Convergence of expectation

Question

I've got a bounded sequence $X_n$ in Lp for some $p>1$ of random real variables which converge in law to a variable $X$. I need to show that the sequence of expected values converge to the expected value of the limit in distribution. I couldn't manage to fo much. I tried to use Skorohod representation to express it for a.e convergence but i still wasn't able to conclude. Ty

Answer 1

Define $f_n(t):=\mathbb P\left(X_n^+\gt t\right)$, where $X_n^+$ denotes the non-negative part of $X_n$. Then $$\mathbb E\left[X_n^+\right]=\int_0^\infty f_n(t)\mathrm dt,$$ $0\leqslant f_n(t)\leqslant \min\left\{1,\sup_i\mathbb E\left\lvert X_l\right\rvert^p t^{-p} \right\}$ and $f_n(t)\to\mathbb P\left(X^+\gt t\right)$ for almost every $t$ (in fact, the set of $t$ for which the previous convergence holds is at most countable) hence we can use the dominated convergence theorem. A similar treatment for $X_n-X_n^+$ gives the result.