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Haar measure and quotient integration

Original online course · Return to the programme (English)

Course text language: English

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Lessons

  1. measure-and-hilbert-space-tools: Measure and Hilbert space tools for Haar integration
    PDF page 4 · EPUB: EPUB/text/ch002.xhtml#measure-and-hilbert-space-tools
    Named locations in this document (11)
    • Measure and Hilbert space tools for Haar integration
      measure-and-hilbert-space-tools-for-haar-integration
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      EPUB: EPUB/text/ch002.xhtml#measure-and-hilbert-space-tools
    • 1. From an outer measure to a measure
      1-from-an-outer-measure-to-a-measure
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    • 2. Integration and convergence without countability assumptions
      2-integration-and-convergence-without-countability-assumptions
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    • 3. The complete spaces of integrable functions
      3-the-complete-spaces-of-integrable-functions
      Open exact PDF location · PDF page 7
      EPUB: EPUB/text/ch002.xhtml#measure-and-hilbert-space-tools--3-the-complete-spaces-of-integrable-functions
    • 4. Densities and bounded functionals
      4-densities-and-bounded-functionals
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      EPUB: EPUB/text/ch002.xhtml#measure-and-hilbert-space-tools--4-densities-and-bounded-functionals
    • Why sigma-finiteness matters for a density
      radon-nikodym-countability
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      EPUB: EPUB/text/ch002.xhtml#measure-and-hilbert-space-tools--radon-nikodym-countability
    • 5. Tensor products as Hilbert–Schmidt operators
      5-tensor-products-as-hilbert-schmidt-operators
      Open exact PDF location · PDF page 9
      EPUB: EPUB/text/ch002.xhtml#measure-and-hilbert-space-tools--5-tensor-products-as-hilbert-schmidt-operators
    • 6. One-variable integral identities
      6-one-variable-integral-identities
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    • 7. Exercises with complete solutions
      7-exercises-with-complete-solutions
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    • What this enables
      what-this-enables
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    • References
      references
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  2. haar-measure-on-locally-compact-groups: Haar measure on locally compact groups
    PDF page 11 · EPUB: EPUB/text/ch003.xhtml#haar-measure-on-locally-compact-groups
    Named locations in this document (39)
  3. quotient-measures-and-weils-integration-formula: Quotient measures and Weil's integration formula
    PDF page 39 · EPUB: EPUB/text/ch004.xhtml#quotient-measures-and-weils-integration-formula
    Named locations in this document (16)
    • Quotient measures and Weil's integration formula
      quotient-measures-and-weil-s-integration-formula
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      EPUB: EPUB/text/ch004.xhtml#quotient-measures-and-weils-integration-formula
    • What to know first
      what-to-know-first
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    • Local topology and compactly supported functions
      local-topology-and-compactly-supported-functions
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    • Averaging along the subgroup
      averaging-along-the-subgroup
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    • Exactly when the quotient measure is invariant
      exactly-when-the-quotient-measure-is-invariant
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    • A cutoff over all cosets
      a-cutoff-over-all-cosets
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    • The modular density and weighted integration
      the-modular-density-and-weighted-integration
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    • Quasi-invariance and the direction of translation
      quasi-invariance-and-the-direction-of-translation
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    • Relative invariance and a character
      relative-invariance-character
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    • The quasi-regular representation
      quasi-regular-representation
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    • Finite covolume forces unimodularity
      finite-covolume-unimodularity
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    • Integration of quotient densities
      quotient-density-integral
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    • Examples that test the formula
      examples-that-test-the-formula
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    • Exercises with solutions
      exercises-with-solutions
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    • What this enables
      what-this-enables
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    • Sources and credit
      sources-and-credit
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  4. finite-covolume-and-arithmetic-quotients: Finite covolume and arithmetic quotients
    PDF page 51 · EPUB: EPUB/text/ch005.xhtml#finite-covolume-and-arithmetic-quotients
    Named locations in this document (11)
    • Finite covolume and arithmetic quotients
      finite-covolume-and-arithmetic-quotients
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    • What to know first
      what-to-know-first
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    • 1. Two integer reductions
      1-two-integer-reductions
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    • Borel fundamental domains and covolume
      borel-fundamental-domain
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    • 2. A fundamental region in the upper half-plane
      2-a-fundamental-region-in-the-upper-half-plane
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    • 3. From area to group covolume
      3-from-area-to-group-covolume
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    • 4. Additive rational adèles
      4-additive-rational-adèles
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    • 5. Norm-one rational idèles
      5-norm-one-rational-idèles
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    • Heisenberg lattice volume
      heisenberg-lattice-volume
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    • 6. Exercises with complete solutions
      6-exercises-with-complete-solutions
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    • References and relation to the programme
      references-and-relation-to-the-programme
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      EPUB: EPUB/text/ch005.xhtml#finite-covolume-and-arithmetic-quotients--references-and-relation-to-the-programme
  5. weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian: Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian
    Prerequisite chapter
    PDF page 57 · EPUB: EPUB/text/ch006.xhtml#weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian
    Named locations in this document (19)
    • Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian
      weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-šmulian
      Open exact PDF location · PDF page 57
      EPUB: EPUB/text/ch006.xhtml#weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian
    • Conventions
      conventions
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    • 1. Weak topologies
      OA-FND-WT-01
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    • oa-fnd-wt-01
      oa-fnd-wt-01
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    • 2. Tychonoff's theorem
      OA-FND-WT-02
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    • oa-fnd-wt-02
      oa-fnd-wt-02
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    • 3. The Banach–Alaoglu theorem
      OA-FND-WT-03
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    • oa-fnd-wt-03
      oa-fnd-wt-03
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    • 4. Mazur's theorem
      OA-FND-WT-04
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    • oa-fnd-wt-04
      oa-fnd-wt-04
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    • 5. Annihilators, bipolars and Goldstine's theorem
      OA-FND-WT-05
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    • oa-fnd-wt-05
      oa-fnd-wt-05
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    • 6. The Krein–Milman theorem
      OA-FND-WT-06
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    • oa-fnd-wt-06
      oa-fnd-wt-06
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    • 7. The Eberlein–Šmulian theorem
      OA-FND-WT-07
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    • oa-fnd-wt-07
      oa-fnd-wt-07
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    • Exercises
      exercises
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    • Where this leads
      where-this-leads
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    • References
      references
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  6. hilbert-spaces-and-compact-operators: Hilbert spaces and compact operators
    Prerequisite chapter
    PDF page 65 · EPUB: EPUB/text/ch007.xhtml#hilbert-spaces-and-compact-operators
    Named locations in this document (26)
  7. the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity: The Stone–Weierstrass theorem for functions vanishing at infinity
    Prerequisite chapter
    PDF page 74 · EPUB: EPUB/text/ch008.xhtml#the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity
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    • The Stone–Weierstrass theorem for functions vanishing at infinity
      the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity
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    • 1. Conventions
      1-conventions
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    • 2. Spaces of bounded and continuous functions
      OA-FND-SW-01
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    • oa-fnd-sw-01
      oa-fnd-sw-01
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    • 3. The one-point compactification
      OA-FND-SW-02
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    • oa-fnd-sw-02
      oa-fnd-sw-02
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    • OA-FND-SW-03
      OA-FND-SW-03
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    • oa-fnd-sw-03
      oa-fnd-sw-03
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    • 4. Compact and locally compact spaces
      OA-FND-SW-15
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    • oa-fnd-sw-15
      oa-fnd-sw-15
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    • 5. Urysohn's lemma and metrizability
      OA-FND-SW-16
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    • oa-fnd-sw-16
      oa-fnd-sw-16
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    • 6. The polynomial step: absolute values without constant terms
      OA-FND-SW-04
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    • oa-fnd-sw-04
      oa-fnd-sw-04
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    • 7. Interpolation at two points
      OA-FND-SW-05
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    • oa-fnd-sw-05
      oa-fnd-sw-05
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    • 8. Stone's lattice lemma
      OA-FND-SW-06
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    • oa-fnd-sw-06
      oa-fnd-sw-06
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    • 9. The real Stone–Weierstrass theorem
      OA-FND-SW-07
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    • oa-fnd-sw-07
      oa-fnd-sw-07
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    • OA-FND-SW-08
      OA-FND-SW-08
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    • oa-fnd-sw-08
      oa-fnd-sw-08
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    • 10. The complex Stone–Weierstrass theorem
      OA-FND-SW-09
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    • oa-fnd-sw-09
      oa-fnd-sw-09
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    • 11. Worked examples
      OA-FND-SW-13
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    • oa-fnd-sw-13
      oa-fnd-sw-13
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    • 12. The role of each hypothesis
      OA-FND-SW-10
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    • oa-fnd-sw-10
      oa-fnd-sw-10
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    • OA-FND-SW-11
      OA-FND-SW-11
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    • oa-fnd-sw-11
      oa-fnd-sw-11
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    • 13. The closure of an arbitrary self-adjoint subalgebra
      OA-FND-SW-12
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    • 14. The Weierstrass theorem: Bernstein polynomials and Korovkin's theorem
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    • 15. Further density theorems on compact spaces
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    • oa-fnd-sw-17
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    • 16. Compact sets, extension of functions, and bounded functions
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    • oa-fnd-sw-18
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    • 17. Convex functions, singular functions, and the disc algebra
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    • oa-fnd-sw-19
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    • 18. Integer coefficients and simultaneous approximation of derivatives
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    • oa-fnd-sw-20
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    • Exercises
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    • Results used from other lessons
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  8. NT-ADL-01: Restricted products and profinite completions
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    • Characters see only finitely many integral tails
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    • Integral residues and finite adèles
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  9. NT-ADL-02: The adèle ring of a number field
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  10. NT-ADL-03: Idèles and the idèle class group
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  11. NT-LOC-02: Completions, the p-adic numbers and complete discretely valued fields
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  12. fremlin-neighbourhood-uniqueness: Haar uniqueness through shrinking neighborhoods
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Edition and sources

Original source notice

Includes all four lessons, all seven bundled prerequisite chapters and the Fremlin-derived supplementary proof. It is not the complete harmonic-analysis course, and further reading listed by the source remains outside this companion. The four course lessons and seven bundled programme chapters are CC0 1.0, with all original human source credits and AI author statements retained. The supplementary proof Haar uniqueness through shrinking neighborhoods adapts D. H. Fremlin and remains under the Design Science License. Its source and change notices accompany the proof, and the complete licence is included. This mixed collection does not relicense Fremlin’s work as CC0.

Original attribution and format-conversion attribution

Claude Opus 5.5 (Anthropic; original Haar lesson, effort not recorded); OpenAI Codex — GPT-6 Astra, Ultra (quotient lesson); GPT-6.1 Sol, Ultra (measure tools, arithmetic integration, corrections and self-check). Each bundled chapter retains its own author statement. OpenAI Codex — GPT-6 Astra, Ultra effort (source-preserving format conversion)

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