Equality of strong topologies for related dual pairs?

Question

Let $H$ be a locally compact Hausdorff space and $C_c(H)$ the space of continuous functions with compact support. Equip $C_c(H)$ with the locally convex inductive limit topology $\tau$ of the Banach spaces $C_K(H)$ of continuous functions with support contained in a compact $K \subseteq H$, where $K$ runs over all compact subsets of $H$. Then $(C_c(H), \tau)' = M(H)$ is the space of real-valued Radon measures defined on the collection of relatively compact Borel sets. [$C_c(H)$ equipped with the (coarser) supremum norm $\lVert \cdot \rVert$ gives rise to the subspace $(C_c(H), \lVert \cdot \rVert)' = M_b(H) \subseteq M(H)$ of real-valued Radon measures defined on the whole Borel $\sigma$-algebra; the measures in $M_b(H)$ have bounded variation (in contrast to $M(H)$).]

Consider the dual pair $\langle C_c(H), M(H) \rangle$. For the strong topology it holds $\beta(C_c(H), M(H)) = \tau\,$ (because $C_c(H)$ is ultrabornological as an inductive limit of Banach spaces, hence barrelled). The linear hull $D := \textrm{span}(\delta_x \mid x \in H)$ of Dirac measures is (weakly-)dense in $M(H)$ and we can consider the dual pairs $\langle C_c(H), D \rangle$ and $\langle C_c(H), M_b(H) \rangle$. I want to know whether

$$\beta(C_c(H), D) = \beta(C_c(H), M(H)) \quad \textrm{or at least} \quad \beta(C_c(H), M_b(H)) = \beta(C_c(H), M(H)).$$

I have the following steps:

1. For a dual pair $\langle Y, X \rangle$ of vector spaces $X, Y$ and a (weakly-)dense subspace $S \subseteq X$ consider the dual pair $\langle Y, S \rangle$. For the strong topologies on $Y$ it then holds $\beta(Y, S) \subseteq \beta(Y, X)$. For the equality $\beta(Y,S) = \beta(Y, X)$ it is sufficient and necessary that every bounded set $B \subseteq X$ is contained in the (weak) closure of a bounded set $A \subseteq S$.

2. For simplicity let us first restrict to the case of a compact Hausdorff space $K = H$. Then $\tau = \lVert \cdot \rVert$ and $C(K)' = M(K)$. Equip $M(K)$ with the weak$^*$-topology. In order to apply 1., let $B \subseteq M(H)$ be a bounded set and show that there exists $A \subseteq D$ with $B \subseteq \overline{A}$. It holds $B \subseteq r \overline{B}_1(0)$ where $r > 0$ and $\overline{B}_1(0) \subseteq M(H)$ is the closed unit ball. It is enough to show that $\overline{B}_1(0) \subseteq \overline{A}$ for some bounded $A \subseteq D$. I know that the closed unit ball $\overline{B}_1(0) \subseteq M(K)$ is a weakly$^*$-compact convex set (Banach-Alaoglu). Also, the mapping $K \to M(K)$, $x \mapsto \delta_x$ is an embedding, so that we can identify $K$ as a compact subspace of $M(K)$. By Krein-Milman, $\overline{B}_1(0)$ is the closed convex hull of its extreme points, but the set of all the extreme points is the unit sphere $\{ \mu \in M(H) \mid \lVert \mu \rVert = 1 \}$ and this set is not contained in $D$. So, is it possible to find a suitable bounded set $A \subseteq D$? A canonical candidate for $A$ would be the absolutely convex hull $\textrm{aco}(K)$. If the strong topologies are not equal, what is then the dual of $(C_c(H), \beta(C_c(H), D))$ and $(C_c(H), \beta(C_c(H), M_b(H)))$?

3. If we can prove the conjecture for a compact space $K$, then I think it should also work for a more general locally compact space $H$, by relating $C_c(H)$ to its Banach space components $C_K(H)$ and considering polars of open neighborhoods in place of balls.

Note that for the special case $H = \mathbb{N}$ the conjecture is true: $C_c(H) = \varphi$ is the space of sequences that are eventually $0$, $M(H) = C_c(H)' = \omega$ is the space of all sequences, $D = \textrm{span}(\mathbb{N}) = \varphi$ and for this example, one can show that $\beta(\varphi, \varphi) = \beta(\varphi, \omega)$, because $\beta(\varphi, \varphi)$ is barrelled.

Answer 1

I believe that your conjecture is true if $H$ is locally compact and $\sigma$-compact. Then the topology of the inductive limit $C_c(K)=\lim\limits_\to C(K,H)$ (where $C(K,H)$ denotes the Banach space of continuous functions with support in $K$) can be described by quite simple semi-norms of the form $$\|f\|_v=\sup\lbrace|f(x)|v(x): x\in H\rbrace$$ where $v$ is any strictly positive function on $H$ which is bounded on compact sets. It is clear that the restriction of $\|\cdot\|_v$ to each $C(H,K)$ is continuous there. It remains to show that these semi-norms indeed give the inductive limit topology. Fix a sequence $K_n$ of compact sets such that $K_n$ is contained in the interior of $K_{n+1}$ and $H=\bigcup_n K_n$. Then $C_c(H)=\lim\limits_\to C(K_n,H)$. Given any continuous semi-norm $p$ on $C_c(H)$ there are constants $c_n>c_{n-1}>0$ such that $p(f)\le c_n \sup\lbrace|f(x)|:x\in H\rbrace$ for all $f\in C(K_n,X)$. Next choose a continuous locally finite partition of unity $\varphi_n$ subordinated to the covering $\mathring K_n$ and set $v=2^nc_n$ on $K_n\setminus K_{n-1}$. Given any $f\in C_c(H)$ we can write $f=\sum_{n\in \mathbb N} \varphi_nf$ (where the sum is in fact finite) to obtain $$p(f)=\sum_{n\in \mathbb N} p(\varphi_nf) \le \sum_{n\in \mathbb N} 2^{-n} \|f\varphi_n\|_v\le \|f\|_v.$$

Now the proof should be as the one you suggested for the compact case: Given a $0$-neighbourhood $U$ of $C_c(H)$ it contains a unit ball with respect to some $\|\cdot\|_v$ and the extreme points of $U^\circ$ should be $\pm v(x)\delta_x$.