a.s convergence on an event $E$

Question

Consider a probability space $(\Omega,\mathcal{F},P),$ and sequence $(X_k)_k.$

What does it mean that $X_k$ converges a.s to a finite limit, on an event $E$ ?

Is there a rigorous definition for this notion?

Answer 1

Equivalently to what Math1000 says:

You know what it means for a sequence of real numbers to converge to a finite limit, of course.

Saying "$X$ converges to a finite limit on $E$" would mean that for every $\omega \in E$, the real-number sequence $X_n(\omega)$ converges to a finite limit.

Saying "$X$ converges a.s. to a finite limit on $E$" would mean that there exists an event $N$ with $P(N) = 0$ such that for every $\omega \in E \setminus N$, the real-number sequence $X_n(\omega)$ converges to a finite limit.

This is consistent with the usual way the phrase "foo happens almost surely" is used; i.e. there is an event of probability 0 on which foo potentially doesn't happen, but otherwise it does.

Answer 2

Your state is not very clearly posed, but I would interpret it as meaning there exists a random variable $X$ with $\mathbb P(|X|<\infty)=0$ such that $$ \mathbb P\left(\lim_{k\to\infty} X_k\mathsf 1_E = X \right) = 1, $$ where $\mathsf 1_E$ denotes the indicator function of $E$.