I have the following, seemingly simple question: 


Consider a stochastic process $(X\_t)$ satisfying $X\_t\le X\_s$ a.s. for all $t\le s.$ My question is: Does there exist a modification $\tilde{X}$ of $X$, which almost surely has increasing sample paths $t\mapsto\tilde{X}\_t(\omega)$?


I assume such a modification exists, but I did not manage to proof it.


Thanks in advance! 



Increasing stochastic process
Yes, this works. As suggested by fedja, focus for a moment on the rational times. The function (on $\mathbb Q$) $f\_{\omega}(t)=X\_t(\omega)$ is almost surely increasing. Set $\widetilde{X\_t}=0$ for all $t\in\mathbb R$ on the exceptional set. Now if $t\in\mathbb R$ is arbitrary, then $f\_{\omega}(t-)\le X\_t(\omega) \le f\_{\omega}(t+)$ a.s. because only countably many times are involved (of course, the null set will depend on $t$). Set $\widetilde{X\_t}(\omega)=f\_{\omega}(t+)$ (say) on the null set where this fails.


All paths are increasing now because the required inequalities hold if at least one rational time is involved.


