Exercise 3, § 6

Let $\mu$ be a measure on a compact space $X$, such that $X$ is equal to the support of $\mu$. Show that if every measurable numerical function that is bounded in measure is equal almost everywhere to a continuous function, then $X$ is a Stone space (cf. Ch. II, §1, Exer. 13) (consider the characteristic function of a compact set). Conversely, if $X$ is a Stone space, in order that every measurable function that is bounded in measure be equal almost everywhere to a continuous function, it is necessary and sufficient that every nowhere dense set in $X$ be negligible (cf. §5, Exer. 18).