Is $\int_{\mathbb{R}^{d}}{e^{-d(x,C)}dP}$ finite?

Question

Given a probability measure $P$ on $\mathbb{R}^{d}$ a closed subset $C\subset\mathbb{R}^{d}$ , and the function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$ defined by $f(x)=e^{-d(x,C)}$ where $d(x,C)=\inf\limits_{y\in C}{|x-y|}$ , I would like to know if $$\int_{\mathbb{R}^{d}}{e^{-d(x,C)}dP}< +\infty$$

Thank you.

Answer 1

What were your efforts?

However, since I don’t have enough reputation to comment, I will give an answer.

Since $C$ is closed, $f(x) = e^{-d(x,C)}$ is bounded, we have that $\int_{\mathbb{R}^{d}} e^{-d(x,C)} dP \leq \overset{1}{P(\mathbb{R}^{d})} \cdot \sup e^{-d(x,C)} < +\infty$, hence $f$ is summable, as desired.