Lemma 9

The Fourier transform is a continuous mapping of $\mathscr{M}^1(G)$ endowed with the weak topology into $\mathscr{C}_b(\widehat{G})$ endowed with the topology induced by the weak topology $\sigma (L^{\infty}(\widehat{G}),L^1(\widehat{G}))$.

Let $\mathfrak{F}$ be a filter on $\mathscr{M}^1(G)$ converging weakly to a bounded measure $\nu$ on G. For every $\varphi \in L^1(\widehat{G})$, one has $\mathscr{F}(\varphi )\in \mathscr{C}_0(G)$ (Prop. 4 of II, p. 209), hence

= lim$_{\eta ,\mathfrak{F}}\int_G\mathscr{F}(\varphi )d\eta =$ lim$_{\eta ,\mathfrak{F}}\int_{\widehat{G}}\varphi \mathscr{F}(\eta )d\widehat{\mu}$,

by the transposition property of the Fourier transform (Prop. 13 of II, p. 221). The lemma follows.