Assume that we have a metric space $(A,\rho)$ and a sequence of probabilistic Borel measures $\mu\_{n}$ on $A$ that converges weakly to the probabilistic Borel measure $\mu$. Assume also that one is given a sequence $A\_{j}$ of closed subsets of $A$. Let $\limsup\_{n\to\infty} A\_{n}=\bigcap\_{j=1}^{\infty}\bigcup\_{i=j}^{\infty} A\_{i}$ be the usual set-theoretic upper limit of the sequence $A\_{j}$. An easy combination of Portmanteau theorem and Fatou lemma gives one that
$$\limsup\_{m\to\infty}\limsup\_{n\to\infty}\mu\_{n}(A\_m)\leq \mu(\limsup\_{m\to\infty} A\_{m}).$$


I am interested under what additional assumptions on the measures $\mu\_n$ one can get the stronger statement:


$$\limsup\_{n\to\infty}\mu\_{n}(A\_n)\leq \mu(\limsup\_{n\to\infty} A\_{n})?$$


Note that this is true if $\mu\_{n}$ converges strongly to $\mu$ and in this case the assumption that $A\_{n}$ are closed is unnecessary. The inequality also trivially holds if we assume that $A\_{n}$ is a nested family of closed sets, that is $A\_{n+1}\subseteq A\_{n}$. On the other hand the inequality does not hold for an arbitrary weakly convergent sequence of probability measures as the following trivial counterexample shows:


$$A=\mathbb R, \rho(x,y)=|x-y|, \mu\_{n}=\delta\_{\frac{1}{n}}, \mu=\delta\_{0}, A\_{n}=\{\frac{1}{n}\}.$$



Weak convergence of measures and a sequence of closed sets
There is a version of a generalized Fatou lemma, under the condition that, for all measurable set $E$, $\liminf\_n \mu\_n(E)\leq\mu(E)$, see


O. Hernández-Lerma, J-B. Lasserre,
Fatou's lemma and Lebesgue's convergence theorem for measures, J. Appl. Math. Stochastic Anal. 13 (2000), no. 2, 137–146. 


There is also the possibility to modify the right-hand side of the seeked inequality as shown in 


E. Feinberg, P. Kasyanov, N. Zadoianchuk, 
Fatou's lemma for weakly converging probabilities,
Theory Probab. Appl. 58 (2014), no. 4, 683-689 (Arxiv1206.4073).


Theorem 1.1 p.2, stated for measurable functions instead of sets, says


Let $\{\mu\_{n}\}\subset\mathbb{P}(A)$ converge weakly to 
$\mu\in\mathbb{P}(A)$,
and let $\{f\_{n}\}\_{n\geq1}$ be a sequence of measurable nonnegative $\overline{\mathbb{R}}$-valued functions on $A$. Then
$$
\int \liminf\_{n\to\infty,~s'\to s}f \_ { n } ( s ^ { \prime } ) \mu ( d s ) \leq 
\liminf \_ { n \rightarrow \infty } \int f \_ { n } ( s ) \mu \_ { n } ( d s ).
$$
or, equivalently, with $\limsup$ instead of $\liminf$, and a sequence $\{f\_{n}\}$ of measurable functions uniformly bounded above, 
$$
\limsup\_ { n \rightarrow \infty } \int f \_ { n } ( s ) \mu \_ { n } ( d s )\leq
\int \limsup\_{n\to\infty,~s'\to s}f \_ { n } ( s ^ { \prime } ) \mu ( d s ).
$$


