Exercise 6, § 5

a) Let $\mu$ be a positive measure on $T$. If $A \subset T$ is universally measurable and not locally $\mu$-negligible, show that there exists a positive measure $\nu$ carried by $A$ such that $\nu \neq 0$ and $\nu \leq \mu$. (Take $\nu = \varphi_K \cdot \mu$, where $K$ is a compact subset of $A$ that is not $\mu$-negligible.)

b) Let $M$ be a universally measurable subset of $T$. In order that a positive measure $\lambda$ on $T$ be carried by $M$, it is necessary and sufficient that $\lambda$ be alien to every positive measure carried by $\mathbf{C}M$ (make use of $a$).

c) Deduce from $b$) that the measures $\rho$ on $T$, such that $|\rho|$ is carried by $M$, form a band in $\mathscr{M}(T)$. This band is vaguely closed in $\mathscr{M}(T)$ if $M$ is a closed subset of $T$ (Ch. III, §2, Prop. 6).

d) Show that Lebesgue measure on $I = [0,1]$ is the vague limit of a sequence of atomic measures carried by a fixed countable set $A \subset I$ (cf. Ch. III, §2, Th. 1 and Prop. 13). The band of measures carried by $A$ is therefore not vaguely closed.