What is the definition of "allmost all" in probability theory?

Question

Assume $X_1,X_2,\dots$ are identically distributed random variables. Assume $A(X_1,X_2,\dots)$ is an event which depends on the $X_i$. What does it mean to say "For almost all $X_1,X_2,\dots$ we have $A(X_1,X_2,\dots)$"?

Edit: See theorem 2.1 of: https://projecteuclid.org/download/pdf_1/euclid.aos/1176345637

Answer 1

"Almost all" unambiguously means that it happens with probability $1$; contrary to the comments, this doesn't depend on the source, and it doesn't mean that it's true for all but finitely members of a sequence. The event in the paper you link is more complicated, but here's a more elementary / classic example:

Let $(X_i)$ be a sequence of i.i.d. random variables with $\mathbb{E}[X_1]< \infty$. Then for almost all sequences $(X_i)$, $$\frac{\sum_{k = 1}^n X_k}{n} \to \mathbb{E}[X_1]\,.$$

This is the strong law of large numbers, and can be rephrased as: $$\mathbb{P}\left[\frac{\sum_{k = 1}^n X_k}{n} \to \mathbb{E}[X_1] \right] = 1\,.$$