A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable

Let $n\ge1$ be a natural number, let $d$ denote the Euclidean distance, a metric on each Euclidean space, let $\lVert\,\cdot\,\rVert$ be the Euclidean norm on $\mathbb{R}^n$, and let $\lambda_n$ be Lebesgue measure on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$. Topological notions on $\mathbb{R}^n$ refer to the topology of the open sets of $d$, which is a topology by The Open Subsets of a Metric Space Form a Topology.

Let $g:\mathbb{R}^n\to\mathbb{R}$ be continuous from $(\mathbb{R}^n,d)$ to $(\mathbb{R},d)$ and compactly supported.

1. (Boundedness) $g$ is bounded.

2. (Integrability and a bound) The function $g$ is measurable with respect to $\mathcal{B}(\mathbb{R}^n)$ and the Borel $\sigma$-algebra of the real line, and it is integrable with respect to $\lambda_n$. Moreover, if $M$ is a bound for $g$ and $R>0$ is a real number such that $g(x)=0$ for every $x$ with $\lVert x\rVert>R$, then the closed ball $\bar B$ of centre $0$ and radius $R$ in $(\mathbb{R}^n,d)$ satisfies $\lambda_n(\bar B)<\infty$ and

$$\int_{\mathbb{R}^n}|g|\,d\lambda_n\le M\,\lambda_n(\bar B).$$

3. (Positivity) If in addition $g(x)\ge0$ for every $x\in\mathbb{R}^n$ and $g(x_1)>0$ for at least one $x_1\in\mathbb{R}^n$, then

$$0<\int_{\mathbb{R}^n}g\,d\lambda_n<\infty.$$