I know that if a sequence of non-negative random variables $(X_n)_{n \in \mathbb{N}}$ satisifies
$$\mathbb{E}(X_n) \rightarrow 0 $$
as $n \rightarrow \infty$ implies that a subsequence converges almost surely to $0$.
But suppose we have a non-negative process $(X_t)_{t \geq 0}$ satisfying
$$\mathbb{E}(X_t) \rightarrow 0 $$
as $t \rightarrow \infty$.
What is the strongest thing we can say about almost sure convergence in this case?
(Of course, for instance, you can get a subsequence $q_{n_k} \uparrow \infty$ of any sequence of rationals $q_n \uparrow \infty$ such that $X_{q_{n_k}} \rightarrow 0$ a.s., but I'd like something stronger....)
Many thanks for your help.