retain the standard Indonesian technical noun and use the second form for the section title. Preserve the source operation `f*g` and its domain clauses exactly. |"
          },
          {
            "cells": {
              "Concept or construction": "circle group",
              "Decision and boundary": "Use for (S^1) as a topological group; do not confuse it with a geometric disk or with the interval model used to represent it.",
              "Preferred O007 form": "`grup lingkaran`"
            },
            "line": 201,
            "raw_markdown": "| circle group | `grup lingkaran` | Use for (S^1) as a topological group; do not confuse it with a geometric disk or with the interval model used to represent it. |"
          },
          {
            "cells": {
              "Concept or construction": "translation/reflection invariance",
              "Decision and boundary": "Keep the transformation named explicitly and distinguish measure invariance from pointwise invariance of a function.",
              "Preferred O007 form": "`invariansi terhadap translasi`; `invariansi terhadap refleksi`"
            },
            "line": 202,
            "raw_markdown": "| translation/reflection invariance | `invariansi terhadap translasi`; `invariansi terhadap refleksi` | Keep the transformation named explicitly and distinguish measure invariance from pointwise invariance of a function. |"
          },
          {
            "cells": {
              "Concept or construction": "Lebesgue set",
              "Decision and boundary": "Use for the set of Lebesgue points of an integrable function in the approximation exercises.",
              "Preferred O007 form": "`himpunan Lebesgue`"
            },
            "line": 203,
            "raw_markdown": "| Lebesgue set | `himpunan Lebesgue` | Use for the set of Lebesgue points of an integrable function in the approximation exercises. |"
          },
          {
            "cells": {
              "Concept or construction": "non-decreasing / non-increasing",
              "Decision and boundary": "Use unhyphenated standard orthography. The right-hand approximate-identity condition in 255Ya is a separately ledgered source correction from non-decreasing to non-increasing.",
              "Preferred O007 form": "`tak menurun`; `tak menaik`"
            },
            "line": 204,
            "raw_markdown": "| non-decreasing / non-increasing | `tak menurun`; `tak menaik` | Use unhyphenated standard orthography. The right-hand approximate-identity condition in 255Ya is a separately ledgered source correction from non-decreasing to non-increasing. |"
          },
          {
            "cells": {
              "Concept or construction": "corollary",
              "Decision and boundary": "Continue the established compact heading form. Reject `korolari`, which is not the selected Indonesian form in this edition.",
              "Preferred O007 form": "`akibat`"
            },
            "line": 205,
            "raw_markdown": "| corollary | `akibat` | Continue the established compact heading form. Reject `korolari`, which is not the selected Indonesian form in this edition. |"
          }
        ]
      },
      {
        "headers": [
          "Concept or construction",
          "Preferred O007 form",
          "Decision and boundary"
        ],
        "heading": "Section 256 — Radon measures on $\\BbbR^r$ (2026-08-28)",
        "rows": [
          {
            "cells": {
              "Concept or construction": "Radon measure / Euclidean space",
              "Decision and boundary": "Retain the proper name and use the standard Indonesian geometric noun.",
              "Preferred O007 form": "`ukuran Radon`; `ruang Euklides`"
            },
            "line": 218,
            "raw_markdown": "| Radon measure / Euclidean space | `ukuran Radon`; `ruang Euklides` | Retain the proper name and use the standard Indonesian geometric noun. |"
          },
          {
            "cells": {
              "Concept or construction": "topological measure / locally finite",
              "Decision and boundary": "Keep the topology and finiteness properties distinct; do not shorten the second form to an ambiguous bare `lokal`.",
              "Preferred O007 form": "`ukuran topologis`; `berhingga secara lokal`"
            },
            "line": 219,
            "raw_markdown": "| topological measure / locally finite | `ukuran topologis`; `berhingga secara lokal` | Keep the topology and finiteness properties distinct; do not shorten the second form to an ambiguous bare `lokal`. |"
          },
          {
            "cells": {
              "Concept or construction": "inner regular / inner regularity",
              "Decision and boundary": "Attach the governing compact, closed, or K$_{\\sigma}$ class explicitly. Use `regularitas luar` for the complementary open-set notion.",
              "Preferred O007 form": "`regular dalam`; `regularitas dalam`"
            },
            "line": 220,
            "raw_markdown": "| inner regular / inner regularity | `regular dalam`; `regularitas dalam` | Attach the governing compact, closed, or K$_{\\sigma}$ class explicitly. Use `regularitas luar` for the complementary open-set notion. |"
          },
          {
            "cells": {
              "Concept or construction": "Borel sigma-algebra / K-sigma set",
              "Decision and boundary": "Continue the locale-first lexical sigma convention without altering formula-level sigma symbols.",
              "Preferred O007 form": "`aljabar-sigma Borel`; `himpunan K$_{\\sigma}$`"
            },
            "line": 221,
            "raw_markdown": "| Borel sigma-algebra / K-sigma set | `aljabar-sigma Borel`; `himpunan K$_{\\sigma}$` | Continue the locale-first lexical sigma convention without altering formula-level sigma symbols. |"
          },
          {
            "cells": {
              "Concept or construction": "Lusin's theorem / lower semi-continuous",
              "Decision and boundary": "Retain the named theorem and the established one-word technical compound.",
              "Preferred O007 form": "`teorema Lusin`; `semikontinu bawah`"
            },
            "line": 222,
            "raw_markdown": "| Lusin's theorem / lower semi-continuous | `teorema Lusin`; `semikontinu bawah` | Retain the named theorem and the established one-word technical compound. |"
          },
          {
            "cells": {
              "Concept or construction": "locally integrable / indefinite-integral measure",
              "Decision and boundary": "Continue the admitted Chapter 23 wording and avoid a new hyphenated compound.",
              "Preferred O007 form": "`terintegralkan secara lokal`; `ukuran integral tak tentu`"
            },
            "line": 223,
            "raw_markdown": "| locally integrable / indefinite-integral measure | `terintegralkan secara lokal`; `ukuran integral tak tentu` | Continue the admitted Chapter 23 wording and avoid a new hyphenated compound. |"
          },
          {
            "cells": {
              "Concept or construction": "image measure / point-supported measure",
              "Decision and boundary": "Distinguish pushforward construction from support terminology.",
              "Preferred O007 form": "`ukuran citra`; `ukuran yang bertumpu pada titik`"
            },
            "line": 224,
            "raw_markdown": "| image measure / point-supported measure | `ukuran citra`; `ukuran yang bertumpu pada titik` | Distinguish pushforward construction from support terminology. |"
          },
          {
            "cells": {
              "Concept or construction": "bounded support / support / self-supporting",
              "Decision and boundary": "Reserve `dukungan` for the ordinary support. Render Fremlin's coined predicate as a full reader-first clause and retain its English key only in the backend.",
              "Preferred O007 form": "`dukungan terbatas`; `dukungan`; `menopang dirinya sendiri`"
            },
            "line": 225,
            "raw_markdown": "| bounded support / support / self-supporting | `dukungan terbatas`; `dukungan`; `menopang dirinya sendiri` | Reserve `dukungan` for the ordinary support. Render Fremlin's coined predicate as a full reader-first clause and retain its English key only in the backend. |"
          },
          {
            "cells": {
              "Concept or construction": "Radon--Nikodým derivative / Lebesgue--Stieltjes measure",
              "Decision and boundary": "Preserve both proper names and their exact mathematical roles.",
              "Preferred O007 form": "`turunan Radon--Nikodým`; `ukuran Lebesgue--Stieltjes`"
            },
            "line": 226,
            "raw_markdown": "| Radon--Nikodým derivative / Lebesgue--Stieltjes measure | `turunan Radon--Nikodým`; `ukuran Lebesgue--Stieltjes` | Preserve both proper names and their exact mathematical roles. |"
          },
          {
            "cells": {
              "Concept or construction": "upwards/downwards-directed",
              "Decision and boundary": "Continue the established order-theoretic forms.",
              "Preferred O007 form": "`terarah ke atas`; `terarah ke bawah`"
            },
            "line": 227,
            "raw_markdown": "| upwards/downwards-directed | `terarah ke atas`; `terarah ke bawah` | Continue the established order-theoretic forms. |"
          },
          {
            "cells": {
              "Concept or construction": "c.l.d. product measure / subspace measure",
              "Decision and boundary": "Expand c.l.d. at first use in the section, then retain `ukuran produk c.l.d.`.",
              "Preferred O007 form": "`ukuran produk lengkap yang ditentukan secara lokal (c.l.d.)`; `ukuran subruang`"
            },
            "line": 228,
            "raw_markdown": "| c.l.d. product measure / subspace measure | `ukuran produk lengkap yang ditentukan secara lokal (c.l.d.)`; `ukuran subruang` | Expand c.l.d. at first use in the section, then retain `ukuran produk c.l.d.`. |"
          }
        ]
      },
      {
        "headers": [
          "Concept or construction",
          "Preferred O007 form",
          "Decision and boundary"
        ],
        "heading": "Section 257 — Convolutions of measures (2026-08-28)",
        "rows": [
          {
            "cells": {
              "Concept or construction": "convolution of measures / convolution of functions",
              "Decision and boundary": "Distinguish the pushforward construction on measures from the integral construction on functions while retaining the shared operation symbol `*`.",
              "Preferred O007 form": "`konvolusi ukuran`; `konvolusi fungsi`"
            },
            "line": 242,
            "raw_markdown": "| convolution of measures / convolution of functions | `konvolusi ukuran`; `konvolusi fungsi` | Distinguish the pushforward construction on measures from the integral construction on functions while retaining the shared operation symbol `*`. |"
          },
          {
            "cells": {
              "Concept or construction": "totally finite Radon measure",
              "Decision and boundary": "Continue the Section 256 form and do not silently strengthen it to a probability measure.",
              "Preferred O007 form": "`ukuran Radon berhingga total`"
            },
            "line": 243,
            "raw_markdown": "| totally finite Radon measure | `ukuran Radon berhingga total` | Continue the Section 256 form and do not silently strengthen it to a probability measure. |"
          },
          {
            "cells": {
              "Concept or construction": "image measure / product measure",
              "Decision and boundary": "Preserve the distinct pushforward and product constructions in 257A.",
              "Preferred O007 form": "`ukuran citra`; `ukuran produk`"
            },
            "line": 244,
            "raw_markdown": "| image measure / product measure | `ukuran citra`; `ukuran produk` | Preserve the distinct pushforward and product constructions in 257A. |"
          },
          {
            "cells": {
              "Concept or construction": "characteristic function",
              "Decision and boundary": "Use for `\\chi E`; preserve the set argument and all integration variables.",
              "Preferred O007 form": "`fungsi karakteristik`"
            },
            "line": 245,
            "raw_markdown": "| characteristic function | `fungsi karakteristik` | Use for `\\chi E`; preserve the set argument and all integration variables. |"
          },
          {
            "cells": {
              "Concept or construction": "absolutely continuous with respect to",
              "Decision and boundary": "Keep both measures visible and preserve the direction of the relation.",
              "Preferred O007 form": "`kontinu mutlak terhadap`"
            },
            "line": 246,
            "raw_markdown": "| absolutely continuous with respect to | `kontinu mutlak terhadap` | Keep both measures visible and preserve the direction of the relation. |"
          },
          {
            "cells": {
              "Concept or construction": "Banach algebra / multiplicative identity",
              "Decision and boundary": "Use in 257Ya for the normed convolution algebra and its identity element.",
              "Preferred O007 form": "`aljabar Banach`; `identitas perkalian`"
            },
            "line": 247,
            "raw_markdown": "| Banach algebra / multiplicative identity | `aljabar Banach`; `identitas perkalian` | Use in 257Ya for the normed convolution algebra and its identity element. |"
          },
          {
            "cells": {
              "Concept or construction": "countably additive functional",
              "Decision and boundary": "Retain the functional viewpoint of the exercise rather than silently relabeling every element as a positive measure.",
              "Preferred O007 form": "`fungsional aditif terhitung`"
            },
            "line": 248,
            "raw_markdown": "| countably additive functional | `fungsional aditif terhitung` | Retain the functional viewpoint of the exercise rather than silently relabeling every element as a positive measure. |"
          },
          {
            "cells": {
              "Concept or construction": "locally compact Hausdorff abelian topological group",
              "Decision and boundary": "Preserve all four structural qualifiers in the final note.",
              "Preferred O007 form": "`grup topologis abelian Hausdorff yang kompak lokal`"
            },
            "line": 249,
            "raw_markdown": "| locally compact Hausdorff abelian topological group | `grup topologis abelian Hausdorff yang kompak lokal` | Preserve all four structural qualifiers in the final note. |"
          }
        ]
      },
      {
        "headers": [
          "Concept or construction",
          "Preferred O007 form",
          "Decision and boundary"
        ],
        "heading": "Section 265 — Hausdorff measure of surfaces (2026-08-29)",
        "rows": [
          {
            "cells": {
              "Concept or construction": "ball / unit ball",
              "Decision and boundary": "Reserve these forms for sets defined by a norm inequality, such as $B(x,r)$ or $\\{x:|x|\\le 1\\}$.",
              "Preferred O007 form": "`bola`; `bola satuan`"
            },
            "line": 261,
            "raw_markdown": "| ball / unit ball | `bola`; `bola satuan` | Reserve these forms for sets defined by a norm inequality, such as $B(x,r)$ or $\\{x:\\|x\\|\\le 1\\}$. |"
          },
          {
            "cells": {
              "Concept or construction": "sphere / unit sphere / $r$-sphere",
              "Decision and boundary": "Use the explicit surface phrase for norm-level sets such as $\\partial B(y,t)$ and $\\{x:|x|=1\\}$. This prevents the English distinction between *ball* and *sphere* from collapsing into the single word `bola`; it does not alter any formula or source identifier.",
              "Preferred O007 form": "`permukaan bola`; `permukaan bola satuan`; `permukaan bola berdimensi $r$`"
            },
            "line": 262,
            "raw_markdown": "| sphere / unit sphere / $r$-sphere | `permukaan bola`; `permukaan bola satuan`; `permukaan bola berdimensi $r$` | Use the explicit surface phrase for norm-level sets such as $\\partial B(y,t)$ and $\\{x:\\|x\\|=1\\}$. This prevents the English distinction between *ball* and *sphere* from collapsing into the single word `bola`; it does not alter any formula or source identifier. |"
          }
        ]
      },
      {
        "headers": [
          "Concept or construction",
          "Preferred O007 form",
          "Decision and boundary"
        ],
        "heading": "Chapter 27 — Probability theory (2026-08-29)",
        "rows": [
          {
            "cells": {
              "Concept or construction": "probability theory / probability space",
              "Decision and boundary": "Continue the admitted noun `probabilitas` and do not alternate the chapter title with `teori peluang`.",
              "Preferred O007 form": "`teori probabilitas`; `ruang probabilitas`"
            },
            "line": 275,
            "raw_markdown": "| probability theory / probability space | `teori probabilitas`; `ruang probabilitas` | Continue the admitted noun `probabilitas` and do not alternate the chapter title with `teori peluang`. |"
          },
          {
            "cells": {
              "Concept or construction": "distribution / joint distribution",
              "Decision and boundary": "Use for laws of random variables and tuples; preserve each image-measure formula defining the distribution.",
              "Preferred O007 form": "`distribusi`; `distribusi bersama`"
            },
            "line": 276,
            "raw_markdown": "| distribution / joint distribution | `distribusi`; `distribusi bersama` | Use for laws of random variables and tuples; preserve each image-measure formula defining the distribution. |"
          },
          {
            "cells": {
              "Concept or construction": "independent / independence",
              "Decision and boundary": "Use the compact established probability adjective and noun. Reserve `saling lepas` for disjoint sets, not independent events.",
              "Preferred O007 form": "`bebas`; `kebebasan`"
            },
            "line": 277,
            "raw_markdown": "| independent / independence | `bebas`; `kebebasan` | Use the compact established probability adjective and noun. Reserve `saling lepas` for disjoint sets, not independent events. |"
          },
          {
            "cells": {
              "Concept or construction": "random variable / expectation / variance",
              "Decision and boundary": "Apply consistently in prose while preserving `\\Expn`, `\\Var`, and all source formulas.",
              "Preferred O007 form": "`variabel acak`; `ekspektasi`; `varians`"
            },
            "line": 278,
            "raw_markdown": "| random variable / expectation / variance | `variabel acak`; `ekspektasi`; `varians` | Apply consistently in prose while preserving `\\Expn`, `\\Var`, and all source formulas. |"
          },
          {
            "cells": {
              "Concept or construction": "strong law of large numbers",
              "Decision and boundary": "Use this word order in titles and cross-references for all six forms in the chapter.",
              "Preferred O007 form": "`hukum kuat bilangan besar`"
            },
            "line": 279,
            "raw_markdown": "| strong law of large numbers | `hukum kuat bilangan besar` | Use this word order in titles and cross-references for all six forms in the chapter. |"
          },
          {
            "cells": {
              "Concept or construction": "central limit theorem",
              "Decision and boundary": "Use the conventional Indonesian statistical term; retain every normalization and limiting distribution exactly.",
              "Preferred O007 form": "`teorema limit pusat`"
            },
            "line": 280,
            "raw_markdown": "| central limit theorem | `teorema limit pusat` | Use the conventional Indonesian statistical term; retain every normalization and limiting distribution exactly. |"
          },
          {
            "cells": {
              "Concept or construction": "martingale / martingale difference sequence",
              "Decision and boundary": "Use the localized technical noun consistently. A sequence of martingale increments is a `barisan selisih`, not a generic sequence of numerical differences.",
              "Preferred O007 form": "`martingal`; `barisan selisih martingal`"
            },
            "line": 281,
            "raw_markdown": "| martingale / martingale difference sequence | `martingal`; `barisan selisih martingal` | Use the localized technical noun consistently. A sequence of martingale increments is a `barisan selisih`, not a generic sequence of numerical differences. |"
          },
          {
            "cells": {
              "Concept or construction": "conditional expectation / stopping time",
              "Decision and boundary": "Continue the Chapter 23 conditional-expectation term and use the standard stochastic-process term for stopping times.",
              "Preferred O007 form": "`ekspektasi bersyarat`; `waktu henti`"
            },
            "line": 282,
            "raw_markdown": "| conditional expectation / stopping time | `ekspektasi bersyarat`; `waktu henti` | Continue the Chapter 23 conditional-expectation term and use the standard stochastic-process term for stopping times. |"
          },
          {
            "cells": {
              "Concept or construction": "exchangeable / predictable sequence",
              "Decision and boundary": "The first names invariance under finite reindexing; the second names adaptation to past information. Do not collapse either into ordinary independence.",
              "Preferred O007 form": "`dapat dipertukarkan`; `barisan terprediksi`"
            },
            "line": 283,
            "raw_markdown": "| exchangeable / predictable sequence | `dapat dipertukarkan`; `barisan terprediksi` | The first names invariance under finite reindexing; the second names adaptation to past information. Do not collapse either into ordinary independence. |"
          },
          {
            "cells": {
              "Concept or construction": "almost everywhere / almost surely",
              "Decision and boundary": "Use the measure-theoretic form for a.e. statements and the probability idiom for events asserted with probability one.",
              "Preferred O007 form": "`hampir di mana-mana`; `hampir pasti`"
            },
            "line": 284,
            "raw_markdown": "| almost everywhere / almost surely | `hampir di mana-mana`; `hampir pasti` | Use the measure-theoretic form for a.e. statements and the probability idiom for events asserted with probability one. |"
          },
          {
            "cells": {
              "Concept or construction": "sigma-additive",
              "Decision and boundary": "Keep the symbolic prefix before the Indonesian adjective, matching the established mathematical compound; do not reverse it to `aditif-$\\sigma$`.",
              "Preferred O007 form": "`$\\sigma$-aditif`"
            },
            "line": 285,
            "raw_markdown": "| sigma-additive | `$\\sigma$-aditif` | Keep the symbolic prefix before the Indonesian adjective, matching the established mathematical compound; do not reverse it to `aditif-$\\sigma$`. |"
          }
        ]
      },
      {
        "headers": [
          "Concept or construction",
          "Preferred O007 form",
          "Decision and boundary"
        ],
        "heading": "Chapter 28 — Fourier analysis (2026-08-29)",
        "rows": [
          {
            "cells": {
              "Concept or construction": "Fourier analysis / Fourier series",
              "Decision and boundary": "Retain the proper name and distinguish the whole field from a particular trigonometric series.",
              "Preferred O007 form": "`analisis Fourier`; `deret Fourier`"
            },
            "line": 301,
            "raw_markdown": "| Fourier analysis / Fourier series | `analisis Fourier`; `deret Fourier` | Retain the proper name and distinguish the whole field from a particular trigonometric series. |"
          },
          {
            "cells": {
              "Concept or construction": "Fourier coefficient / Fourier sum",
              "Decision and boundary": "Use `jumlah` for partial/summability expressions and `koefisien` for the indexed scalar data.",
              "Preferred O007 form": "`koefisien Fourier`; `jumlah Fourier`"
            },
            "line": 302,
            "raw_markdown": "| Fourier coefficient / Fourier sum | `koefisien Fourier`; `jumlah Fourier` | Use `jumlah` for partial/summability expressions and `koefisien` for the indexed scalar data. |"
          },
          {
            "cells": {
              "Concept or construction": "Fourier transform / inverse Fourier transform",
              "Decision and boundary": "Continue the existing `mt02.tex` forms; preserve hats, checks, normalizations, and domains exactly.",
              "Preferred O007 form": "`transformasi Fourier`; `transformasi Fourier invers`"
            },
            "line": 303,
            "raw_markdown": "| Fourier transform / inverse Fourier transform | `transformasi Fourier`; `transformasi Fourier invers` | Continue the existing `mt02.tex` forms; preserve hats, checks, normalizations, and domains exactly. |"
          },
          {
            "cells": {
              "Concept or construction": "Stone--Weierstrass theorem",
              "Decision and boundary": "Use the proper-name compound in headings and prose; use `hampiran`/`menghampiri` for the associated approximation statements.",
              "Preferred O007 form": "`teorema Stone--Weierstrass`"
            },
            "line": 304,
            "raw_markdown": "| Stone--Weierstrass theorem | `teorema Stone--Weierstrass` | Use the proper-name compound in headings and prose; use `hampiran`/`menghampiri` for the associated approximation statements. |"
          },
          {
            "cells": {
              "Concept or construction": "Weyl equidistribution theorem",
              "Decision and boundary": "Continue the contents wording and avoid alternating with the less established literal phrase `ketersebaran seragam`.",
              "Preferred O007 form": "`teorema ekuidistribusi Weyl`"
            },
            "line": 305,
            "raw_markdown": "| Weyl equidistribution theorem | `teorema ekuidistribusi Weyl` | Continue the contents wording and avoid alternating with the less established literal phrase `ketersebaran seragam`. |"
          },
          {
            "cells": {
              "Concept or construction": "Dirichlet / Fejér kernel",
              "Decision and boundary": "Retain both proper names and the conventional technical noun `kernel`.",
              "Preferred O007 form": "`kernel Dirichlet`; `kernel Fejér`"
            },
            "line": 306,
            "raw_markdown": "| Dirichlet / Fejér kernel | `kernel Dirichlet`; `kernel Fejér` | Retain both proper names and the conventional technical noun `kernel`. |"
          },
          {
            "cells": {
              "Concept or construction": "rapidly decreasing test function / tempered function",
              "Decision and boundary": "Keep the test-function property explicit; retain the established borrowed adjective `tempered` used in `mt02.tex`.",
              "Preferred O007 form": "`fungsi uji yang menurun cepat`; `fungsi tempered`"
            },
            "line": 307,
            "raw_markdown": "| rapidly decreasing test function / tempered function | `fungsi uji yang menurun cepat`; `fungsi tempered` | Keep the test-function property explicit; retain the established borrowed adjective `tempered` used in `mt02.tex`. |"
          },
          {
            "cells": {
              "Concept or construction": "characteristic function of a distribution",
              "Decision and boundary": "In probability contexts this is a Fourier transform of a law, not the indicator `\\chi E`; keep the argument and normalization visible.",
              "Preferred O007 form": "`fungsi karakteristik suatu distribusi`"
            },
            "line": 308,
            "raw_markdown": "| characteristic function of a distribution | `fungsi karakteristik suatu distribusi` | In probability contexts this is a Fourier transform of a law, not the indicator `\\chi E`; keep the argument and normalization visible. |"
          },
          {
            "cells": {
              "Concept or construction": "random variable / joint distribution",
              "Decision and boundary": "Continue the admitted Chapter 27 forms even where the older contents surface says `peubah acak`.",
              "Preferred O007 form": "`variabel acak`; `distribusi bersama`"
            },
            "line": 309,
            "raw_markdown": "| random variable / joint distribution | `variabel acak`; `distribusi bersama` | Continue the admitted Chapter 27 forms even where the older contents surface says `peubah acak`. |"
          },
          {
            "cells": {
              "Concept or construction": "vague topology",
              "Decision and boundary": "Continue the admitted Chapter 27/contents term for weak convergence of probability distributions.",
              "Preferred O007 form": "`topologi samar`"
            },
            "line": 310,
            "raw_markdown": "| vague topology | `topologi samar` | Continue the admitted Chapter 27/contents term for weak convergence of probability distributions. |"
          },
          {
            "cells": {
              "Concept or construction": "Carleson's theorem / Hardy--Littlewood maximal theorem",
              "Decision and boundary": "Preserve the named results and every stated function-space/domain qualifier.",
              "Preferred O007 form": "`teorema Carleson`; `teorema maksimal Hardy--Littlewood`"
            },
            "line": 311,
            "raw_markdown": "| Carleson's theorem / Hardy--Littlewood maximal theorem | `teorema Carleson`; `teorema maksimal Hardy--Littlewood` | Preserve the named results and every stated function-space/domain qualifier. |"
          }
        ]
      },
      {
        "headers": [
          "Source concept",
          "Preferred O007 form",
          "Decision and boundary"
        ],
        "heading": "Volume II appendix — functional-analysis foundations (2026-08-29)",
        "rows": [
          {
            "cells": {
              "Decision and boundary": "Continue the admitted Chapter 24 forms and retain Banach as a proper name.",
              "Preferred O007 form": "ruang bernorma; ruang Banach",
              "Source concept": "normed space / Banach space"
            },
            "line": 324,
            "raw_markdown": "| normed space / Banach space | ruang bernorma; ruang Banach | Continue the admitted Chapter 24 forms and retain Banach as a proper name. |"
          },
          {
            "cells": {
              "Decision and boundary": "Match the localized Volume II contents surface while keeping the two synonymous source labels distinct.",
              "Preferred O007 form": "ruang topologis linear; ruang vektor topologis",
              "Source concept": "linear topological space / topological vector space"
            },
            "line": 325,
            "raw_markdown": "| linear topological space / topological vector space | ruang topologis linear; ruang vektor topologis | Match the localized Volume II contents surface while keeping the two synonymous source labels distinct. |"
          },
          {
            "cells": {
              "Decision and boundary": "Put the F qualifier first and localize seminorm consistently with seminorma.",
              "Preferred O007 form": "F-seminorma; F-norma",
              "Source concept": "F-seminorm / F-norm"
            },
            "line": 326,
            "raw_markdown": "| F-seminorm / F-norm | F-seminorma; F-norma | Put the F qualifier first and localize seminorm consistently with seminorma. |"
          },
          {
            "cells": {
              "Decision and boundary": "Continue the established Chapter 25 term selubung, not the geometric-body term lambung.",
              "Preferred O007 form": "selubung konveks; selubung konveks tertutup",
              "Source concept": "convex hull / closed convex hull"
            },
            "line": 327,
            "raw_markdown": "| convex hull / closed convex hull | selubung konveks; selubung konveks tertutup | Continue the established Chapter 25 term selubung, not the geometric-body term lambung. |"
          },
          {
            "cells": {
              "Decision and boundary": "Preserve the star as part of the technical label and do not conflate the actions of U* and U**.",
              "Preferred O007 form": "topologi lemah; topologi lemah*",
              "Source concept": "weak topology / weak* topology"
            },
            "line": 328,
            "raw_markdown": "| weak topology / weak* topology | topologi lemah; topologi lemah* | Preserve the star as part of the technical label and do not conflate the actions of U* and U**. |"
          },
          {
            "cells": {
              "Decision and boundary": "Retain Cauchy as a proper name; use the compact established adaptation angelik for the sequential-compactness property.",
              "Preferred O007 form": "filter Cauchy; ruang angelik",
              "Source concept": "Cauchy filter / angelic space"
            },
            "line": 329,
            "raw_markdown": "| Cauchy filter / angelic space | filter Cauchy; ruang angelik | Retain Cauchy as a proper name; use the compact established adaptation angelik for the sequential-compactness property. |"
          }
        ]
      },
      {
        "headers": [
          "Source concept",
          "Preferred O007 form",
          "Decision and boundary"
        ],
        "heading": "Chapter 28 B — Sections 282–283 (2026-08-29)",
        "rows": [
          {
            "cells": {
              "Decision and boundary": "Use `terjumlah mutlak` rather than a literal `benar-benar dapat dijumlahkan`; this is a terminology-only repair.",
              "Preferred O007 form": "`barisan yang terjumlah mutlak`",
              "Source concept": "absolutely summable sequence"
            },
            "line": 341,
            "raw_markdown": "| absolutely summable sequence | `barisan yang terjumlah mutlak` | Use `terjumlah mutlak` rather than a literal `benar-benar dapat dijumlahkan`; this is a terminology-only repair. |"
          },
          {
            "cells": {
              "Decision and boundary": "Continue the established measure-theory term; do not use `benar-benar kontinu`.",
              "Preferred O007 form": "`kontinu mutlak`",
              "Source concept": "absolute continuity"
            },
            "line": 342,
            "raw_markdown": "| absolute continuity | `kontinu mutlak` | Continue the established measure-theory term; do not use `benar-benar kontinu`. |"
          },
          {
            "cells": {
              "Decision and boundary": "Keep the proper-name order used throughout the chapter and contents surface.",
              "Preferred O007 form": "`transformasi Fourier invers`",
              "Source concept": "inverse Fourier transform"
            },
            "line": 343,
            "raw_markdown": "| inverse Fourier transform | `transformasi Fourier invers` | Keep the proper-name order used throughout the chapter and contents surface. |"
          },
          {
            "cells": {
              "Decision and boundary": "Use the standard Indonesian calculus term in proofs and hints.",
              "Preferred O007 form": "`integrasi parsial`",
              "Source concept": "integration by parts"
            },
            "line": 344,
            "raw_markdown": "| integration by parts | `integrasi parsial` | Use the standard Indonesian calculus term in proofs and hints. |"
          },
          {
            "cells": {
              "Decision and boundary": "Preserve both named theorems and their joint role in the proof.",
              "Preferred O007 form": "`teorema Fubini dan Tonelli`",
              "Source concept": "Fubini/Tonelli theorem"
            },
            "line": 345,
            "raw_markdown": "| Fubini/Tonelli theorem | `teorema Fubini dan Tonelli` | Preserve both named theorems and their joint role in the proof. |"
          },
          {
            "cells": {
              "Decision and boundary": "Retain the proper names and distinguish the modified kernel explicitly.",
              "Preferred O007 form": "`kernel Dirichlet` / `kernel Fejér`",
              "Source concept": "Fourier kernel"
            },
            "line": 346,
            "raw_markdown": "| Fourier kernel | `kernel Dirichlet` / `kernel Fejér` | Retain the proper names and distinguish the modified kernel explicitly. |"
          }
        ]
      }
    ]
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