Exercise 8, § 1

Let $X$ be a locally compact space; for every compact subset $K$ of $X$ and every integer $n > 0$, let $S_{K,n}$ be the set of functions $f \in \mathscr{H}(X; \mathbf{C})$ with support contained in $K$ and such that $\|f\| \leq n$. One calls quasi-strong topology on $\mathscr{M}(X; \mathbf{C})$ the topology of uniform convergence in the sets $S_{K,n}$. It is coarser than the strong topology (when $\mathscr{M}(X; \mathbf{C})$ is regarded as the dual of the space $\mathscr{H}(X; \mathbf{C})$, cf. TVS, III, §3, No. 1) and is identical to it when $X$ is paracompact.

a) Show that the quasi-strong topology on $\mathscr{M}(X; \mathbf{C})$ is defined by the semi-norms $\mu \mapsto |\mu|(f)$, where $f$ runs over $\mathscr{H}_+(X)$; the space $\mathscr{M}(X; \mathbf{C})$ is complete for the quasi-strong topology. The bounded subsets of $\mathscr{M}(X; \mathbf{C})$ for the quasi-strong topology are the equicontinuous subsets of $\mathscr{M}(X; \mathbf{C})$.

b) Show that on $\mathscr{M}(X; \mathbf{R})$, the quasi-strong topology is compatible with the ordered vector space structure (TVS, II, §2, No. 7). From this, deduce that if $H$ is an increasing directed set in $\mathscr{M}(X; \mathbf{R})$ that is bounded above, then the supremum of $H$ in $\mathscr{M}(X; \mathbf{R})$ is identical to the limit, for the quasi-strong topology, of its section filter (loc. cit.).