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 (ii)), 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 Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square. 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's and Fubini's 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 Measurable Rectangles and the Product σ-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 (iii): 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 of Measurable Real- and [0,∞]-Valued Maps). 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 (i), claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps, Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii)); 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 (i), 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 (ii) and The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails (iii), 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 (ii) 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 Functions of Class Cᵏ on an Open Subset of ℝⁿ, 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 (i). 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 (iii); and $g_{s}*\mu\ge0$ because its integrand is nonnegative (claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, 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's and Fubini's Theorems (Tonelli) the function $y\mapsto\int g_{s}(y-x)\,\mu(dx)=(g_{s}*\mu)(y)$ is Borel and

$$\int_{\mathbb{R}^{q}}g_{s}*\mu\,d\lambda_{q}=\int_{\mathbb{R}^{q}}\Bigl(\int_{\mathbb{R}^{q}}g_{s}(y-x)\,dy\Bigr)\mu(dx)=\int_{\mathbb{R}^{q}}1\,\mu(dx)=1,$$

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 of Measurable Real- and [0,∞]-Valued Maps), 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, and the Integrability Criterion; so by Tonelli it is integrable with respect to $\lambda_{q}\otimes\mu$, and by Fubini (Tonelli's and Fubini's Theorems) its two iterated integrals coincide:

$$\int_{\mathbb{R}^{q}}\phi(y)\,(g_{s}*\mu)(y)\,dy=\int_{\mathbb{R}^{q}}\Bigl(\int_{\mathbb{R}^{q}}\phi(y)\,g_{s}(y-x)\,dy\Bigr)\mu(dx)=\int_{\mathbb{R}^{q}}(g_{s}*\phi)(x)\,\mu(dx),$$

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 the Integrability Criterion 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, and the Integrability Criterion gives

$$\bigl|(g_{s}*\phi)(x)-\phi(x)\bigr|=\Bigl|\int_{\mathbb{R}^{q}}g_{s}(x-y)\bigl(\phi(y)-\phi(x)\bigr)\,dy\Bigr|\le\int_{\mathbb{R}^{q}}g_{s}(x-y)\,|\phi(y)-\phi(x)|\,dy .$$

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, and the Integrability Criterion), with $A_{r}=\{z:r<\lVert z\rVert\}$ and $\mathbf{1}_{A_{r}}(x-y)$ the indicator of the second set of $y$'s,

$$\int g_{s}(x-y)\,|\phi(y)-\phi(x)|\,dy\le\epsilon\int g_{s}(x-y)\,dy+2M\int\mathbf{1}_{A_{r}}(x-y)\,g_{s}(x-y)\,dy=\epsilon+2M\int\mathbf{1}_{A_{r}}(z)\,g_{s}(z)\,dz\le\epsilon+\frac{2Mqs}{r^{2}},$$

where the substitution $z=x-y$ is the reflection and translation invariance of Translation and Reflection Invariance of Lebesgue Measure on ℝⁿ, and the last inequality is The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails (v).

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 (iii),

$$\int_{\mathbb{R}^{q+q}}g_{s}(y-\mathrm{pr}_{1}(z))\,g_{s}(y-\mathrm{pr}_{2}(z))\,(\mu\boxtimes\nu)(dz)=\int\Bigl(\int g_{s}(y-x)\,g_{s}(y-x')\,\nu(dx')\Bigr)\mu(dx)=(g_{s}*\mu)(y)\,(g_{s}*\nu)(y),$$

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 Integrability Criterion, 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 (iv). 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

$$\int_{\mathbb{R}^{q}}(g_{s}*\mu)(g_{s}*\nu)\,d\lambda_{q}=\int_{\mathbb{R}^{q+q}}g_{2s}\bigl(\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\bigr)\,(\mu\boxtimes\nu)(dz),$$

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 (iii), $2s\le1$, 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 of Measurable Real- and [0,∞]-Valued Maps) 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 (iii).