Exercise 7, § 4

Let $\mu$ be a positive measure on $X$. For every subset $A \subset X$, the lower integral (Exer. 5) of the characteristic function $\varphi_A$ is called the inner measure of $A$ and is denoted $\mu_*(A)$.

a) Show that $\mu_*(A)$ is the supremum of the measures of the compact sets contained in $A$ (argue as in Th. 4).

b) For every subset $A$ of $X$ of finite outer measure, show that there exist two integrable sets $A_1, A_2$ such that $A_1 \subset A \subset A_2$ and $\mu_*(A) = \mu(A_1), \mu^*(A) = \mu(A_2)$. For $A$ to be integrable, it is necessary and sufficient that $\mu^*(A)$ and $\mu_*(A)$ be finite and equal. With the same notations, show that

and

c) Let $A$ be an integrable set; show that for every set $B \subset A$, one has $\mu(A) = \mu^*(B) + \mu_*(A \cap \mathbf{C}B)$.

d) Let $A$ and $B$ be two sets of finite outer measure that are disjoint. Show that if $C = A \cup B$, then

and that

(for the latter inequality, reduce to the case that $\mu_*(A) = \mu_*(B) = 0$ with the help of b); if $A_2$ and $B_2$ are integrable sets such that $A \subset A_2, B \subset B_2, \mu^*(A) = \mu(A_2), \mu^*(B) = \mu(B_2)$, show that $\mu_*(C) \leq \mu(A_2 \cap B_2)$).