Test Functions Approximate Bounded Continuous Functions Pointwise, Determine a Finite Borel Measure, and Detect a Vanishing Density

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $C_{c}^{\infty}(\mathbb{R}^{d})$ be the set of test functions on $\mathbb{R}^{d}$, let $\mathcal{B}(\mathbb{R}^{d})$ be the Borel $\sigma$-algebra of $\mathbb{R}^{d}$, which is that of the metric space $(\mathbb{R}^{d},d_{E})$, and call a Borel measure $\mu$ on $(\mathbb{R}^{d},d_{E})$ finite if $\mu(\mathbb{R}^{d})<\infty$. Continuous and bounded are as fixed in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii) and Bounded Real-Valued Function on a Set, and Borel and integrable as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii) and Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii).

1. (Pointwise approximation of a bounded continuous function)(i) Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be continuous on $\mathbb{R}^{d}$ and let $M$ be a real number with $|f(x)|\le M$ for every $x\in\mathbb{R}^{d}$. Then there is a sequence $(\psi_{n})_{n\in\mathbb{N}}$ in $C_{c}^{\infty}(\mathbb{R}^{d})$ such that $|\psi_{n}(x)|\le M$ for every $n\in\mathbb{N}$ and every $x\in\mathbb{R}^{d}$, and such that for every $x\in\mathbb{R}^{d}$ the sequence $(\psi_{n}(x))_{n\in\mathbb{N}}$ converges to $f(x)$.

2. (Determination of a finite Borel measure)(ii) Let $\mu$ and $\nu$ be finite Borel measures on $(\mathbb{R}^{d},d_{E})$ such that

$$\int_{\mathbb{R}^{d}}\psi\,d\mu=\int_{\mathbb{R}^{d}}\psi\,d\nu\qquad\text{for every }\psi\in C_{c}^{\infty}(\mathbb{R}^{d}),$$

the integrals existing because each $\psi$ is Borel and, being continuous with compact support, bounded, so that claim 6 of Borel Measurability and Bounded Integration on a Metric Space applies. Then $\mu(B)=\nu(B)$ for every $B\in\mathcal{B}(\mathbb{R}^{d})$.

3. (A density detected by test functions)(iii) Let $\mu\in\mathcal{P}(\mathbb{R}^{d})$ and let $g:\mathbb{R}^{d}\to\mathbb{R}$ be Borel and integrable with respect to $\mu$. For $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ the product $g\psi$ is Borel and integrable with respect to $\mu$, being dominated in absolute value by a real multiple of $|g|$, since $\psi$ is bounded. Suppose that

$$\int_{\mathbb{R}^{d}}g\,\psi\,d\mu=0\qquad\text{for every }\psi\in C_{c}^{\infty}(\mathbb{R}^{d}).$$

Then the set $\{x\in\mathbb{R}^{d}:g(x)\ne0\}$, which belongs to $\mathcal{B}(\mathbb{R}^{d})$, satisfies

$$\mu\bigl(\{x\in\mathbb{R}^{d}:g(x)\ne0\}\bigr)=0 .$$