Exercise 11, § 1

Let $G$ be a nondiscrete locally compact group countable at infinity, $\beta$ a left Haar measure on $G$. Show that every compact neighborhood $V$ of $e$ contains a normal subgroup $H$ of $G$, $\beta$-negligible and such that $G/H$ is a Polish locally compact group. (One can proceed as follows: let $L$ be the open subgroup of $G$ generated by $V$; let $b_1, b_2, \ldots$ be representatives of the left cosets with respect to $L$. Form a decreasing sequence $(V_n)$ of symmetric neighborhoods of $e$ such that: 1) $V_n^2 \subset V_{n-1}$; 2) $xV_nx^{-1} \subset V_{n-1}$ for every $x \in V$; 3) $b_i V_n b_i^{-1} \subset V_{n-1}$ for $1 \leq i \leq n$; 4) $\beta(V_n) \leq 1/n$. Set $H = V_1 \cap V_2 \cap \cdots$.

Let $(f_n)$ be a sequence of numerical functions uniformly continuous for the left uniform structure of $G$. Show that $H$ can be constructed in such a way that, in addition, the $f_n$ are constant on the cosets of $H$. (In the preceding construction, take $V_n$ to be such that $|f_i(x) - f_i(y)| \leq 1/n$ for $x^{-1}y \in V_n$ and $1 \leq i \leq n$.)

Let $(g_n)$ be a sequence of $\beta$-integrable numerical functions on $G$. Show that $H$ can be constructed in such a way that, in addition, each $g_n$ is equal almost everywhere to a function that is constant on the cosets of $H$.