Proposition 1

a) Let M be a measurable subset of $\widehat{G}$. For every $x\in G$, the space $E_M$ is stable under the mapping $f\mapsto \varepsilon_x*f$;

b) Let M and N be measurable subsets of $\widehat{G}$. We have $E_M= E_N$ if and only if M and N are equal up to a locally negligible set;

c) Every subspace of $L^2(G)$ stable under the mappings $f\mapsto \varepsilon_x*f$ for every $x\in G$ is of the form $E_M$ for some measurable subset M of $\widehat{G}$.

This result will be proved later (cf. V, to appear).