Exercise 7, § 5

a) Let $\mu$ be a positive measure on $T$, and let $A$ be a $\mu$-integrable set such that $\mu(A) > 0$. Show that if, for every $\mu$-integrable set $B \subset A$, either $\mu(B) = 0$ or $\mu(B) = \mu(A)$, then there exists a point $a \in A$ such that $\mu(\{a\}) = \mu(A)$. (Consider the intersection of the compact sets $K \subset A$ such that $\mu(K) = \mu(A)$; show that it is nonempty, has measure equal to $\mu(A)$, and reduces to a single point.)

b) Suppose that $\mu$ is a diffuse measure. For every $\mu$-integrable set $A$ such that $\mu(A) > 0$, and every $\varepsilon > 0$, show that there exists a $\mu$-integrable subset $B$ of $A$ such that $0 < \mu(B) \leq \varepsilon$ (observe, with the help of $a$), that there exists a $\mu$-integrable subset $C$ of $A$ such that $0 < \mu(C) \leq \frac{1}{2} \mu(A)$). From this, deduce that when $X$ runs over the set of $\mu$-integrable subsets of $A$, the set of values of $\mu(X)$ is the closed interval $[0, \mu(A)]$ (for every number $\beta$ such that $0 < \beta < \mu(A)$, let $\gamma$ be the supremum of the measures of the measurable subsets $X$ of $A$ such that $\mu(X) \leq \beta$; first show that $\gamma = \beta$, using the preceding result, then prove that there exists an increasing sequence $(X_n)$ of measurable subsets of $A$ such that $\lim_{n \to \infty} \mu(X_n) = \beta$).