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, 
$$
\limsup_ { n \rightarrow \infty } \int  f _ { n } ( s ) \mu _ { n } ( d s )\leq
\int  \limsup_{n\to\infty,~s'\to s}f _ { n } ( s ^ { \prime } ) \mu ( d s ).
$$