Let

• $X$ be a metric space,
• $\mathcal M(X)$ the space of all finite signed Borel measures on $X$,
• $\mathcal C_b(X)$ be the space of real-valued bounded continuous functions on $X$, and
• $\mathcal C_0(X)$ be the space of real-valued continuous functions on $X$ that vanish at infinity.

Then $\mathcal C_b(X)$ and $\mathcal C_0(X)$ are 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 $\mathcal M(X)^* := (\mathcal M(X))^*$ and $\mathcal C_b(X)^* := (\mathcal C_b(X))^*$ be the continuous duals. 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.

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.