the definition of a.s. convergence

Question

One of the definitions of a.s. convergence is $X_n \overset{a.s.}{\to} a$ if for any $\varepsilon > 0$

$$ P( \liminf_{n\to\infty} A_n) =1 $$ where $A_n = \{ \omega \in \Omega: |X_{n}(\omega) - a| < \varepsilon\}$.

Therefore, $X_n \overset{a.s.}{\to} a$ means that for any $\varepsilon > 0$ there exists finite $n_0$, which depends ONLY on $a$ and $\varepsilon$ such that for all $n > n_{0}$ one has

$$ P(A_n) =1 $$

Is this correct? If not, what is the correct description in words of this definition?

Answer 1

This result is not true. The statement $$ P\left(\liminf_{n\to\infty}A_n\right)=1 $$ means that there exists a set $\Omega'\subseteq \Omega$ such that $P(\Omega') = 1$ and for each $\omega\in\Omega'$ there exists an $n_0(\omega)$ such that $\omega\in A_n$ for all $n\ge n_0(\omega)$.

Answer 2

Not true. Let $E_n$ be any disjoint collection of events each having positive probability. Then $I_{E_n} \to 0$ a.s. but $P(A_n)<1$ for all $n$.