We know that if $X_t$ is a right continuous sub-martingale with $\sup_t \mathbb{E}[X_t^+] < \infty$ then $\lim_{t \rightarrow \infty} X_t$ exists almost surely, but I haven't been able to find a counterexample showing why the right continuity is necessary. I see why the proof used in the right continuous case doesn't generalize, but is there an example of a sub-martingale with $\sup_t \mathbb{E}[X_t^+] < \infty$ and $\mathbb{P}(\{\lim_{t \rightarrow \infty} X_t \text{ exists}\}) < 1$?
Edit: I think I was able to find an example that's almost what I wanted. If we let $Z_n$ be i.i.d. exponentially distributed random variables and set $X_t := -\sum_{n=1}^\infty 1_{Z_n = t}$, then $X_t$ is a sub-martingale with respect to its natural filtration and $\mathbb{E}[X_t^+] = 0$ for all $t$, but $\limsup_{t \rightarrow \infty} X_t = 0$ and $\liminf_{t \rightarrow \infty} X_t = -1$ almost surely so $X_t$ does not converge almost surely. However, the constant process $\tilde X_t = 0$ is a modification of $X_t$ and does converge almost surely. That's not quite what I wanted, so I guess the question should be if there's an example of a sub-martingale with $\sup_t \mathbb{E}[X_t^+] < \infty$ and for any modification $\tilde X_t$ of $X_t$ we have $\mathbb{P}(\{\lim_{t \rightarrow \infty} \tilde X_t \text{ exists}\}) < 1$?