Proposition 17

— Let G be a locally compact group, A a subset of G, measurable and not locally negligible for a Haar measure. Then $A \cdot A^{-1}$ is a neighborhood of e.

Let $\beta$ be a left Haar measure. There exists a compact subset K of G such that $B = A \cap K$ is integrable with measure > 0 for $\beta$. Let us apply the Cor. of Prop. 15 with $f = g = \varphi_B$. The function $F = \varphi_B * \check{\varphi}_B$ is continuous and > 0 at e. Therefore there exists a neighborhood V of e such that $F(x) > 0$ for $x \in V$. Now,

Therefore, for $x \in V$, one has $B \cap xB \neq \emptyset$, whence $x \in B \cdot B^{-1}$. Thus $V \subset B \cdot B^{-1} \subset A \cdot A^{-1}$.