Let $(X_n)$ be an increasing sequence of real valued integrable rvs on a probability space $(\Omega,\mathcal{F},P)$, such that $(X_n)$ converges ae to some rv $X$. Is it true that $E(X_n)\rightarrow E(X)$?
Some thoughts: Ι guess not. If $X_n$'s were positive then we could apply the monotone convergence theorem, but now we are looking for a counterexample. I am working on $[0,1]$, equipped with the Lebesgue measure and looking for discrete rvs that do the job, but no matter how I change their formula, I seem not to be getting the desired non-convergence.
Thanks a lot in advance!