Exercise 25, § 3

Let $G$ be a locally compact group, $\beta$ a left Haar measure on $G$, $\mu$ a positive measure on $G$. If $A$ is a $\mu$-integrable set and $B$ a Borel set in $G$, the function

is $\beta$-measurable in $G$; if in addition $B$ is $\beta$-integrable, the same is true of $u$, and one has $\int \mu(A \cap sB) d\beta(s) = \mu(A)\beta(B^{-1})$. Give an example of a measure $\mu$ such that $s \to \mu(A \cap sB)$ is not continuous; if $\mu$ is a basis measure $\beta$ and if $A$ is $\mu$-integrable and $B$ is Borel, the function $s \to \mu(A \cap sB)$ is continuous in $G$.