Exercise 10, § 5

Let $F$ be a Banach space, $f$ a measurable mapping of $X$ into $F$, such that the set $A$ of $x \in X$ where $f(x) \neq 0$ is a countable union of integrable sets. Show that there exists a sequence $(f_n)$ of continuous functions with compact support, with values in $F$, such that the sequence $(f_n(x))$ converges to $f(x)$ almost everywhere in $X$ (note on the one hand that $A$ is the union of a negligible set $N$ and a sequence of pairwise disjoint compact sets $K_n$ such that the restriction of $f$ to each $K_n$ is continuous; and on the other hand, that there exists a decreasing sequence $(U_n)$ of open sets containing $A$ such that $\mu(U_n \cap \complement A)$ tends to 0 as $n$ tends to infinity).