Proof of The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure

Each result cited is universally quantified over the data in its own statement. The topology of Test Functions on Euclidean Space, Their Gradient Maps and Laplacians (i) is the topology of the open subsets of $\mathbb{R}^{q}$, which is also the topology of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background (ii) used in Integration by Parts on Euclidean Space Against a Compactly Supported Function of Class $C^{1}$ and the metric topology of $(\mathbb{R}^{q},d_{E})$ (Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$); compact support is the same notion for all three. By $\mathbb{R}^1$ with the Euclidean Distance Is Isometric to the Real Line the Euclidean distance on $\mathbb{R}$ is $d_{\mathbb{R}}$, so continuity in the sense of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (iv) is the continuity hypothesis of A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable. A smooth function is of class $C^{1}$ and of class $C^{2}$ (Test Functions on Euclidean Space, Their Gradient Maps and Laplacians (ii)), and the partial derivatives of a smooth function are smooth by claim 3 of Basic Properties of Cᵏ Functions. A continuous function $\mathbb{R}^{q}\to\mathbb{R}$ is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii). Finite sums of real numbers are those of Finite Sum Notation in a Field, with Properties of Finite Sums and Comparison and Absolute Value Bounds for Finite Sums of Real Numbers in force.

Step 1: claim 1. Fix $i\in[q]$. The function $\psi$ is of class $C^{1}$ on $\mathbb{R}^{q}$ and compactly supported, so by Integration by Parts on Euclidean Space Against a Compactly Supported Function of Class $C^{1}$ (i) (applied with $h=\psi$) the function $\partial_{i}\psi$ is continuous on $\mathbb{R}^{q}$ and compactly supported; by claim 1 of A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable it is bounded, and we choose a bound $K_{i}\ge0$ with $|\partial_{i}\psi(x)|\le K_{i}$ for every $x$. Being continuous, $\partial_{i}\psi$ is Borel, so $\nabla\psi$, whose components are the $\partial_{i}\psi$, is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii). Put $K=\sqrt{\sum_{i=1}^{q}K_{i}^{2}}$, the nonnegative square root (Probability Measures on Euclidean Space and Random Vectors: Standing Notation (i)) of a sum of nonnegative terms (claim 2 of Nonnegativity of Squares in an Ordered Field and claim 5 of Properties of Finite Sums). For every $x$, claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square gives $\lVert\nabla\psi(x)\rVert^{2}=\sum_{i=1}^{q}\partial_{i}\psi(x)^{2}$; each term satisfies $\partial_{i}\psi(x)^{2}=|\partial_{i}\psi(x)|^{2}\le K_{i}^{2}$ by claim 1 of Nonnegativity of Squares in an Ordered Field and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, so $\lVert\nabla\psi(x)\rVert^{2}\le\sum_{i=1}^{q}K_{i}^{2}=K^{2}$ by claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, and hence $\lVert\nabla\psi(x)\rVert\le K$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, both numbers being nonnegative. The function $x\mapsto\lVert\nabla\psi(x)\rVert^{2}$ is the composition of the Borel map $\nabla\psi$ with the Borel map $y\mapsto\lVert y\rVert^{2}$ of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v), hence Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii); it is bounded by $K^{2}$, so for every $\mu\in\mathcal{P}(\mathbb{R}^{q})$ it is integrable with respect to $\mu$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii), and claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space gives $\bigl|\int\lVert\nabla\psi\rVert^{2}\,d\mu\bigr|\le K^{2}\mu(\mathbb{R}^{q})=K^{2}$, whence $\int\lVert\nabla\psi\rVert^{2}\,d\mu\le K^{2}$ by claim 3 of Properties of the Absolute Value in an Ordered Field.

Step 2: claim 2. For $i\in[q]$, the function $\partial_{i}\psi$ is smooth, hence of class $C^{1}$, and compactly supported (Step 1), so by Integration by Parts on Euclidean Space Against a Compactly Supported Function of Class $C^{1}$ (i) (applied with $h=\partial_{i}\psi$) the function $\partial_{i}\partial_{i}\psi$ is continuous and compactly supported, and by claim 1 of A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable it is bounded; choose a bound $B_{i}\ge0$, and by claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set a real number $R_{i}>0$ with $\partial_{i}\partial_{i}\psi(x)=0$ whenever $\lVert x\rVert>R_{i}$. For $m\in[q]$ let $s_{m}:\mathbb{R}^{q}\to\mathbb{R}$ be $s_{m}(x)=\sum_{i=1}^{m}\partial_{i}\partial_{i}\psi(x)$, so that $\Delta\psi=s_{q}$ by The Laplacian of a Twice Continuously Differentiable Function (i). We show by induction that $s_{m}$ is continuous for every $m\in[q]$: let $T$ be the set of $m\in\mathbb{N}$ such that either $q<m$, or $m\in[q]$ and $s_{m}$ is continuous on $\mathbb{R}^{q}$. Then $1\in T$, since $1\in[q]$ (claim 4 of Properties of the Order on the Natural Numbers) and $s_{1}=\partial_{1}\partial_{1}\psi$ by claim 1 of Properties of Finite Sums; and if $m\in T$ then $m+1\in T$: by trichotomy (claim 3 of Properties of the Order on the Natural Numbers), either $q<m$ or $q=m$, in which case $q<m+1$ by claims 5 and 1 there, or $m<q$, in which case $q=m+k$ for some $k\in\mathbb{N}$ (claim 7 there), $1\le k$ (claim 4 there) and so $m+1\le m+k=q$ (claim 6 there); in the last case $q<m$ fails by claim 3 there and $m\le q$ by claim 1 there, so $m\in[q]$ (Initial Segment of the Natural Numbers), $s_{m}$ is defined and, since $m\in T$, continuous, and $s_{m+1}=s_{m}+\partial_{m+1}\partial_{m+1}\psi$ by the recursion in claim 1 of Properties of Finite Sums is continuous by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space. By Principle of Induction for the Natural Numbers, $T=\mathbb{N}$, so $\Delta\psi=s_{q}$ is continuous, hence Borel. Let $R=\sum_{i=1}^{q}R_{i}$; then $0<R_{1}\le R$ and $R_{i}\le R$ for every $i$ by claim 6 of Properties of Finite Sums, so for $\lVert x\rVert>R$ every term $\partial_{i}\partial_{i}\psi(x)$ vanishes and $\Delta\psi(x)=\sum_{i=1}^{q}0=0$ by claim 3 of Properties of Finite Sums with the scalar $0$ and claim 1 of Zero Products and Elementary Identities in a Field; thus $\Delta\psi$ is compactly supported by claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set. It is bounded, by $\sum_{i=1}^{q}B_{i}$, by claims 2 and 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers. Finally, a bounded Borel function is integrable with respect to every $\mu\in\mathcal{P}(\mathbb{R}^{q})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii).

Step 3: claim 3. The zero map on $\mathbb{R}^{q}$ is smooth by claim 2 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions and vanishes for $\lVert x\rVert>1$, hence is compactly supported by claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set; so it is a test function. Let $\psi,\phi\in C_{c}^{\infty}(\mathbb{R}^{q})$ and $a,b\in\mathbb{R}$. By claim 3 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions, $a\psi+b\phi$ is smooth; by claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set there are $R_{\psi},R_{\phi}>0$ with $\psi(x)=0$ for $\lVert x\rVert>R_{\psi}$ and $\phi(x)=0$ for $\lVert x\rVert>R_{\phi}$, so $(a\psi+b\phi)(x)=0$ for $\lVert x\rVert>R_{\psi}+R_{\phi}$, and $a\psi+b\phi$ is compactly supported by the same claim. Hence $C_{c}^{\infty}(\mathbb{R}^{q})$ is closed under pointwise sums and scalar multiples and contains the zero map, and is a linear subspace by The Real Vector Space of Real-Valued Functions on a Set (ii). For $i\in[q]$ and every $x$, the sum and scalar-multiple rules of claim 1 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions give $\partial_{i}(a\psi+b\phi)(x)=a\,\partial_{i}\psi(x)+b\,\partial_{i}\phi(x)$; since sums and scalar multiples of points of $\mathbb{R}^{q}$ are formed coordinatewise (Sum of Points of $\mathbb{R}^n$, Scalar Multiple of a Point of $\mathbb{R}^n$), a point is determined by its coordinates (claim 1 of Euclidean Points as Tuples of Real Numbers), and the coordinates of $\nabla\psi(x)$ are the $\partial_{i}\psi(x)$ (Test Functions on Euclidean Space, Their Gradient Maps and Laplacians (ii)), this is $\nabla(a\psi+b\phi)(x)=a\,\nabla\psi(x)+b\,\nabla\phi(x)$. Applying claim 1 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions once more to the identity $\partial_{i}(a\psi+b\phi)=a\,\partial_{i}\psi+b\,\partial_{i}\phi$ of functions gives $\partial_{i}\partial_{i}(a\psi+b\phi)(x)=a\,\partial_{i}\partial_{i}\psi(x)+b\,\partial_{i}\partial_{i}\phi(x)$, and summing over $i\in[q]$ with claims 2 and 3 of Properties of Finite Sums gives $\Delta(a\psi+b\phi)(x)=a\,\Delta\psi(x)+b\,\Delta\phi(x)$.