Proposition 9

The measure $\nu$ characterized by lemma 4 is a Haar measure on $\widehat{G}$.

Let $\chi \in \widehat{G}$. For $f$ and $g$ in A(G), let us apply formula (21) to $\chi f$ and $\chi g$. The right-hand side is unchanged, whence

From formulas (12) of II, p. 208 and (13) of II, p. 208, one gets

One then deduces from INT, VIII, §4, No.$^o3$, prop. 7 that

that is to say

Since the space $\widehat{A}(G)$ is dense in $\mathscr{C}_0(\widehat{G})$ (cor. 1), it follows that one has the equality

The measures $\nu$ and $\varepsilon_{\chi^{-1}}*\nu$ therefore coincide on the open set $\Omega_f$ where $\mathscr{F}(f)$ is not zero. By corollary 2 and INT, III, §2, No.$^o1$, cor. of prop. 1, these measures are therefore equal.

This shows that the measure $\nu$ is proportional to a Haar measure on $\widehat{G}$. Let $f\in A(G)$. Take $g=\widetilde{f}$ in formula (21). It then takes the form

which shows that the measure $\nu$ is not zero. The measure $\nu$ is therefore a Haar measure on $\widehat{G}$.