Exercise 12, § 5

A compact Stone space $H$ is said to be hyperstonian if the union of the supports of the normal positive measures on $H$ (Exer. 11) is dense.

a) Let $T$ be a locally compact space, $\mu$ a positive measure on $T$. Show that there exists an isomorphism $u \mapsto \theta_u$ of the Banach space $L^\infty(T, \mu)$ onto the space $\mathscr{C}(H)$ of continuous real-valued functions on a hyperstonian space $H$, such that $\theta_u^+ = \theta_{u^+}$ (cf. Prop. 14 and Ch. II, §1, Exer. 13).

b) Let $H$ be a hyperstonian compact space, and let $(\mu_\iota)_{\iota \in I}$ be a family of normal positive measures on $H$ such that the union of the supports of the $\mu_\iota$ is dense in $H$. Show that, for a set to be nowhere dense in $H$, it is necessary and sufficient that it be $\mu_\iota$-negligible for all $\iota \in I$ (observe that if $A$ is $\mu_\iota$-negligible, then so is $\overline{A}$ (Exer. 11 b)). Deduce from this that in a hyperstonian space, every meager set is nowhere dense.

c) The hypotheses being the same as in b), show that if $f$ is a numerical function $\mu_\iota$-measurable for all $\iota \in I$, there exists a continuous function $g$ on $H$ such that $f$ and $g$ coincide on the complement of a nowhere dense set (use b) and Exer. 11 b)).

d) Let $H$ be a hyperstonian space. Show that there exists a family $(G_\alpha)_{\alpha \in A}$ of nonempty open and closed subsets of $H$, pairwise disjoint, whose union $T$ is dense in $H$, and such that for every $\alpha \in A$, there exists a normal positive measure $\mu_\alpha$ on $G_\alpha$ with support $G_\alpha$ (apply Zorn's lemma to the set of families of normal positive measures on $H$ whose supports are mutually disjoint, and use Exer. 11 a)). Let $\mu$ be the measure on $T$ whose restriction to each $G_\alpha$ is $\mu_\alpha$ (Ch. III, §2, Prop. 1). Show that the mapping that, to every function $f \in \mathscr{C}(H)$, makes correspond the class in $L^\infty(T, \mu)$ of its restriction to $T$, is an isomorphism of the Banach space $\mathscr{C}(H)$ onto $L^\infty(T, \mu)$ (use c) to show that the mapping is surjective).