Proof of The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral

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 $\mathbb{R}^{q}$ 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 $d_{E}(y-x,y-x')=\lVert x'-x\rVert=d_{E}(x',x)=d_{E}(x,x')$ 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 and the symmetry of the metric $d_{E}$ ($\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence), so that the map satisfies the condition of Continuous Map Between Metric Spaces with the same $\delta$ as $\varepsilon$. Continuity of a function on $\mathbb{R}^{q}$ at a point is equivalent to sequential continuity there by Continuity Between Metric Spaces is Equivalent to Sequential Continuity.

Claim 1. For each $y$ the function $x\mapsto H(y-x)$ is Borel as a composition, and bounded by $C$, hence integrable with respect to $\mu$ with $|\int H(y-x)\,\mu(dx)|\le C\,\mu(\mathbb{R}^{q})=C$, by claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space. Let $y_{n}\to y$ in $\mathbb{R}^{q}$. For each $x$, $d_{E}(y_{n}-x,y-x)=d_{E}(y_{n},y)\to0$, so $H(y_{n}-x)\to H(y-x)$ by the sequential continuity of $H$; the functions $x\mapsto H(y_{n}-x)$ are Borel and dominated by the $\mu$-integrable constant $C$ (claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space); so $\int H(y_{n}-x)\,\mu(dx)\to\int H(y-x)\,\mu(dx)$ by Dominated Convergence Theorem. Thus $H*\mu$ is sequentially continuous at $y$, hence continuous on $\mathbb{R}^{q}$.

Claim 2. $G$ and each $\partial_{i}G$ are continuous (clause 1 of Functions of Class Cᵏ on an Open Subset of ℝⁿ read through its clause 3) and bounded by $C$, so by claim 1 the functions $G*\mu$ and $(\partial_{i}G)*\mu$ are defined and continuous. Fix $y\in\mathbb{R}^{q}$ and $i\in[q]$, let $e_{i}$ be the $i$th standard basis vector, so that $y+te_{i}$ is the point whose $i$th coordinate is $y_{i}+t$ and whose other coordinates are those of $y$, and define $f:\mathbb{R}\times\mathbb{R}^{q}\to\mathbb{R}$ by $f(t,x)=G(y+te_{i}-x)$. Let $U=(-1,1)$, the open interval, every point of which is interior to it (Basic Facts about Intervals of the Real Line (iv)). For every $x$ the function $t\mapsto f(t,x)$, restricted to $U$, is differentiable at every $t\in U$ with derivative $\partial_{i}G(y+te_{i}-x)$: the defining condition of Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ for $G$ at the point $y+te_{i}-x$, in which the displaced point $(a_{1},\dots,a_{i}+h,\dots,a_{q})$ is $y+(t+h)e_{i}-x$, is the defining condition of Derivative of a Real Function on an Interval for $t'\mapsto f(t',x)$ at $t$, once $\delta$ is decreased so that $|h|<\delta$ forces $t+h\in U$. Moreover $|\partial_{i}G(y+te_{i}-x)|\le C$ for all $t\in U$ and $x$, with the constant $C$ integrable with respect to $\mu$ (claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space), and $x\mapsto f(t,x)$ is integrable for each $t$ by claim 1. By Differentiation under the Integral Sign, applied on the open interval $U$, the function $F(t)=\int f(t,x)\,\mu(dx)=(G*\mu)(y+te_{i})$, $t\in U$, is differentiable at every $t\in U$ with

$$F'(t)=\int_{\mathbb{R}^{q}}\partial_{i}G(y+te_{i}-x)\,\mu(dx)=\bigl((\partial_{i}G)*\mu\bigr)(y+te_{i}).$$

At $t=0$ the defining condition of Derivative of a Real Function on an Interval for $F$ (with $\delta\le1$) is the defining condition of Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ for $G*\mu$ at $y$ (the displaced point being $y+he_{i}$, which lies in the open set $\mathbb{R}^{q}$); so $\partial_{i}(G*\mu)(y)$ exists and equals $((\partial_{i}G)*\mu)(y)$. As $y$ and $i$ were arbitrary, $\partial_{i}(G*\mu)=(\partial_{i}G)*\mu$, a continuous function, and $G*\mu$ is continuous; hence $G*\mu$ is of class $C^{1}$ on $\mathbb{R}^{q}$ by clauses 1 and 3 of Functions of Class Cᵏ on an Open Subset of ℝⁿ.