Exercise 19, § 5

Let $\mu$ be a positive measure on a locally compact space $X$. Let $f$ be a $\mu$-measurable mapping of $X$ into a complete metric space $F$. In order that, in a compact subset $K$ of $X$, $f$ may be uniformly approximated by measurable step functions, it is necessary and sufficient that $f(K)$ be relatively compact in $F$.