Is there some probability space $ (Ω, \mathcal F, \mathbb P)$ and a sequence $(X_n)$ of positive random variables on $Ω$ for which $X_n → ∞ $ almost surely, but the sequence $\mathbb E[X_n]$ of expectations is bounded (or more generally does not converge to $∞$)?
The monotone convergence theorem tells us no monotone sequence of this kind exists, and I think this is also not possible when $Ω$ is finite or countable. Intuitively, the a.s. convergence of the $X_n$ would have to be very non-uniform, but I have not managed to come up with an example.
Best Answer
As pointed out by user fwd in the comments, no such sequence exists as a consequence of Fatou's lemma, according to which
$$ \mathbb ∞ = \mathbb E[\liminf_{n → ∞} X_n] \leq \liminf_{n → ∞} \mathbb E[X_n]. $$
Edit: as also pointed out in the comments, if we drop the requirement $X_n \geq 0$, such a sequence exists, e.g. with $ Ω = (0,1]$ equipped with the Borel-σ-algebra and Lebesgue measure and $$ X_n = -n^2 I_{(0,1/n)} + n I_{[1/n,1]}. $$