Exercise 18, § 5

Let $X$ be a locally compact space such that, in $X$, the closure of every relatively compact open set is again an open set (a Stone space; cf. Ch. II, §1, Exer. 13); let $\mu$ be a positive measure on $X$ with support identical to $X$ and such that every bounded integrable numerical function on $X$ is equivalent to a finite continuous function (§4, Exer. 10).

a) Show that, in $X$, every nowhere dense set $N$ is locally negligible (for every compact set $K$, consider the continuous function equivalent to $\varphi_{K \cap \overline{N}}$).

b) Let $f$ be a measurable numerical function (finite or not) on $X$, and let $g$ be the largest lower semi-continuous function on $X$ that is $\leq f$. Show that $f$ and $g$ are equal locally almost everywhere (note that if the restriction of $f$ to a compact set $K$ is continuous then $f$ and $g$ are equal on the interior of $K$, and make use of a)).

c) Deduce from b) that in $X$, every set locally negligible for $\mu$ is a nowhere dense set. In particular, every meager set in $X$ is nowhere dense.