Exercise 9, § 3

a) Let G be a locally compact group, B a bounded subset of $\mathscr{M}(G)$, C an equicontinuous subset of $\mathscr{C}'(G)$; show that if $\mathscr{C}'(G)$ is equipped with the weak topology $\sigma(\mathscr{C}'(G), \mathscr{C}(G))$, and $\mathscr{M}(G)$ with the vague topology $\sigma(\mathscr{M}(G), \mathscr{K}(G))$, then the mapping $(\mu, \nu) \mapsto \mu * \nu$ of $B \times C$ into $\mathscr{M}(G)$ is continuous. (Observe that all of the measures $\nu \in C$ have their support in a common compact set, and make use of Ch. III, §4, No. 3, remarks following Prop. 6.)

b) If, in $\mathscr{M}^1(\mathbf{R})$, one sets $\mu_n = \varepsilon_n$, $\nu_n = \overline{\varepsilon_{-n}}$, the sequences $(\mu_n)$ and $(\nu_n)$ tend vaguely to 0 for the weak topology $\sigma(\mathscr{M}^1(\mathbf{R}), \mathscr{K}(\mathbf{R}))$, but the sequence $(\mu_n * \nu_n)$ does not tend to 0 for the vague topology.