Convergence of sequence of integrable random variables Let $\left(X_{n}\right)_n$ be a sequence of positive integrable random variables which converges almost everywhere to an integrable random variable $X$. Suppose $E\left(X_{n}\right) \to E(X)$. I am not able to show  that for all  bounded random variables $Y$, $E\left(YX_{n}\right)\to E\left(YX\right)$. And also  I need an example that the conclusion may be false if $X_{n}$ are not positive. 
Can someone give me hints to solve the above problem?

We can apply Fatou's lemma with the sequence $\left(Y_n\right)_{n\geqslant 1}$ defined by 
$$Y_n:=\left|X_n-X\right|-X_n+X.$$
For non-negativeness: define $X_n:=-n\mathbf 1_{\left(-1/n,0\right)}+n\mathbf 1_{\left(0,1/n\right)}$ on $[-1,1]$ endowed with $\lambda/2$, where $\lambda$ is the Lebesgue measure and choose $Y=\mathbf 1_{(0,1)}$. 
