Let $X_n$ be a sequence of positive integrable random variables, which converges almost everywhere to an integrable random variable $X$. Suppose
$$E(X_n) \rightarrow E(X)$$
I showed that all bounded random variables Y, $E(YX_n) \rightarrow E(YX)$.
But I am unable to get an example, in which the above conclusion is false, if $X_n$ is not positive. Can someone give me one?
Take $\Omega = (0,1)$ with Borel sets and Lebesgue measure and define the random variables:
$$ X_n(\omega) = \left\{\begin{matrix} n, & \omega \in (0,\frac{1}{n})\\ -n, &\omega \in (1-\frac{1}{n}, 1) \\ 0, &else \end{matrix}\right. $$
Also let $X(\omega)=0$, then $X_n \to X$ almost surely, $E[X_n]=0$ and $E[X]=0$, so that the limiting result $E[X_n] \to E[X]$ also holds.
Now let $Y(\omega) = 1$ for $\omega < \frac{1}{2}$ and $Y(\omega) = -1$ for $\omega \geq \frac{1}{2}$.
Then $E[XY]=0$ still but $E[X_nY] = 1$, which of course does not converge to 0.