What is a.e. a.s

Question

I am reading a paper which uses almost everywhere almost surely (a.e.,a.s.) simultaneously, I am not quite sure what it means then. To be specific, they consider a stochastic process $\{X_t\}$ such that $|X_t| < +\infty$ a.e., a.s. Does it mean for all $t$, almost surely $\omega \in \Omega$? Why do they need the a.e. in that case?