Exercise 5, § 1

Let $\mu\in \mathscr{M}^1(G)$. Suppose that $\mathscr{F}(\mu)\in L^1(\widehat{G})$. Then $\mu$ is a measure based on the Haar measure on G (INT, V, § 5, n$^o2$, def. 2), and admits a continuous density with respect to this measure. (Use the filter basis $\mathfrak{B}$ of prop. 8 of II, p. 211. Now prove that lim$_{\varphi ,\mathfrak{B}}\varphi *\mu=\mu$ for the vague topology, and for $\varphi$ in an element of $\mathfrak{B}$, one has $\mathscr{F}(\varphi *\mu) =\mathscr{F}(\varphi )\mathscr{F}(\mu)$ ; but $\mathscr{F}(\mu)\in$ $L^1(\widehat{G})$, hence $\varphi *\mu=\overline{\mathscr{F}}(\mathscr{F}(\varphi *\mu))$ tends to $\overline{\mathscr{F}}(\mathscr{F}(\mu))$ in $\mathscr{C}_0(G)$. Thus $d\mu(x) =f(x)dx$ where $f=\overline{\mathscr{F}}(\mathscr{F}(\mu))$.)