Proof of Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity

Each result cited is universally quantified over the data in its own statement. Borel is as in the preamble of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets; a continuous function on a Euclidean space is Borel (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps), a composition of Borel maps is Borel (claim 4 of Borel Measurability and Bounded Integration on a Metric Space), and for $y\in\mathbb{R}^{q}$ the map $x\mapsto y-x$ is continuous, since $d_{E}(y-x,y-x')=\lVert x'-x\rVert=d_{E}(x,x')$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$. Throughout, $\lambda_{q}$ is $\sigma$-finite by claim 1 of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$ and $\mu$, $\nu$ are finite, so Tonelli and Fubini Theorems applies to the product measures $\lambda_{q}\otimes\mu$, $\lambda_{q}\otimes\lambda_{q}$, $\mu\otimes\nu$ and $\lambda_{q}\otimes(\mu\boxtimes\nu)$ (Existence and Uniqueness of the Product Measure). A function of the first variable alone, $(y,x)\mapsto\phi(y)$ with $\phi$ Borel, is measurable for the product $\sigma$-algebra, the first coordinate projection being measurable by Product Sigma-Algebra. For measurability with respect to a product $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{a})\otimes\mathcal{B}(\mathbb{R}^{b})$ we use Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product: for a Borel $F:\mathbb{R}^{a+b}\to[0,\infty]$ the function $F\circ\iota^{a,b}$ on $\mathbb{R}^{a}\times\mathbb{R}^{b}$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{a})\otimes\mathcal{B}(\mathbb{R}^{b})$, and a real-valued Borel $F$ is handled through its positive and negative parts (claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions). The map $\mathbb{R}^{q+q}\to\mathbb{R}^{q}$, $z\mapsto\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)$, is Borel, its components being differences of the Borel components of the projections (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps); hence for every Borel $G:\mathbb{R}^{q}\to\mathbb{R}$ the function $z\mapsto G(\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z))$ is Borel on $\mathbb{R}^{q+q}$, and $(y,x)\mapsto G(y-x)$, which is this function composed with $\iota^{q,q}$, is measurable with respect to $\mathcal{B}(\mathbb{R}^{q})\otimes\mathcal{B}(\mathbb{R}^{q})$.

Claim 1. For a continuous bounded $G:\mathbb{R}^{q}\to\mathbb{R}$ we write $G*\mu$ for the function $y\mapsto\int G(y-x)\,\mu(dx)$ of The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral §continuous, so that $g_{s}*\mu$ is the function of the statement. By The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails §derivatives and The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails §bounds, each of $g_{s}$, $\partial_{i}g_{s}$ and $\partial_{j}\partial_{i}g_{s}$ is smooth, in particular of class $C^{1}$, and is bounded together with its partial derivatives by the constant $C=\max\{A_{0},A_{1},A_{2},A_{3}\}\,s^{-(q+3)/2}$ (as $s\le1$, $s^{-m/2}\le s^{-(q+3)/2}$ for $m\le q+3$, by claim 2 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities applied to the base $1/\sqrt{s}\ge1$ and the exponents $m\le q+3$). The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral §derivative applied to $G=g_{s}$ shows that $g_{s}*\mu$ is of class $C^{1}$ with $\partial_{i}(g_{s}*\mu)=(\partial_{i}g_{s})*\mu$; applied to $G=\partial_{i}g_{s}$ it shows that $(\partial_{i}g_{s})*\mu$ is of class $C^{1}$ with $\partial_{j}\partial_{i}(g_{s}*\mu)=(\partial_{j}\partial_{i}g_{s})*\mu$; and applied to $G=\partial_{j}\partial_{i}g_{s}$ it shows that this is of class $C^{1}$ with $\partial_{k}\partial_{j}\partial_{i}(g_{s}*\mu)=(\partial_{k}\partial_{j}\partial_{i}g_{s})*\mu$, continuous. By clause 2 of C^k Maps on a Euclidean Open Set, applied twice, $g_{s}*\mu$ is of class $C^{3}$. The displayed formulas are the identities just obtained, and the integrands are bounded Borel functions of $x$ by The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral §continuous. The bounds follow from claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space and the pointwise bounds of The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails §bounds; and $g_{s}*\mu\ge0$ because its integrand is nonnegative (claim 2 of Linearity and Monotonicity of the Lebesgue Integral, monotonicity applied against the zero function).

Claim 2. The function $(y,x)\mapsto g_{s}(y-x)$ is nonnegative and $\mathcal{B}(\mathbb{R}^{q})\otimes\mathcal{B}(\mathbb{R}^{q})$-measurable (preamble). By Tonelli and Fubini Theorems (Tonelli) the function $y\mapsto\int g_{s}(y-x)\,\mu(dx)=(g_{s}*\mu)(y)$ is Borel and

using $\int g_{s}(y-x)\,dy=1$ for every $x$ (claim 1 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder, with the evenness $g_{s}(y-x)=g_{s}(x-y)$) and claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space. Being nonnegative, Borel and of finite integral, $g_{s}*\mu$ is integrable. Now let $\phi$ be bounded Borel with $|\phi|\le M$. The function $(y,x)\mapsto\phi(y)g_{s}(y-x)$ is measurable for the product $\sigma$-algebra (product of two such functions, claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), and $\int\int|\phi(y)|g_{s}(y-x)\,\mu(dx)\,dy\le M\int g_{s}*\mu\,d\lambda_{q}=M<\infty$ by claim 1 of Linearity and Monotonicity of the Lebesgue Integral; so by Tonelli it is integrable with respect to $\lambda_{q}\otimes\mu$, and by Fubini (Tonelli and Fubini Theorems) its two iterated integrals coincide:

the inner integral being $(g_{s}*\phi)(x)$ by the evenness of $g_{s}$. Here $\phi\cdot(g_{s}*\mu)$ is integrable, being Borel and dominated by $M\,g_{s}*\mu$; $g_{s}*\phi$ is Borel, being the difference of the Tonelli section integrals of $\phi^{+}g_{s}$ and $\phi^{-}g_{s}$; and $|(g_{s}*\phi)(x)|\le\int g_{s}(x-y)|\phi(y)|\,dy\le M$ by claim 2 of Linearity and Monotonicity of the Lebesgue Integral and $\int g_{s}(x-y)\,dy=1$.

Claim 3. Fix $x$. Since $\int g_{s}(x-y)\,dy=1$, claim 2 of Linearity and Monotonicity of the Lebesgue Integral gives

For $y$ with $\lVert x-y\rVert\le r$ the integrand is at most $\epsilon\,g_{s}(x-y)$; for $y$ with $\lVert x-y\rVert>r$ it is at most $2M\,g_{s}(x-y)$ (claim 5 of Properties of the Absolute Value in an Ordered Field). Hence, by monotonicity and additivity (claim 1 of Linearity and Monotonicity of the Lebesgue Integral), with $A_{r}=\{z:r<\lVert z\rVert\}$ and $\mathbf{1}_{A_{r}}(x-y)$ the indicator of the second set of $y$'s,

where the substitution $z=x-y$ is the reflection and translation invariance of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$, and the last inequality is The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails §moments.

Claim 4. Let $0<s\le\tfrac12$ and $\nu\in\mathcal{P}(\mathbb{R}^{q})$. The function $(y,z)\mapsto g_{s}(y-\mathrm{pr}_{1}(z))\,g_{s}(y-\mathrm{pr}_{2}(z))$ on $\mathbb{R}^{q}\times\mathbb{R}^{q+q}$ is nonnegative and measurable for $\mathcal{B}(\mathbb{R}^{q})\otimes\mathcal{B}(\mathbb{R}^{q+q})$: it is the composition with $\iota^{q,q+q}$ of a function on $\mathbb{R}^{q+(q+q)}$ that is a product of two functions of the form $G\circ(\text{Borel map})$ with $G=g_{s}$ (the maps $w\mapsto\mathrm{pr}^{q,q+q}_{1}(w)-\mathrm{pr}_{1}(\mathrm{pr}^{q,q+q}_{2}(w))$ and $w\mapsto\mathrm{pr}^{q,q+q}_{1}(w)-\mathrm{pr}_{2}(\mathrm{pr}^{q,q+q}_{2}(w))$ are Borel, as compositions and differences of Borel maps). For fixed $y$, by Tonelli on $\mu\otimes\nu$ and the image-measure identity of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product,

the inner integral being $g_{s}(y-x)\,(g_{s}*\nu)(y)$ by claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the outer one then $(g_{s}*\nu)(y)\,(g_{s}*\mu)(y)$. For fixed $z$, with $x=\mathrm{pr}_{1}(z)$ and $x'=\mathrm{pr}_{2}(z)$, $\int g_{s}(y-x)g_{s}(y-x')\,dy=g_{2s}(x-x')$ by The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails §convolution. Tonelli on $\lambda_{q}\otimes(\mu\boxtimes\nu)$, applied through the image measure as before (or directly, $\mu\boxtimes\nu$ being a probability measure on $\mathbb{R}^{q+q}$), therefore gives

and the right side is at most $A_{0}(2s)^{-q/2}$ (The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails §bounds, $2s\le1$, claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space), so the nonnegative Borel product $(g_{s}*\mu)(g_{s}*\nu)$ (claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) is integrable. That $z\mapsto g_{2s}(\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z))$ is Borel and bounded by $A_{0}(2s)^{-q/2}$ was noted in the preamble and in The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails §bounds.