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](https://math.stackexchange.com/questions/4440407/the-space-of-finite-signed-measures-with-the-total-variation-norm-is-a-banach-sp). 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](https://math.stackexchange.com/questions/4568256/does-mu-n-overset1-rightharpoonup-mu-necessarily-imply-mu-n-overset) $\mu\_n \overset{1}{\rightharpoonup} \mu \implies [\mu] \le \liminf\_n [\mu\_n]$.



> 
> 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!




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I posted [this](https://math.stackexchange.com/questions/4568256/does-mu-n-overset1-rightharpoonup-mu-necessarily-imply-mu-n-overset) question on MSE, but it seems to receive no answer. So I post it here.



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?
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 $[0,1]$.


