Exercise 22, § 4

Notations are those of Exer. 21 of §4 and Exer. 11 of §3.

a) Let (μ_n) be a sequence of bounded measures on G. Assume that for every function f ∈ L^1(G, β), the sequence (μ_n \* f) converges to 0 in $\mathscr{P}_I$. Show that the sequence of norms (||μ_n||) is bounded (use the Banach–Steinhaus theorem for the family of mappings f ↦ μ_n \* f of L^1(G, β) into itself, as well as Exer. 15). Deduce from this that the sequence (μ_n) tends to 0 in $\mathscr{P}_I$ (use Exer. 20).

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

c) Let (μ_n) be a sequence of bounded measures on G such that for every function f ∈ L^1(G, β), the sequence (μ_n \* f) converges to 0 in $\mathscr{P}_{II}$; show that the sequence (μ_n) converges to 0 in $\mathscr{P}_{II}$ (make use of a)).