Exercise 10, § 3

For a locally compact space T, one denotes by $\mathscr{C}^\infty(T)$ the Banach space of continuous bounded numerical functions on T. A subset H of $\mathscr{M}^1(T)$ is said to be cramped if, for every $\varepsilon > 0$, there exists a compact subset K of T such that $|\mu|(T - K) \leq \varepsilon$ for all $\mu \in H$.

a) Show that if a subset H of $\mathscr{M}^1(G)$ is cramped and is bounded for the topology defined by the norm of $\mathscr{M}^1(T)$, then it is relatively compact for the topology $\sigma(\mathscr{M}^1(T), \mathscr{C}^\infty(T))$ (observe that H is relatively compact for $\sigma(\mathscr{M}^1(T), \mathscr{K}(T))$).

b) Assuming in addition that T is paracompact, show that, conversely, if H is a subset of $\mathscr{M}^1(T)$ relatively compact for $\sigma(\mathscr{M}^1(T), \mathscr{C}^\infty(T))$, then H is bounded for the norm of $\mathscr{M}^1(T)$ and is cramped. (Consider first the case that $T = \mathbf{N}$, applying Exer. 15 of Ch. V, §5. Then contemplate the case that T is countable at infinity, the union of a sequence $(U_n)$ of relatively compact open sets such that $\overline{U}_n \subset U_{n+1}$. Arguing by contradiction, show that one can reduce to the case that there would exist for each n a continuous numerical function $f_n$ defined on T, with support contained in $U_{n+1} - \overline{U}_n$, such that $\|f_n\| \leq 1$, and a sequence $(\mu_n)$ of measures belonging to H and for which $\mu_n(f_n) \geq a > 0$ for all n. Consider then the continuous mapping $u : L^1(\mathbf{N}) \to \mathscr{C}^\infty(T)$ such that $u((\xi_n)) = \sum_{n=0}^\infty \xi_n f_n$ and obtain a contradiction to what was proved for $T = \mathbf{N}$, by considering the transpose of u. Finally, when T is an arbitrary paracompact locally compact space, T is the topological sum of a family $(T_\alpha)$ of locally compact spaces that are countable at infinity; for every $\alpha$, let $m_\alpha = \sup_{\mu \in H} |\mu|(T_\alpha)$; show, arguing by contradiction and making use of the preceding case, that necessarily $m_\alpha = 0$ except for a countable infinity of indices $\alpha$.)

c) Show that the conclusion of b) is not valid for the non-paracompact locally compact space defined in Exer. 16 h) of Ch. IV, §4.