I have the following stupid question in my mind while i am studying for exams. Does $X<\infty \ a.s$, implies that $\mathbb E(X)<\infty$?
Further on this, is the converse of the above statement true? Do give me a bit summary on this. Thanks very much.
I thought this was true until I realize the following example: Let's consider a simple symmetric random walk, we know that each state is null-recurrent. Let $\tau_L$ be a stopping time when the walk first hits $L$ started from $0$. then $$\mathbb P(\tau_L<\infty)=1$$ so $\tau_L<\infty \ a.s.$ but we also know that $$\mathbb E(\tau_L)=\infty$$ Is this a counter-example? Thanks, i am a bit weak on measure theory.