Let $X$ be a locally compact Hausdorff space, $C_0(X)$ the Banach space of continuous functions vanishing at infinity, $M(X) := C_0(X)'$ the space of Radon measures and $M^+(X) \subseteq M(X)$ the positive finite Radon measures. On $M(X)$, denote by $w^*$ the weak$^*$ topology (relative to $C_0(X)$) and by $\tau$ the topology of uniform convergence on norm compact sets of $C_0(X)$, so that $w^* \subseteq \tau$. It is known that $\tau$ coincides with the topology of uniform convergence on norm null sequences (by a theorem of Grothendieck, every norm compact set in a Banach space is contained in the absolutely convex closure of a norm null sequence).

Is it true that on $M^+(X)$ it holds $w^* = \tau$?

So, we have to show that for nets $\mu_\alpha, \mu \in M^+(X)$ with $\mu_\alpha f \to \mu f$ for each $f \in C_0(X)$ it also holds $\sup_n |(\mu_\alpha - \mu) f_n| \to 0$ for each sequence $f_n \in C_0(X)$ with $f_n \geq 0$ and $\lVert f_n \rVert \to 0$.

I have read somewhere that this is true for a compact space $X$. So, it may be also true for a locally compact Hausdorff space. But since the above condition involves sequences, I think, one has to restrict to $\sigma$-compact or paracompact spaces $X$.

**Edit**: Here is the proof for compact $X$:

Let $\mu_\alpha \to \mu$ for $w^*$ in $M^+(X)$. Since $X$ is compact, $1_X \in C_0(X) = C(X)$. From $\mu_\alpha 1_X \to \mu 1_X$, there is $\alpha_0$ such that $0 \leq \mu_\alpha 1_X \leq \mu 1_X + 1 =: c$ for all $\alpha \geq \alpha_0$. Then for any $f \in C(X)$: $|\mu_\alpha f| \leq \lVert \mu_\alpha \rVert \cdot \lVert f \rVert = \mu_\alpha 1_X \cdot \lVert f \rVert \leq c \lVert f \rVert$ for all $\alpha \geq \alpha_0$. Therefore, $\{ \mu_\alpha \mid \alpha \geq \alpha_0 \}$ is $w^*$-bounded. By Banach-Alaouglu, this set is $w^*$-relatively compact and since $\tau$ and $w^*$ coincide on $w^*$-compact sets (because $C(X)$ is complete) it follows that $\mu_\alpha \to \mu$ for $\tau$.

For non-compact $X$, I think, $1_X$ should be replaced by some strictly positive function in $C_0(X)$, and these do exist, if $X$ is paracompact - have to think about it.

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