Exercise 22, § 3

The notations are those of Exercise 21 of § 4 and Exercise 11 of § 3.

a) Let $(\mu_n)$ be a sequence of bounded measures on G. Suppose that for every function $f \in L^1(G, \beta)$, the sequence $(\mu_n * f)$ converges to 0 in $\mathscr{M}_I$. Now prove that the sequence of norms $(\|\mu_n\|)$ is bounded (use the Banach-Steinhaus theorem for the family of mappings $f \to \mu_n * f$ from $L^1(G, \beta)$ into itself, as well as Exercise 15). Deduce that the sequence $(\mu_n)$ tends to 0 in $\mathscr{M}_I$ (use Exercise 20).

b) Let $(\mu_n)$ be a sequence of bounded measures on G such that the sequence $(\mu_n * f)$ tends vaguely to 0 for every function $f \in L^1(G, \beta)$; now prove that the sequence $(\mu_n)$ tends vaguely to 0 (for every compact subset K of G, prove, reasoning as in a), that the sequence $(|\mu_n|(K))$ is bounded). Give an example (with $G = \mathbf{Z}$) where the sequence $(\|\mu_n\|)$ is not bounded.

c) Let $(\mu_n)$ be a sequence of bounded measures on G such that for every function $f \in L^1(G, \beta)$, the sequence $(\mu_n * f)$ converges to 0 in $\mathscr{M}_{II}$; now prove that the sequence $(\mu_n)$ converges to 0 in $\mathscr{M}_{II}$ (use a).