If $X_n\geq 0,~X_n\rightarrow X$ ae and $E(X_n)\leq c,$ then $E(X)\leq c.$

Question

Let $(X_n)$ be a sequence of positive valued rvs on a probability space $(\Omega,\mathcal{F},P),$ such that $(X_n)$ converges ae to a rv $X.$ If $E(X_n)\leq c<+\infty$ for all $n$, then $X$ is integrable and $E(X)\leq c.$

Some thoughts: the dominated (bounded) convergence theorem can not be applied (at least not at first glance), since the $(X_n)$ are not uniformly bounded by some integrable rv $Y$ (by a constant, respectively).

Thanks a lot for the help!

Answer 1

As proposed by @T.Bongers, it follows by Fatou' lemma. Indeed, $X_n,~n\geq 1$ are non negative and converge ae to $X$, so by Fatou's lemma, we get $Ε(X)\leq \liminf E(X_n)$. Since $E(X_n)\leq c$ for all $n,$ we get $Ε(X)\leq \liminf E(X_n)\leq c,$ as desired.

Answer 2

(From my comment above.)

This follows from Fatou's lemma.