$P(\liminf S_n)=1$ what does it mean?

Question

$P(\cdot)$ is the probability measure.

$S_n$ is a sequence of events.

$P(\liminf S_n)=1$ does it mean that $S_n$ always happen after some large n?

Can I say that it must be true that $\liminf P(S_n)=1$ given $P(\liminf S_n)=1$ ?

Answer 1

Recall that $S = \liminf S_n = \bigcup_{k=1}^\infty \left(\bigcap_{n \ge k} S_n\right)$; saying that $P(S)=1$ means that (almost) all outcomes $\omega$ belong to $S$, that is belong to $\bigcap_{n \ge k} S_n$ for at least one $k$; in other terms, $\omega\in S_n$ for all sufficiently large $n$'s.

In short, this means that that all events $S_n$ happen, except for a finite number of them.