Almost sure boundedness

Question

I want to know if there are any well known sufficient conditions for almost sure boundedness of a sequence of (not necessarily independent) non-negative random variables $\{X_n\}_{n\geq 1}$. I say that the sequence $\{X_n\}_{n\geq 1}$ of non-negative random variables is bounded almost surely, if the following holds:
$$\mathbb{P}\left(\sup_{n\geq 1} X_n < \infty\right) = 1.$$
For example, does $\sup_{n\geq 1} \mathbb{E} (X_n) < \infty$ imply that $\{X_n\}_{n\geq 1}$ is bounded almost surely? Can someone at least provide a counterexample to this?

Answer 1

Let $I_1,I_2...$ be an arrangement of the intervals $[\frac {i-1} {2^{n}}, \frac i {2^{n}})$ in a sequence. If $X_n=n$ on $I_n$ and $0$ elsewhere then $sup_nEX_n<\infty$ but $P(\sup_n X_n <\infty)=0$. My basic space is $[0,1]$ with Lebesgue measure.