Let $\mu$ be a probability measure on $X$ (closed but unbounded), so that $\int_X \mu(dx) = 1$.
Let functions $f_i:X \rightarrow \mathbb{R}_{\geq 0}$, $i = 1,2,...$, be Uniformly Integrable.
Prove that
$ \limsup_{i\rightarrow \infty} \int_X f_i(x) \mu(dx) \ - \ \lim_{n \rightarrow \infty} \limsup_{i \rightarrow \infty} \int_{X_n} f_i(x) \mu(dx) \ \ = \ \ 0 $
for some sequence of compact sets $\{X_n\}_{n=1}^{\infty}$ converging to $X$ ($\lim_{n \rightarrow \infty} X_n = X$).