Exercise 11, § 5

Let $S$ be a compact Stone space (Ch. II, §1, Exer. 13 f)); a positive measure $\mu$ on $S$ is said to be normal if, for every increasing directed family $(f_\alpha)_{\alpha \in A}$ of continuous functions on $S$ that is bounded above in $\mathscr{C}(S)$, and whose supremum in the fully lattice-ordered space $\mathscr{C}(S)$ is $f$ (which supremum is not necessarily equal to the upper envelope of the family $(f_\alpha)$), one has $\mu(f) = \sup_{\alpha \in A} \mu(f_\alpha)$. A real measure $\lambda$ on $S$ is said to be normal if the positive measures $\lambda^+$ and $\lambda^-$ are normal.

a) For a positive measure $\mu$ on $S$ to be normal, it is necessary and sufficient that every nowhere dense subset $A$ be $\mu$-negligible (to see that the condition is necessary, consider the sets in $S$ containing $A$ that are open and closed; to see that the condition is sufficient, observe that the upper envelope and the supremum in $\mathscr{C}(S)$ of an increasing directed family $(f_\alpha)$ that is bounded above are equal on the complement of a meager set (cf. Ch. II, §1, Exer. 13 f)). From this, deduce that the support of $\mu$ is both open and closed.

b) Let $\mu$ be a normal positive measure on $S$, $f$ a $\mu$-measurable numerical function, and $g$ the largest lower semi-continuous function on $S$ that is $\leq f$. Show that $f$ and $g$ are equal almost everywhere for $\mu$ (cf. Ch. IV, §5, Exer. 16 b)). From this, deduce that for every $\mu$-measurable subset $A$ of $S$, $A - \overset{\circ}{A}$ and $\overline{A} - A$ are $\mu$-negligible.

c) Show that, in the Banach space $\mathscr{M}(S)$ of measures on $S$, the set of normal measures is a closed linear subspace and is a band for the fully lattice-ordered space structure of $\mathscr{M}(S)$.