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

Adopt Probability Measures on Euclidean Space and Random Vectors: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, and fix a dimension $q$; a scalar written $\mu$ in the latter setting is written $t$ here, and the letter $\mu$ is reserved for a probability measure on $\mathbb{R}^{q}$. Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{q})$ be a test function, with gradient map $\nabla\psi$, whose components are the partial derivatives $\partial_{i}\psi$, $i\in[q]$, and Laplacian $\Delta\psi$. Continuity is that of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (iv), compact support that of Test Functions on Euclidean Space, Their Gradient Maps and Laplacians (i), and for $\mu\in\mathcal{P}(\mathbb{R}^{q})$ integrable with respect to $\mu$ is integrable on the measure space $(\mathbb{R}^{q},\mathcal{B}(\mathbb{R}^{q}),\mu)$. Sums and scalar multiples of points of $\mathbb{R}^{q}$ are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (i).

1. (Gradient)(i) For every $i\in[q]$ the function $\partial_{i}\psi$ is continuous, compactly supported and bounded. The map $\nabla\psi$ is Borel, and there is a real number $K\ge0$ with $\lVert\nabla\psi(x)\rVert\le K$ for every $x\in\mathbb{R}^{q}$. Consequently the function $x\mapsto\lVert\nabla\psi(x)\rVert^{2}$ is Borel and bounded, and for every $\mu\in\mathcal{P}(\mathbb{R}^{q})$ it is integrable with respect to $\mu$, with

$$\int_{\mathbb{R}^{q}}\lVert\nabla\psi\rVert^{2}\,d\mu\le K^{2}.$$

2. (Laplacian)(ii) The function $\Delta\psi$ is continuous, compactly supported, bounded and Borel; hence it is integrable with respect to every $\mu\in\mathcal{P}(\mathbb{R}^{q})$.

3. (Linearity)(iii) The set $C_{c}^{\infty}(\mathbb{R}^{q})$ is a linear subspace of the vector space of real-valued maps on $\mathbb{R}^{q}$, and for all $\psi,\phi\in C_{c}^{\infty}(\mathbb{R}^{q})$, $a,b\in\mathbb{R}$ and $x\in\mathbb{R}^{q}$,

$$\nabla(a\psi+b\phi)(x)=a\,\nabla\psi(x)+b\,\nabla\phi(x),\qquad \Delta(a\psi+b\phi)(x)=a\,\Delta\psi(x)+b\,\Delta\phi(x).$$