For a sequence of positive random variables $X_n$ (which i assumed is not nessesarily monotonically non decreasing). Assuming that $\lim\limits_{n\rightarrow \infty} X_n = X$
I don't understand why $ \mathbb{E} \limsup\limits_{n\rightarrow\infty}X_n = \mathbb{E} \lim\limits_{n\rightarrow \infty} X_n$.
Because according to my understanding, $\lim\limits_{n\rightarrow \infty}X_n(\omega)$ for disjoint $\omega$ is not necessarily increasing, hence for disjoint set of $\omega$ $A$ and $B$, it could be either $\limsup\limits_{n\rightarrow \infty} X_n (\omega_A)= X(\omega_A)$ or $\liminf\limits_{n\rightarrow \infty} X_n (\omega_B)= X(\omega_B)$