Exercise 4, § 1

Let $X$ be a locally compact space, $\alpha$ a finite numerical function $\geqslant 0$ on $X$ such that, for every compact subset $K$ of $X$, $\sum_{x \in K} \alpha(x)$ is finite; let $\mu$ be the positive measure on $X$ defined by the masses $\alpha(x)$ (Ch. III, §1, No. 3, Example I).

a) Show that for every function $f \geqslant 0$, lower semi-continuous on $X$, one has $\mu^*(f) = \sum_{x \in X} \alpha(x)f(x)$, with the convention that $\alpha(x)f(x) = 0$ when $\alpha(x) = 0$ and $f(x) = +\infty$.

b) Let $f \geqslant 0$ be any numerical function defined on $X$. Show that if $\mu^*(f) < +\infty$, then $\mu^*(f) = \sum_{x \in X} \alpha(x)f(x)$, with the same convention as in a). (Cf. Exer. 5.)