Exercise 8, § 5

a) Let $\nu$ be a positive atomic measure on $T$, and let $A$ be a $\nu$-integrable subset of $T$. Show that the set of values of $\nu(X)$, as $X$ runs over the set of $\nu$-integrable subsets of $A$, is closed in $\mathbf{R}$. (Let $M$ be the smallest set carrying $\nu$. Assuming $A \cap M$ infinite and arranging the points of $A \cap M$ in a sequence $(a_n)$, consider the mapping $\varphi$ of the product space $\{0,1\}^\mathbf{N}$ into $\mathbf{R}$ defined by $\varphi((\varepsilon_n)) = \sum_n \varepsilon_n \nu(\{a_n\})$, and show that it is continuous.)

b) Deduce from $a$) and Exer. 7 b) that if $\mu$ is any positive measure on $T$, and $A$ is a $\mu$-integrable subset of $T$, then the set of values of $\mu(X)$, as $X$ runs over the set of $\mu$-integrable subsets of $A$, is closed in $\mathbf{R}$. Extend this result to the case that $\mu$ is any real measure on $T$.

c) Deduce from Exer. 7 b) that if $\mu$ is a diffuse real measure on $T$, then the set of values of $\mu(X) = \mu^+(X) - \mu^-(X)$, where $X$ runs over the set of $|\mu|$-integrable subsets of $T$, is a closed interval of $\mathbf{R}$ (bounded or not).

d) Give an example of an atomic positive measure $\nu$ on a noncompact locally compact space $T$, such that the set of values of $\nu(X)$, where $X$ runs over the set of $\nu$-integrable subsets of $T$, is not closed (take $\nu$ to be such that $\inf_{t \in T} \nu(\{t\}) > 0$).