Exercise 20, § 1

Let G be a locally compact group, $\mu$ a left Haar measure on G, A and B two subsets of G. a) Assume that one of the following two conditions holds: $\alpha)$ A is $\mu$-integrable; $\beta)$ $\mu^*(A) < +\infty$ and B is $\mu$-measurable.

² Régulier, renamed simplifiable in the bound edition of Algèbre. ³ In the cited exercise, monoïde of the French original is translated as "semigroup"; but semi-groupe refers to another concept (A, I, §2, Exer. 11).

Show that, in each of these two cases, the function $f(s) = \mu^*(sA \cap B)$ is uniformly continuous on $G$ for the right uniform structure of $G$. (For any two subsets $M, N$ of $G$, one sets

First consider the case that $A$ is compact; making use of Ch. IV, §4, No. 6, Th. 4, show that for every $\varepsilon > 0$, there exists a neighborhood $U$ of $e$ in $G$ such that, for every $s \in G$ and $t \in U$, Then apply Exer. 13 of Ch. IV, §5. If $B$ is $\mu$-measurable, $\mu^*(A) < +\infty$ and $(A_n)$ is a decreasing sequence of integrable subsets of $G$ containing $A$ such that $\inf (\mu(A_n)) = \mu^*(A)$, show that (Ch. V, 1st edn., §2, No. 2, Lemma 1);⁴ note on the other hand that $\mu(A_n - A_{n+1})$ tends to 0 with $1/n$.

b) If $A^{-1}$ is $\mu$-integrable and $\mu^*(B) < +\infty$, then the function $f$ is uniformly continuous for both the right and left uniform structures of $G$; moreover, one then has $\int_G f(s) d\mu(s) = \mu(A^{-1})\mu^*(B)$. (Reduce to the case that $B$ is integrable; observe then that $\mu(sA \cap B) = \mu(A \cap s^{-1}B)$, that $\varphi_{sA \cap B} = \varphi_{sA}\varphi_B$ and that $\varphi_{sA}(t) = \varphi_{tA^{-1}}(s)$.)

c) Deduce from a) that in the two cases considered, the interiors of $AB$ and $BA$ are not empty if $A$ and $B$ are not negligible. (Cf. Ch. VIII, §4, No. 6, Prop. 17.)

d) In the group $G = \mathbf{SL}_2(\mathbf{R})$, give an example of a compact set $A$ and a $\mu$-measurable set $B$ such that $f(s) = \mu(sA \cap B)$ is not uniformly continuous for the left uniform structure. (Observe that there exist a sequence $(t_n)$ of elements of $G$ tending to $e$ and a sequence $(s_n)$ of elements of $G$ such that the sequence $(s_n^{-1}t_n s_n)$ tends to the point at infinity.)

e) Give an example of two nonmeasurable sets $A, B$ in a locally compact group $G$, of finite outer measure and such that the function $s \mapsto \mu^*(sA \cap B)$ is not continuous (cf. Ch. IV, §4, Exer. 8).