I'm reading a proof relating the Wasserstein topology to the weak* topology (of measures) and there's a part I'm confused about.
"Let $K$ be a compact subset of a locally compact separable space. Since $X$ is locally compact, we can find a family of nonnegative functions $(\phi_i)_{i\in I}\subset C_c(X)$ such that for any $\delta>0$.
$$\begin{cases}\sum_{i\in I} \phi_i\le 1 \text{ in $K$}\\\sum_{i\in I} \phi_i(x) = 1 \text{ for all $x\in K$}\\ \operatorname{diam}(\operatorname{supp}(\phi_i)) \le \delta \text{ for all $i\in I$}\end{cases}$$
I feel like I can believe that this is true since $K$ is compact and $I$ is only finite, so we can kind of just carefully choose functions so that the three properties just work out, but I have no idea what that would look like. I'm also not sure how this is a consequence of local compactness.
Thank you for any help!