Exercise 23, § 1

Let $G$ be a locally compact abelian group, written additively, $\mu$ a Haar measure on $G$, and $A, B$ two integrable subsets of $G$.

⁴Suppressed from the second edition, the lemma asserts that if $f, g$ are numerical functions $\geq 0$ with $g$ measurable, then $\int^* fg d\mu = \inf \int^* \varphi g d\mu$ as $\varphi$ runs over the set of measurable functions $\geq f$.

a) For every $s \in G$, let

Show that $\mu(A') + \mu(B') = \mu(A) + \mu(B)$ and $A' + B' \subset A + B$.

b) Assume that 0 belongs to $A \cap B$. A pair $(A', B')$ of integrable subsets of $G$ is said to be derived from $(A, B)$ if there exist a sequence $(s_k)_{1 \leq k \leq n}$ of elements of $G$ and two sequences $(A_k)_{0 \leq k \leq n}, (B_k)_{0 \leq k \leq n}$ of subsets of $G$ such that $A_0 = A, B_0 = B, A_k = \sigma_{s_k}(A_{k-1}, B_{k-1}), B_k = \tau_{s_k}(A_{k-1}, B_{k-1})$ for $1 \leq k \leq n, s_k \in A_{k-1}$ for $1 \leq k \leq n$, and $A' = A_n, B' = B_n$. Show that there exists a sequence of pairs $(E_n, F_n)$ such that $E_0 = A, F_0 = B, (E_{n+1}, F_{n+1})$ is derived from $(E_n, F_n)$, and $\mu((E_n - s) \cap F_n) \geq \mu(F_{n+1}) - 2^{-n}$ for every $n$ and every $s \in E_n$. Set $E_\infty = \bigcup_n E_n, F_\infty = \bigcap_n F_n$. Show that for every $s \in E_\infty$,

c) Assume that $\mu(F_\infty) > 0$. Show that the function

can only take on the values 0 and $\mu(F_\infty)$, and that the set $C$ of $s \in G$ such that $f(s) = \mu(F_\infty)$ is open and closed, is such that $\mu(C) = \mu(E_\infty)$, and is the closure of $E_\infty$ (use Exer. 20 a) and b)). On the other hand let $D$ be the set of $s \in F_\infty$ such that the intersection of $F_\infty$ with every neighborhood of $s$ has measure $> 0$. Show that $\mu(D) = \mu(F_\infty)$ and that

from this, deduce that $D$ is contained in the subgroup $H(C)$ defined in Exer. 22, and that $H(C)$ is compact and open in $G$. Finally, show that $C + H(C) = C$, that $\mu(C) \geq \mu(A) + \mu(B) - \mu(H(C))$, and that $C \subset A + B$ (consider the measure of $E_\infty \cap (c - F_\infty)$ for every $c \in C$).

d) Deduce from c) that, for two integrable subsets $A, B$ of $G$: either $\mu_*(A + B) \geq \mu(A) + \mu(B)$; or else there exists a compact open subgroup $H$ of $G$ such that $A + B$ contains a coset of $H$, in which case $\mu_*(A + B) \geq \mu(A) + \mu(B) - \mu(H)$. The case that $G$ is connected.