Exercise 13, § 5

*Let G be a locally compact abelian group. One denotes by $\widehat{G}$ the dual group of G and by $\alpha$ a Haar measure on $\widehat{G}$. For every bounded measure $\mu$ on G, one denotes by $\mathscr{F} \mu$ the Fourier transform of $\mu$, which is a function on the group $\widehat{G}$ (cf. Théor. spec., Ch. II, §1, No. 2).

a) The Fourier transformation $\mathscr{F}$ is a continuous mapping of $\mathscr{M}_+^b(G)$ equipped with the tight topology into $\mathscr{C}^b(\widehat{G})$ equipped with the topology of compact convergence. (Make use of the Cor. of Prop. 13 of No. 6.)

b) Let $\mathscr{F}$ be a filter on $\mathscr{M}_+^b(G)$ and $\Phi$ a bounded continuous function on $\widehat{G}$; assume that $\lim_{\lambda, \mathscr{F}} (\mathscr{F} \lambda) \cdot \alpha = \Phi \cdot \alpha$ (vague convergence) and $\lim_{\lambda, \mathscr{F}} (\mathscr{F} \lambda)(0) = \Phi(0)$. Show that the filter $\mathscr{F}$ converges tightly to a measure $\mu$ such that $\mathscr{F} \mu = \Phi$ (show by application of the Stone–Weierstrass theorem that the set of Fourier transforms of continuous functions on $G$ with compact support is a dense linear subspace of $C^0(\widehat{G})$ for uniform convergence; next observe that $\lim_{\lambda, \mathscr{F}} \int (\mathscr{F} \lambda) \cdot u d\alpha = \int \Phi u d\alpha$ for every $u \in E$ and that the filter $\mathscr{F}$ contains a vaguely compact set of bounded measures; deduce from this that the filter $\mathscr{F}$ has at most one cluster point, then that it converges tightly).

c) Let $\mathscr{F}$ be a filter on $\mathscr{M}_+^b(G)$ having a countable base. In order that $\mathscr{F}$ converge tightly to a bounded positive measure $\mu$, it is necessary and sufficient that there exist a continuous function $\Phi$ on $\widehat{G}$ such that $\mathscr{F} \lambda$ converges pointwise to $\Phi$ with respect to $\mathscr{F}$, in which case $\mathscr{F} \mu = \Phi$ (P. Lévy’s theorem).