Lemma 1

Let $\alpha$ be a positive measure $\neq 0$ on G, S its support; the following two conditions are equivalent: a) S is a closed subgroup of G and the measure induced by $\alpha$ on S is a right Haar measure on S. b) $\delta(s)\alpha = \alpha$ for every $s \in S$. Moreover, when these conditions are satisfied, the set of $t \in G$ such that $\delta(t)\alpha = \alpha$ is equal to S. It is clear that a) implies b); conversely, the relation b) implies that $Sx = S$ for every $x \in S$; in other words, the relations $x \in S$ and $y \in S$ imply that $y \in Sx$, or again that $yx^{-1} \in S$, and since S is nonempty, S is a closed subgroup of G. The set of $t \in G$ such that $St = S$ is then equal to S itself, whence the last assertion.

For the rest of the section, we denote by $\Gamma$ the set of positive measures $\neq 0$ on G satisfying the conditions of Lemma 1, and for every $\alpha \in \Gamma$ we denote by $H_\alpha$ the closed subgroup of G that is the support of $\alpha$.