Convergence of sequence of Lebesgue integrals Suppose we have a sequence of non-negative random variables $\{X_n\}_{n \in \mathbb{N}}$ which are integrable with respect to a probability measure $P$.  Denote $\mu_n \equiv E(X_n)$ and suppose that $\mu_n$ converges to a limit in $\mathbb{R}$, say $\mu$.
Suppose $X_n \xrightarrow{d} X$.  Is it true that $E(X) = \mu$?
It is true that $E(X)$ is well defined, as $|x|$ is a non-negative continuous function so that $E(|X|) \leq \liminf E(|X_n|) = \mu < \infty$ by the Portmanteau lemma.
I know that uniform integrability of the $X_n$ will guarantee the convergence, but I'm not sure if this is true or even needed for the problem.  Any help would be massively appreciated!

That is not true in general.
Take $f_n = n\cdot 1_{[0, 1/n]}$, which converges almost everywhere to $f = 0$. In particular, they converge in distribution as random variables on $[0, 1]$.
However, we have
$$ \lim_{n\to\infty}\int f_n = 1 > 0 = \int f. $$
Aside:
After obtaining a Skorohod representation it boils down to Fatou's lemma. So it is hard to do better.
