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

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, with Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation in force, fix a dimension $q$ and let $\mu\in\mathcal{P}(\mathbb{R}^{q})$. A function $\mathbb{R}^{q}\to\mathbb{R}$ is bounded as defined there, and continuity of a function on $\mathbb{R}^{q}$ is that of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (iv) with $S=\mathbb{R}^{q}$, which is open by claim 1 of Euclidean Space Is Open in Itself. Let $C$ be a nonnegative real number.

1. (Continuity and bound)(i) Let $H:\mathbb{R}^{q}\to\mathbb{R}$ be continuous with $|H(z)|\le C$ for all $z$. Then for every $y\in\mathbb{R}^{q}$ the function $x\mapsto H(y-x)$ is Borel and integrable with respect to $\mu$, the function

$$H*\mu:\mathbb{R}^{q}\to\mathbb{R},\qquad(H*\mu)(y)=\int_{\mathbb{R}^{q}}H(y-x)\,\mu(dx),$$

is continuous, and $|(H*\mu)(y)|\le C$ for all $y$.

2. (Differentiation under the integral)(ii) Let $G:\mathbb{R}^{q}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}^{q}$ with $|G(z)|\le C$ and $|\partial_{i}G(z)|\le C$ for all $z\in\mathbb{R}^{q}$ and $i\in[q]$; then $G$ and each $\partial_{i}G$ are continuous, so $G*\mu$ and $(\partial_{i}G)*\mu$ are defined by claim 1. Then $G*\mu$ is of class $C^{1}$ on $\mathbb{R}^{q}$ and $\partial_{i}(G*\mu)=(\partial_{i}G)*\mu$ for every $i\in[q]$, that is, $\partial_{i}(G*\mu)(y)=\int_{\mathbb{R}^{q}}\partial_{i}G(y-x)\,\mu(dx)$ for every $y$.