Exercise 3, § 3

Let $X$ be a locally compact space, $\alpha$ a finite numerical function $\geqslant 0$ defined on $X$ such that, for every compact subset $K$ of $X$, $\sum_{x \in K} \alpha(x) < +\infty$. Let $\mu$ be the measure on $X$ defined by the masses $\alpha(x)$. Show that, for this measure, $\mathscr{F}_F^p = \mathscr{L}_F^p$ for every finite $p \geqslant 1$ and every Banach space $F$.