Exercise 1, § 5

Let $T$ be a topological space and $\mu$ a positive measure on $T$. Denote by $\mathscr{S}_\mu$ the set of subsets of $T$ whose boundary is $\mu$-negligible.

a) Show that $\mathscr{S}_\mu$ is a clan.

b) If $T$ is completely regular, every point of $T$ has a fundamental system of neighborhoods contained in $\mathscr{S}_\mu$ (observe that if $f$ is a continuous function on $T$, zero outside a $\mu$-integrable set, then the set of real numbers $r$ such that $f(r)^{-1}$ is not $\mu$-negligible is countable).