# Are there some conditions on a metric space $X$ such that these two types of weak converge of finite signed Borel measures on $X$ are related?

Let

• $$X$$ be a metric space,
• $$\mathcal M(X)$$ the space of all finite signed Borel measures on $$X$$, and
• $$\mathcal C_b(X)$$ be the space of real-valued bounded continuous functions on $$X$$.

Then $$\mathcal C_b(X)$$ is a real Banach space with supremum norm $$\|\cdot\|_\infty$$. We endow $$\mathcal M(X)$$ with the total variation norm $$[\cdot]$$. Then $$(\mathcal M(X), [\cdot])$$ is a Banach space. Let $$\mathcal M(X)^* := (\mathcal M(X))^*$$ be the continuous dual. Let $$\mu_n,\mu \in \mathcal M(X)$$.

• We define the first type of weak convergence by $$\mu_n \overset{1}{\rightharpoonup} \mu \overset{\text{def}}{\iff} \int_X f \mathrm d \mu_n \to \int_X f \mathrm d \mu \quad \forall f \in \mathcal C_b(X),$$ Let $$\sigma(\mathcal M(X), \mathcal C_b(X))$$ be the topology induced by $$\overset{1}{\rightharpoonup}$$.

• We define the second type of weak convergence by $$\mu_n \overset{2}{\rightharpoonup} \mu \overset{\text{def}}{\iff} \varphi(\mu_n) \to \varphi (\mu) \quad \forall \varphi \in \mathcal M(X)^*,$$ Let $$\sigma(\mathcal M(X), \mathcal M(X)^*)$$ be the topology induced by $$\overset{2}{\rightharpoonup}$$.

Of course, we have $$\mu_n \overset{2}{\rightharpoonup} \mu \implies [\mu] \le \liminf_n [\mu_n]$$. Also, we can prove that $$\mu_n \overset{1}{\rightharpoonup} \mu \implies [\mu] \le \liminf_n [\mu_n]$$.

Are there some conditions (locally compact, separable, Polish,...) on $$X$$ that ensure [$$\mu_n \overset{1}{\rightharpoonup} \mu \implies \mu_n \overset{2}{\rightharpoonup} \mu$$] or [$$\mu_n \overset{2}{\rightharpoonup} \mu \implies \mu_n \overset{1}{\rightharpoonup} \mu$$]?

Thank you so much for your elaboration!

I posted this question on MSE, but it seems to receive no answer. So I post it here.

• For general metric spaces: What are functions vanishing at infinity? Nov 4, 2022 at 11:53
• @DieterKadelka A function $f \in \mathcal C_b (X)$ vanishing at infinity means that for every $\epsilon>0$ the set $\{x:|f(x)| \geq \epsilon\}$ is compact. Nov 4, 2022 at 11:58
• One direction seems to be simple since $C_0(X) \subset C_b(X) \subset M^*(X)$. We always have that $\mu_n \to^2 \mu$ implies $\mu_n \to^1 \mu$. For the other direction you need strong assumptions, I think. Maybe you find something in the old book of Z. Semadeni, Banach spaces of continuous functions I (1971). Nov 4, 2022 at 12:14
• @DieterKadelka Do you mean by $\mathcal C_b(X) \subset \mathcal M(X)^*$ that there is an isometrically isomorphic embedding from $\mathcal C_b(X)$ to $\mathcal M(X)^*$? Nov 4, 2022 at 12:25
• @JochenGlueck I have removed unnecessary parts. Nov 4, 2022 at 12:34

An example. $$X = [0,1]$$ with the usual metric. $$\mathcal C[0,1] = \mathcal C_b[0,1] = \mathcal C_0[0,1]$$. $$\mathcal C[0,1]^* = \mathcal M[0,1]$$. Let $$\mu_n$$ be the unit point-mass at $$1/n$$ and $$\mu$$ the unit point-mass at $$0$$. Show $$\mu_n \overset{1}{\rightharpoonup} \mu$$ is true but $$\mu_n \overset{2}{\rightharpoonup} \mu$$ is false.
Whatever "some conditions" to insure $$\big[\mu_n \overset{1}{\rightharpoonup} \mu \implies \mu_n \overset{2}{\rightharpoonup} \mu\big]$$ are, they are not satisfied by $$[0,1]$$.
• The space $\mathcal M(X)^*$ is quite hard to imagine. Could you provide a map $\varphi \in \mathcal M(X)^*$ such that $\varphi (\mu_n)$ does not converge to $\varphi (\mu)$? Nov 4, 2022 at 12:39
• @Akira: Let $f$ be any bounded measurable function on $X$. Then $\mu \to \int f~d\mu$ is in $M(X)^*$. Choose $f = 1_{\{0\}}$ f.i. Nov 4, 2022 at 13:15
• The entire space $\mathcal M(X)^*$ is, indeed, hard to imagine. But there are simple examples (as Dieter showed) of elements of $\mathcal M(X)^*$ not in $\mathcal C(X)$. A more difficult question would be: exhibit an infinite comapct $X$ where$\big[\mu_n \overset{1}{\rightharpoonup} \mu \implies \mu_n \overset{2}{\rightharpoonup} \mu\big]$ is true. Surely you should do that before you ask for conditions under which it is true. Nov 4, 2022 at 15:18