Let
- $X$ be a metric space,
- $\mathcal M(X)$ the space of all finite signed Borel measures on $X$,
- $\mathcal M_+(X)$ the space of all finite nonnegative Borel measures on $X$,
- $\mathcal M_1(X)$ the space of all Borel probability 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. For $\mu_n,\mu \in \mathcal M(X)$, we define the weak convergence $$ \mu_n \rightharpoonup \mu \overset{\text{def}}{\iff} \int_X f \mathrm d \mu_n \to \int_X f \mathrm d \mu \quad \forall f \in \mathcal C_b(X). $$
Then we have Portmanteau theorem for finite non-negative Borel measures, i.e.,
Portmanteau theorem Let $\mu_n,\mu \in \mathcal M_+(X)$ such that $\mu_n(X) \to \mu(X)$. Then the following statements are equivalent.
- (1) $\mu_n \rightharpoonup \mu$.
- (2) $\lim_n \int_X f \mathrm d \mu_n = \int_X f \mathrm d \mu$ for all bounded Lipschitz-continuous functions $f:X \to \mathbb R$.
- (3) $\liminf_n \int_X f \mathrm d \mu_n \ge \int_X f \mathrm d \mu$ for all lower semi-continuous functions $f:X \to \mathbb R$ that are bounded from below.
- (4) $\limsup_n \int_X f \mathrm d \mu_n \le \int_X f \mathrm d \mu$ for all upper semi-continuous functions $f:X \to \mathbb R$ that are bounded from above.
- (5) $\liminf_n \mu_n(O) \ge \mu(O)$ for all open subsets $O$ of $X$.
- (6) $\limsup_n \mu_n(C) \le \mu(C)$ for all closed subsets $C$ of $X$.
- (7) $\lim_n \mu_n(A) = \mu(A)$ for all Borel subsets $A$ of $X$ such that $\mu(\partial A) = 0$.
I would like to ask if above theorem extends to $\mathcal M(X)$, i.e., we replace "$\mu_n,\mu \in \mathcal M_+(X)$ s.t. $\mu_n(X) \to \mu(X)$" by "$\mu_n,\mu \in \mathcal M(X)$ s.t. $[\mu_n] \to [\mu]$". If not, are there some mild conditions on $X$ so that such an extension is possoble?
Thank you so much for your elaboration!
