Exercise 15, § 1

Let $X$ be a noncompact locally compact space; on the space $\mathscr{M}^1(X; \mathbf{C})$ of bounded measures on $X$, the dual of the Banach space $\mathscr{C}^0(X; \mathbf{C})$, the topology defined by the norm $\|\mu\|$ is called the ultrastrong topology, and the topology $\sigma(\mathscr{M}^1(X; \mathbf{C}), \mathscr{C}^0(X; \mathbf{C}))$ the weak topology.

a) Show that if $X$ is paracompact, then the weak topology on $\mathscr{M}^1(X; \mathbf{C})$ is strictly finer than the vague topology (compare with Exer. 2).

b) Show that on $\mathscr{M}^1(X; \mathbf{C})$, the ultrastrong topology is strictly finer than the quasi-strong topology (observe that for the latter, $\varepsilon_x$ tends to $0$ as $x$ tends to the point at infinity of $X$).

c) If $X$ is countable at infinity and is not discrete, show that on $\mathscr{M}^1(X; \mathbf{C})$ the weak topology and the quasi-strong topology are not comparable.

d) In $\mathscr{M}^1(X; \mathbf{C})$, every weakly bounded set is bounded for the ultrastrong topology, but there can exist sets that are bounded for the quasi-strong topology but are not weakly bounded.