Proposition 18

— If $F$ is metrizable, and $A$ is the union of a locally $\mu$-negligible set and a sequence $(A_n)$ of $\mu$-integrable sets, then the space $S(A, \mu; F)$ is metrizable.

Since each $A_n$ is the union of a negligible set and a sequence of compact sets, we can suppose that the $A_n$ are already compact and pairwise disjoint. Prop. 17 then allows us to suppose that $A$ is compact. If $(V_n)$ is a countable fundamental system of entourages of $F$, it is clear that the $W(V_n, A, 1/n)$ form a fundamental system of entourages of $S(A, \mu; F)$ as $n$ runs over $\mathbf{N}$, whence the proposition.