# Order, positivity and distributional limits

*Reconstructed by GPT-6 Astra (OpenAI), Ultra reasoning effort, October 2026. The earlier edition was written by GPT-6.1 Sol (OpenAI), Ultra reasoning effort. Original lesson exposition and exercises: CC0. The separately credited programme prerequisites retain their own licences.*

A distribution is controlled by finitely many derivatives on each fixed compact region. The required number can grow as the region changes. Positivity removes this derivative cost: a positive distribution is a measure, and its convergence reaches continuous compact tests. Without positivity, large opposite masses can cancel on smooth tests while diverging on continuous ones.

We supply the test-space, support and finite-net arguments below. The included [functional foundations](../prerequisites/U011-free-foundations/functional-foundations-U008.md) prove complex Hahn–Banach, the complete-metric Baire theorem and completeness of fixed-support smooth test spaces. The included [positive-measure construction](../prerequisites/U011-free-foundations/positive-measure-foundations-U008.md) proves the exact Riesz representation and regularity used here. Scalar calculus, cutoffs and integration have their full proofs in the three accompanying foundation selections listed after the solutions. These are available proof texts, not external citations in place of proofs.

## Local estimates and the meaning of order

Let \(X\subset\mathbb R^n\) be open, \(n\ge1\). All linear forms are complex linear. For compact \(K\subset X\), set

\[
 \begin{aligned}
 \mathcal D(X)&=C_c^\infty(X),\\
 \mathcal D_K&=\{\phi\in\mathcal D(X):\operatorname{supp}\phi\subset K\},\\
 p_{K,m}(\phi)&=\max_{|\alpha|\le m}\sup_K|\partial^\alpha\phi|.
 \end{aligned}
 \tag{1.1}
\]

A multi-index \(\alpha\) is an \(n\)-tuple of nonnegative integers, with \(|\alpha|=\sum_i\alpha_i\). A distribution is a linear form \(u\) such that for every compact \(K\) there are \(C_K<\infty\) and \(m_K\ge0\) with

\[
 |u(\phi)|\le C_Kp_{K,m_K}(\phi),\qquad \phi\in\mathcal D_K.
 \tag{1.2}
\]

It has order at most \(k\) on \(X\) if the same integer \(k\) works for every \(K\); the constant may depend on \(K\). A function in \(\mathcal D_K\), and every derivative of it, extends by zero smoothly to \(\mathbb R^n\). This follows because its support is compactly inside \(X\): near a point of \(K\) use its smooth formula in \(X\), and near any other point of the complement use zero. Thus \(p_{K,m}\) also equals the maximum of the global supremum norms of those zero-extended derivatives.

**The test topology and its bounded families.** Call a seminorm \(q\) on \(\mathcal D(X)\) admissible if its restriction to each \(\mathcal D_K\) is bounded by some \(C p_{K,m}\). A seminorm is nonnegative, absolutely homogeneous and subadditive. Give \(\mathcal D(X)\) the locally convex topology generated by all admissible seminorms: a zero neighborhood specifies finitely many inequalities \(q_i(\phi)<\varepsilon_i\). This is the test-function inductive-limit topology. Indeed the inclusions of the fixed-support spaces are continuous, and every locally convex topology with that property has continuous seminorms satisfying these finite-order bounds. To see the bound on a fixed-support space, continuity supplies a finite intersection of derivative-seminorm balls; since these seminorms increase with order, it contains \(p_{K,m}<\delta\). Rescaling proves a multiple of \(p_{K,m}\) bounds the given seminorm. The same argument shows that (1.2) is exactly continuity of a linear form for this topology.

A family \(\mathcal B\) is bounded in this topology exactly when \(\sup_{\phi\in\mathcal B}q(\phi)<\infty\) for every admissible \(q\). This is equivalent to the usual definition by absorption in every zero neighborhood: a common scalar multiple of a neighborhood controls its finitely many defining seminorms, in either direction. We prove the useful concrete characterization:

\[
 \begin{gathered}
 \mathcal B\text{ is bounded}\quad\Longleftrightarrow\quad
 \mathcal B\subset\mathcal D_K\text{ for some compact }K,\\
 \sup_{\phi\in\mathcal B}p_{K,m}(\phi)<\infty
                  \quad\text{for every }m.
 \end{gathered}
 \tag{T1}
\]

Choose a compact exhaustion \(K_j\subset\operatorname{int}K_{j+1}\) as in (M3) of the measure prerequisite. If no one compact contains all supports, choose \(\phi_j\in\mathcal B\) and \(x_j\notin K_j\) with \(\phi_j(x_j)\ne0\). Such a nonzero point exists outside the closed \(K_j\) whenever the support is not contained in it. The points \(x_j\) escape every compact set. Put

\[
 q(\phi)=\sup_{j\ge1}\frac{j|\phi(x_j)|}{|\phi_j(x_j)|}.
 \tag{T2}
\]

For each compact support only finitely many terms can be nonzero, so this is a finite seminorm on every test. On \(\mathcal D_K\) it is bounded by a finite constant times \(p_{K,0}\). It is therefore admissible, but \(q(\phi_j)\ge j\), contradicting boundedness. Hence a common compact exists. The global supremum seminorm of derivatives through any fixed degree is also admissible, proving the remaining bounds in (T1). Conversely those bounds control every admissible seminorm by its fixed-support estimate, proving (T1).

It follows as well that \(\phi_j\to0\) in this topology exactly when all supports lie in one compact and every derivative tends uniformly to zero. A convergent sequence is bounded, since each of its seminorm values is a bounded scalar sequence; apply (T1), then the admissible global derivative seminorms. The converse follows from the fixed-support bound for each admissible seminorm.

**Proposition 1.1 (the sequential test).** A linear form is a distribution if and only if it sends every sequence tending to zero in \(\mathcal D(X)\) to a scalar sequence tending to zero.

**Proof.** The forward direction follows from (1.2) and the characterization just proved. If (1.2) fails on some fixed \(K\), for every \(j\ge1\) choose \(\phi_j\in\mathcal D_K\) with \(|u(\phi_j)|>j p_{K,j}(\phi_j)\). The value is nonzero. The tests \(\psi_j=\phi_j/u(\phi_j)\) have \(u(\psi_j)=1\) and \(p_{K,j}(\psi_j)<1/j\). Their derivatives of every fixed order tend uniformly to zero, while their values under \(u\) do not. This contradicts sequential continuity. \(\square\)

**Proposition 1.2 (finite differentiability).** An order-at-most-\(k\) distribution extends uniquely to \(C_c^k(X)\), continuously for the \(C^k\) norm on each common compact support. If a test's support is inside the interior of \(K\), the estimate (1.2) retains its original constant \(C_K\). At the boundary of \(K\), an estimate on a larger compact neighborhood suffices.

**Proof.** A function \(f\in C_c^k(X)\) extends by zero to a \(C^k\) function on \(\mathbb R^n\). Choose a nonnegative smooth compact bump \(\rho\) of integral one, using the proved cutoff and integration construction, and put \(\rho_\varepsilon(x)=\varepsilon^{-n}\rho(x/\varepsilon)\). Its convolution with \(f\) is smooth: differentiating the smooth kernel under its compact integral is justified by uniform convergence of its difference quotients. For \(|\alpha|\le k\), ordinary integration by parts on a box containing the support also gives

\[
 \partial^\alpha(f*\rho_\varepsilon)(x)
     =\int\rho(z)\,\partial^\alpha f(x-\varepsilon z)\,dz.
 \tag{1.3}
\]

Uniform continuity of each compactly supported derivative makes this converge uniformly to \(\partial^\alpha f\). All approximants have support in one compact neighborhood \(K_1\Subset X\). Bound (1.2) makes their values under \(u\) a Cauchy family. Define the extension by its limit. Two such approximating families have a common larger compact support and their difference tends to zero in \(C^k\), so the limit is independent of the approximation. This also proves linearity and uniqueness, and passing the estimate to the limit proves continuity. When \(\operatorname{supp}f\subset\operatorname{int}K\), small convolutions remain in \(K\), retaining \(C_K\). For tests supported merely in \(K\), use one fixed larger neighborhood \(K_1\) for the whole family. \(\square\)

## One weighted estimate for every support

**Theorem 2.1 (locally finite derivative weights).** A linear form \(u\) is a distribution if and only if there are continuous nonnegative functions \(\rho_\alpha\) on \(X\) such that only finitely many are nonzero on each compact set and

\[
 |u(\phi)|\le\sum_\alpha
          \|\rho_\alpha\partial^\alpha\phi\|_\infty.
 \tag{2.1}
\]

Every test sees a finite sum. The weights can vanish for all \(|\alpha|>k\) exactly when \(u\) has order at most \(k\).

**Proof.** First construct the partition needed for the forward direction. For the exhaustion \(K_j\) above choose \(0\le b_j\le1\), equal to one near \(K_j\) and compactly supported in \(\operatorname{int}K_{j+1}\), using (M1). Put

\[
 \psi_j=b_j\prod_{i<j}(1-b_i),\qquad j\ge1.
 \tag{P1}
\]

These smooth functions are nonnegative and compactly supported. Their partial sums are \(1-\prod_{j\le N}(1-b_j)\). Near any compact set some \(b_J=1\); all subsequent \(\psi_j\) vanish there and the partial sum is one. Thus \(\sum_j\psi_j=1\), and their supports form a locally finite family.

Write \(L_j=\operatorname{supp}\psi_j\) and choose \(C_j\ge1,m_j\) for (1.2) on \(L_j\). For \(|\beta|\le m_j\), define

\[
 A_{j,\beta}
 =\sum_{\substack{|\alpha|\le m_j\\\beta\le\alpha}}
       \binom\alpha\beta\,
                  |\partial^{\alpha-\beta}\psi_j|,
 \tag{2.2}
\]

and set it to zero otherwise. The relation \(\beta\le\alpha\) is coordinatewise, and \(\binom\alpha\beta\) is the product of the scalar binomial coefficients. Repeated one-coordinate product differentiation gives the multi-index product formula; induction uses the ordinary adjacent-binomial identity. Consequently

\[
 p_{L_j,m_j}(\psi_j\phi)
       \le\sum_\beta\|A_{j,\beta}\partial^\beta\phi\|_\infty.
\]

Each \(A_{j,\beta}\) is continuous, nonnegative and supported in \(L_j\). Define

\[
 \rho_\beta=\sum_{j\ge1}2^j C_j A_{j,\beta}.
 \tag{2.3}
\]

Local finiteness makes these continuous. On a compact set only finitely many \(L_j\) occur, and their finitely many derivative orders leave only finitely many nonzero \(\rho_\beta\). Since \(C_jA_{j,\beta}\le2^{-j}\rho_\beta\), the finite partition of a test gives

\[
 \begin{aligned}
 |u(\phi)|&\le\sum_j|u(\psi_j\phi)|\\
 &\le\sum_j2^{-j}\sum_\beta
                    \|\rho_\beta\partial^\beta\phi\|_\infty\\
 &\le\sum_\beta\|\rho_\beta\partial^\beta\phi\|_\infty.
 \end{aligned}
\]

If the order is at most \(k\), choose every \(m_j=k\), making all higher-degree weights zero. Conversely, on any compact \(K\), finitely many bounded weights occur in (2.1), giving (1.2); if their degrees are at most \(k\), that estimate has order \(k\). \(\square\)

## Distributions are locally finite derivatives of measures

The included functional foundations prove this exact complex Hahn–Banach statement: a bounded complex-linear functional on any subspace of any complex normed space extends with the same norm. The subspace need not be closed and the ambient space need not be complete. Theorem M of the measure prerequisite proves positive Riesz representation on \(X\), together with the necessary local-finiteness, regularity and uniqueness statements. We use its convention that a locally finite complex measure is specified consistently on relatively compact regions; it need not assign a finite complex value to the whole unbounded space.

Let \(C_0(X)\) consist of continuous complex functions vanishing at infinity, meaning that \(\{|f|\ge\varepsilon\}\) is compact for each \(\varepsilon>0\). Its norm is \(\|f\|_\infty\). Such a function is bounded: outside the compact set for \(\varepsilon=1\) it is less than one, and on that compact it has a finite maximum. Compactly supported continuous functions are dense in this norm. Indeed, multiply \(f\) by a cutoff equal to one on \(\{|f|\ge\varepsilon\}\), giving error at most \(\varepsilon\).

**Lemma 3.1 (complex representation on \(C_0\)).** A bounded complex-linear functional \(L\) on \(C_0(X)\) is integration against a unique finite complex Radon measure \(\mu\), with \(|\mu|(X)\le4\|L\|\).

**Proof.** For a bounded real-linear functional \(a\) on real \(C_0(X)\), define on \(f\ge0\)

\[
 a_+(f)=\sup_{0\le g\le f}a(g).
 \tag{3.1}
\]

It is between zero and \(\|a\|\|f\|_\infty\), and is positively homogeneous. Adding admissible functions gives \(a_+(f+h)\ge a_+(f)+a_+(h)\). Conversely, split any \(0\le g\le f+h\) into \(g_1=\min(g,f)\) and \(g_2=g-g_1\). Both are in \(C_0(X)\), with \(0\le g_1\le f\) and \(0\le g_2\le h\). This gives the reverse inequality and hence additivity on the positive cone.

Extend by differences of nonnegative functions. The extension is well-defined: if \(f=g-h=g'-h'\), then \(g+h'=g'+h\), so additivity gives equal differences. Put \(a_-=a_+-a\). It is positive, and for \(f\ge0\), substituting \(f-g\) in (3.1) gives \(a_-(f)=\sup_{0\le g\le f}[-a(g)]\le\|a\|\|f\|_\infty\). Positivity implies \(|a_\pm(f)|\le a_\pm(|f|)\), so both extensions are bounded.

Theorem M applied to their restrictions to \(C_c(X;\mathbb R)\) gives positive Radon measures \(\mu_+,\mu_-\). The open-set formula (M5) bounds each total mass by \(\|a\|\). They represent \(a=a_+-a_-\) on \(C_c\); density and the finite mass bounds extend this equality to \(C_0\).

Apply this to \(\operatorname{Re}L\) and \(\operatorname{Im}L\) on real functions, whose norms are at most \(\|L\|\). Their four positive measures give a complex measure representing \(L\), by complex linearity. The variation construction (M10) bounds its mass by the sum of those four masses. The uniqueness for complex Radon measures proved after (M11) gives uniqueness here. \(\square\)

**Theorem 3.2 (measure-derivative representation).** Each \(u\in\mathcal D'(X)\) has a representation

\[
 \begin{aligned}
 u(\phi)&=\sum_\alpha\int_X\partial^\alpha\phi\,d\nu_\alpha,\\
 u&=\sum_\alpha(-1)^{|\alpha|}\partial^\alpha\nu_\alpha.
 \end{aligned}
 \tag{3.2}
\]

Each \(\nu_\alpha\) is a locally finite complex Radon measure, and each compact subset of \(X\) has a neighborhood meeting only finitely many of their supports. If \(u\) has order at most \(k\), the measures can be zero for \(|\alpha|>k\). Conversely any such finite-degree representation has order at most \(k\). The representing family need not be unique.

**Proof.** Take the weights of Theorem 2.1. Use the normed space

\[
 B=\left\{(g_\alpha):g_\alpha\in C_0(X),\
                 \sum_\alpha\|g_\alpha\|_\infty<\infty\right\},
 \qquad \|g\|_B=\sum_\alpha\|g_\alpha\|_\infty.
\]

The scalar triangle inequalities prove the norm properties by summing their finite partial sums and taking increasing limits. Finite-coordinate truncations are dense, since the tails of the convergent nonnegative series tend to zero. Completeness of \(B\) is not needed.

Map a test to \(J\phi=(\rho_\alpha\partial^\alpha\phi)_\alpha\). Each component is continuous with compact support, and only finitely many components occur. Estimate (2.1) makes \(F(J\phi)=u(\phi)\) a well-defined linear functional of norm at most one on \(J\mathcal D(X)\): a zero value of \(J\phi\) forces \(u(\phi)=0\). Extend \(F\) to \(L\) on \(B\) by the included Hahn–Banach proof. Each coordinate restriction has norm at most one, so Lemma 3.1 represents it by a finite complex measure \(\mu_\alpha\) of variation at most four. Density of finite-coordinate truncations gives

\[
 L(g)=\sum_\alpha\int g_\alpha\,d\mu_\alpha.
\]

The series is absolutely convergent, bounded by \(4\sum_\alpha\|g_\alpha\|_\infty\). Put \(\nu_\alpha=\rho_\alpha\mu_\alpha\). The continuous-weight proof (M12) makes this a locally finite complex Radon measure supported in \(\operatorname{supp}\rho_\alpha\). For compact \(K\), take a larger compact neighborhood \(K_1\Subset X\). All but finitely many weights vanish on \(K_1\), so the corresponding measure supports miss \(\operatorname{int}K_1\). This proves the neighborhood version of local finiteness. Applying the last display to \(J\phi\) proves the first equality of (3.2).

The distributional derivative is defined by \((\partial^\alpha v)(\phi)=(-1)^{|\alpha|}v(\partial^\alpha\phi)\). A finite-order estimate for the right side shows this is again a distribution. This convention gives the second equality of (3.2). If the order is at most \(k\), the higher weights, and hence their measures, are zero. Conversely, a finite-degree representation satisfies on \(\mathcal D_K\)

\[
 |u(\phi)|\le
  \left(\sum_{|\alpha|\le k}|\nu_\alpha|(K)\right)p_{K,k}(\phi).
\]

The sum is finite. Without a degree bound, the assumed locally finite family still leaves finitely many orders on each compact neighborhood, so the same argument gives a distribution. \(\square\)

**Corollary 3.3 (order zero is precisely a measure).** Order-zero distributions are exactly locally finite complex Radon measures. The measure is unique and acts continuously on \(C_c(X)\).

**Proof.** Set \(k=0\) in Theorem 3.2 for existence and its converse. If two measures agree on smooth tests, the uniform approximations of Proposition 1.2 and their compact variation bounds give equality on \(C_c(X)\). The local complex-measure uniqueness proved in the measure prerequisite then identifies them. \(\square\)

## Positivity forces order zero

A linear form \(u\) is positive if \(u(\phi)\) is real and nonnegative for every real nonnegative smooth test. No continuity is assumed in this definition.

**Theorem 4.1 (positive forms are measures).** Every positive linear form on \(\mathcal D(X)\) is an order-zero distribution. Its continuous-test extension is represented by a unique positive Radon measure.

**Proof.** Fix compact \(K\) and a nonnegative compact smooth cutoff \(\chi=1\) on \(K\). For real \(\phi\in\mathcal D_K\), both \(\|\phi\|_\infty\chi+\phi\) and \(\|\phi\|_\infty\chi-\phi\) are nonnegative. Subtracting such functions also shows that \(u\) is real on every real test. Positivity therefore gives

\[
 |u(\phi)|\le u(\chi)\|\phi\|_\infty.
 \tag{4.1}
\]

For complex \(\phi\), if its value is nonzero, take \(\theta=\overline{u(\phi)}/|u(\phi)|\). Complex linearity and reality on real tests imply

\[
 |u(\phi)|=u(\operatorname{Re}(\theta\phi))
                  \le u(\chi)\|\phi\|_\infty.
 \tag{4.2}
\]

A zero value satisfies the same bound. This proves order zero. Proposition 1.2 supplies the continuous-test extension. For a nonnegative continuous test, convolution with the nonnegative mollifier used there preserves nonnegativity. Passing to the limit shows that the extension is positive. Theorem M now gives its positive Radon measure and uniqueness. \(\square\)

This measure may have infinite total mass. Conversely a derivative of a point mass is not positive: take a bump equal to one near zero and supported in \((-1/2,1/2)\), multiply it by \(1+x\) or \(1-x\), and pair its derivative at zero. The two nonnegative tests give opposite signs.

## Limits on smooth tests

**Theorem 5.1 (weak completeness and uniform local order).** Suppose \(u_j\in\mathcal D'(X)\) and \(u_j(\phi)\) converges for each smooth compact test. The limits define a distribution \(u\). For every fixed compact \(K\), one constant \(C_K\) and one order \(m_K\) work in (1.2) for all \(u_j\) and for \(u\).

If \(\phi_j\to\phi\) in the test topology, then \(u_j(\phi_j)\to u(\phi)\). Moreover \(u_j\to u\) uniformly on each bounded test family. This is convergence in the strong dual, whose seminorms are \(v\mapsto\sup_{\phi\in\mathcal B}|v(\phi)|\) for bounded \(\mathcal B\). The assertion concerns sequences; it does not identify the weak and strong topologies or assert the same implication for arbitrary nets.

**Proof.** Scalar limits preserve linearity. On a fixed \(\mathcal D_K\), the functional prerequisite, Sections 14.1–14.2, proves that the increasing derivative seminorms give a complete metric topology. The proof takes uniform limits of every derivative of the zero extensions and uses the fundamental theorem on coordinate segments to identify successive derivatives; no regularity of the boundary of \(K\) is needed.

For \(r=1,2,\ldots\), put

\[
 A_r=\{\phi\in\mathcal D_K:\sup_j|u_j(\phi)|\le r\}.
\]

Each is closed, because every \(u_j\) is continuous there, and their union is \(\mathcal D_K\), because each convergent scalar sequence is bounded. The proved Baire theorem gives a point \(\phi_0\) and a neighborhood \(p_{K,m}(h)<\varepsilon\) with \(\phi_0+h\in A_r\), including \(h=0\). Subtraction gives \(\sup_j|u_j(h)|\le2r\) in that neighborhood. If \(p_{K,m}(h)>0\), apply this to \(\varepsilon h/(2p_{K,m}(h))\), obtaining the common bound \(4r p_{K,m}(h)/\varepsilon\). If the seminorm is zero, \(h=0\), since the zeroth derivative is included. Passing to the scalar limit gives the same bound for \(u\), which proves it is a distribution.

For moving tests, their supports lie in one compact by (T1), and

\[
 u_j(\phi_j)-u(\phi)
   =u_j(\phi_j-\phi)+(u_j-u)(\phi).
 \tag{5.1}
\]

The common derivative bound controls the first term, and fixed-test convergence controls the second.

For the strong conclusion, we prove the finite-net step. Let \(\mathcal B\subset\mathcal D_K\) be bounded and let \(m\) be the common order just obtained. By (T1), the derivatives through degree \(m+1\) have a common global bound \(M\). Each derivative through degree \(m\) of each zero extension is Lipschitz for the coordinate maximum norm, with constant \(nM\): telescope along coordinate segments and apply the fundamental theorem. Enclose \(K\) in a finite cube and choose a finite grid so that every point of the cube is within \(\delta\) of a grid point in that norm. At each grid point and for each derivative through degree \(m\), partition the bounded complex range into finitely many sets of diameter at most \(\eta\). A test is assigned its finite list of range sets. Choose one test from each nonempty list class. There are finitely many such choices. Two tests in the same class differ in any derivative through degree \(m\) by at most

\[
 2nM\delta+\eta
 \tag{T3}
\]

on the cube, by comparing both values with those at a nearest grid point. Off the cube both tests and their derivatives vanish. Taking \(\delta,\eta\) small proves that \(\mathcal B\) has a finite net, with centers in \(\mathcal B\), for \(p_{K,m}\). This also covers \(M=0\), when the family contains only zero.

The already proved bound gives \(|(u_j-u)(h)|\le2C_Kp_{K,m}(h)\). Approximate any member of \(\mathcal B\) by a net center, use this uniform error bound, and then use fixed-test convergence at the finitely many centers. First choose an arbitrarily fine net and then let \(j\to\infty\). The supremum over \(\mathcal B\) tends to zero. \(\square\)

In particular every weakly Cauchy sequence has a weak limit, because its scalar pairings converge by completeness of \(\mathbb C\), after which Theorem 5.1 applies.

**Theorem 5.2 (positive convergence reaches continuous tests).** If the distributions in Theorem 5.1 are positive, the limit is positive and

\[
 u_j(f)\longrightarrow u(f),\qquad f\in C_c(X).
 \tag{5.2}
\]

This is convergence against compactly supported continuous tests, with no assertion about total masses or all bounded continuous tests.

**Proof.** Positivity passes to scalar limits on nonnegative smooth tests. Theorem 4.1 identifies all the functionals with positive measures. For a given \(f\), put its support inside the interior of a compact \(K_1\Subset X\), and choose a nonnegative smooth \(\chi=1\) on \(K_1\). The numbers \(u_j(\chi)\) converge, so they and \(u(\chi)\) have a common bound \(M\). Passing (4.2) through the uniform approximation gives

\[
 |u_j(g)|+|u(g)|\le2M\|g\|_\infty,
 \qquad\operatorname{supp}g\subset K_1.
 \tag{5.3}
\]

Approximate \(f\) uniformly by a smooth \(g\) supported in \(K_1\). Then

\[
 |u_j(f)-u(f)|
      \le2M\|f-g\|_\infty+|(u_j-u)(g)|.
\]

First make the approximation error arbitrarily small, then let \(j\to\infty\) for the fixed smooth test. This proves (5.2). \(\square\)

For contrast, on the line the signed measures

\[
 v_j=j(\delta_{1/j}-\delta_0)
 \tag{5.4}
\]

converge to \(-\delta'_0\) on smooth tests: the fundamental theorem gives \(v_j(\phi)=\int_0^1\phi'(t/j)\,dt\to\phi'(0)\). But a continuous compactly supported \(f\) equal to \(\sqrt{|x|}\) near zero has \(v_j(f)=\sqrt j\) for large \(j\). The local cancellation of these opposite masses prevents continuous-test convergence.

## Exercises

1. **Increasing order — advanced.** On the real line, define \(w=\sum_{j\ge1}\delta_j^{(j)}\). Prove that it is a distribution but has no finite order valid on the whole line. Exhibit tests supported near one integer that defeat any proposed order bound.
2. **A dipole limit — intermediate.** For (5.4), compute the distributional limit and total variation. Evaluate it on a continuous test equal to \(\sqrt{|x|}\) near zero, and prove that the limiting distribution has order exactly one.
3. **Mass leaving compact sets — foundation.** Prove that \(j\delta_j\to0\) on \(C_c(\mathbb R)\), although the total masses diverge. Find a bounded continuous test for which the pairings do not tend to zero.
4. **Two representations of zero — intermediate.** Express the derivative of \(h\in C_c^\infty(\mathbb R)\) in the first form of (3.2). Then use a nonzero \(h\) and \(h'\) to construct a nonzero measure family representing the zero distribution, and check the signs.
5. **Uniform continuous tests — advanced.** Under Theorem 5.2, suppose a family \(\mathcal B\subset C_c(X)\) has common compact support, uniformly bounded values and a common modulus of continuity after zero extension. Prove convergence is uniform over \(\mathcal B\), using one finite net for the whole family.

## Complete solutions

**Solution 1.** A compact set meets only finitely many positive integers. On \(\mathcal D_K\), the pairing is consequently a finite sum, bounded by \(\sum_{j\in K\cap\mathbb N}p_{K,j}(\phi)\). This is a local finite-order estimate, and is zero when the index set is empty.

If a global order \(k\) existed, choose an integer \(j>k\) and a compact interval around \(j\) containing no other integer. Let \(b(t)=t^j\eta(t)\), where \(\eta\) is a compact smooth cutoff equal to one near zero; then \(b^{(j)}(0)=j!\ne0\). For small \(\varepsilon>0\), the tests

\[
 \phi_\varepsilon(x)=\varepsilon^k b((x-j)/\varepsilon)
\]

are supported in that interval. Their derivatives through degree \(k\) have uniform bounds for \(\varepsilon\le1\), whereas \(|w(\phi_\varepsilon)|=\varepsilon^{k-j}j!\to\infty\). This contradicts the proposed bound on one fixed compact set.

**Solution 2.** The fundamental-theorem calculation after (5.4) gives the limit \(-\delta'_0\). The variation of each two-point measure is \(2j\): partitioning into its two atoms attains this value, and the triangle inequality bounds every other partition by it. The continuous test has value \(j(j^{-1/2}-0)=\sqrt j\) for large \(j\).

The derivative of a point mass has order at most one by its defining action. It cannot have order zero. Choose \(b\) compactly supported and smooth with \(b'(0)\ne0\); the functions \(b(x/\varepsilon)\) have bounded supremum and common compact support for \(\varepsilon\le1\), while the absolute value of their pairing with \(\delta'_0\) is \(\varepsilon^{-1}|b'(0)|\). Thus the order is exactly one, as it is for its negative.

**Solution 3.** Each compact support misses the point \(j\) eventually, making its pairing exactly zero for all sufficiently large \(j\). The total mass is nevertheless \(j\). The constant function one is bounded and continuous and has pairing \(j\), so it supplies the requested failure outside the compact-test class.

**Solution 4.** Ordinary integration by parts gives \((\partial h)(\phi)=-\int h\phi'\,dx\). Thus the first form of (3.2) uses \(\nu_1=-h\,dx\), with other measures zero. The second form has \(-\partial\nu_1=\partial(h\,dx)\), preserving the sign. To represent zero, use \(\nu_0=h'\,dx\), \(\nu_1=h\,dx\), and all other measures zero. Their first-form pairing is \(\int h'\phi+\int h\phi'=0\). These compact measures form a nonzero family when \(h\ne0\), whereas the all-zero family represents the same distribution. If desired, their Radon character follows by splitting the real and imaginary continuous densities into positive and negative parts and applying (M12) to Lebesgue measure.

**Solution 5.** Enclose the common support in a finite cube and use a finite grid. The common modulus of continuity makes the change from any point to its nearest grid point uniformly small for every function in the family. Partition the bounded complex range into finitely many small-diameter sets at the grid points and choose one member from each nonempty finite signature class. The same argument as (T3), with the common modulus in place of the Lipschitz bound, gives a finite \(\varepsilon\)-net in supremum norm, with centers \(f_1,\ldots,f_N\) in the family. All supports are still in the original compact set.

Choose a compact neighborhood and cutoff as in (5.3), giving a common bound \(M\) for the measures and their limit. If \(\|f-f_l\|_\infty<\varepsilon\), then

\[
 |(u_j-u)(f)|\le2M\varepsilon+
                 \max_{1\le l\le N}|(u_j-u)(f_l)|.
\]

Each of the finitely many centers converges by (5.2). Taking the supremum and then the limit superior bounds it by \(2M\varepsilon\); let \(\varepsilon\downarrow0\). The single finite net makes this estimate uniform over the entire family.

## Programme proof locations and freely accessible sources

- [Scalar calculus and Euclidean topology](../prerequisites/U011-free-foundations/metric-foundation-bridges.md): Sections 12.1–12.9 and 13.1–13.5 for arithmetic, compactness, continuity and calculus; Section 13.10 for the exact smooth ball cutoff. This selection retains CC0 1.0.
- [Complex scalars and finite algebra](../prerequisites/U011-free-foundations/stable-prerequisite-bridges.md): Sections 10.1–10.6 for the scalar and norm rules used in those proofs. CC0 1.0.
- [Measure, integration and smoothing](../prerequisites/U011-free-foundations/banach-foundation-bridges.md): Sections 15.0–15.4 and 16.1–16.2 for the measure construction, convergence, integral operations, mollification and generating-class argument. CC0 1.0.
- [Functional extension and complete test spaces](../prerequisites/U011-free-foundations/functional-foundations-U008.md): the complete complex Hahn–Banach proof in Section 5, Baire in Section 6, complete test metrics in Sections 14.1–14.2, and the explicit choice consequence of the selected Zorn axiom in Section 19. CC0 1.0.
- [Positive functionals and locally finite measures](../prerequisites/U011-free-foundations/positive-measure-foundations-U008.md): compact cutoffs, finite partitions, Theorem M, Borel regularity, complex variation, uniqueness and continuous weights, including the locally finite complex-measure convention. This adaptation of the earlier CC0 programme measure lesson is CC0.

The free human sources used for comparison and construction are Semyon Dyatlov's [*Lecture notes for 18.155: distributions, elliptic regularity, and applications to PDEs*](https://math.mit.edu/~dyatlov/18.155/155-notes.pdf), Section 4.3, Theorems 4.14 and 4.16 and Propositions 4.15, 4.17–4.18; Paul Garrett's [*Banach Spaces*](https://www-users.cse.umn.edu/~garrett/m/fun/notes_2016-17/02_banach.pdf), Section 9, and [*Review of metric spaces*](https://www-users.cse.umn.edu/~garrett/m/fun/notes_2012-13/01_metric_spaces.pdf), Theorem 4.0.1; and D. H. Fremlin's free author edition of [*Measure Theory*, Chapter 43](https://www1.essex.ac.uk/maths/people/fremlin/chap43.pdf), 436J. Their citations do not replace any of the programme proofs listed above.