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!

  • 3
    $\begingroup$ This follows from Fatou's lemma. $\endgroup$
    – user296602
    Feb 9 '16 at 20:52
  • 2
    $\begingroup$ @T.Bongers: You should make that an answer... $\endgroup$
    – copper.hat
    Feb 9 '16 at 20:55
  • $\begingroup$ Definately! Thanks! $\endgroup$ Feb 9 '16 at 20:56

(From my comment above.)

This follows from Fatou's lemma.


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.


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