Why can we restate the standard definition of almost sure convergence in terms of the limit infimum of sets?

Question

It is stated in this Wikipedia article that convergence almost surely may be restated using the notion of the limit inferior of a sequence of sets, in that:

$$ \operatorname{Pr}\Big( \omega \in \Omega : \lim_{n \to \infty} X_n(\omega) = X(\omega) \Big) = 1. $$

can be restated as:

$$ \operatorname{Pr}\Big( \liminf_{n\to\infty} \big\{\omega \in \Omega : | X_n(\omega) - X(\omega) | < \varepsilon \big\} \Big) = 1 \quad\text{for all}\quad \varepsilon>0. $$

I am wondering why this is the case. I know that generally, it is NOT the case that:

$$ \omega\in\left\{\lim_{n\to\infty} X_n= X\right\}\iff \omega\in\bigcup_{n=1}^\infty\bigcap_{k=n}^\infty\{|X_k-X|<\varepsilon \}, \text{ for all }\varepsilon > 0. $$

So why are the two statements above equal?

Answer 1

Fix $\varepsilon>0$. Denote $$A=\{\omega \in \Omega:\lim_{n \to \infty}X_n(\omega)=X(\omega)\},$$ $$A_n^\varepsilon=\{\omega \in \Omega \, : \,|X_k(\omega)-X(\omega)|< \varepsilon\text{ for all } k \ge n\}.$$ Then the original statement can be written as $P(A)=1$, and the second statement can be written either as $\lim\limits_{n \to \infty} P(A_n^\varepsilon)=1$ or as $P(A^\varepsilon)=1$, where $A^\varepsilon \equiv \lim\limits_{n \to \infty}A_n^\varepsilon=\bigcup\limits_{n=1}^{\infty}A_n^\varepsilon$. (Notice that $\{A_n^\varepsilon\}$ is an increasing sequence of events. By the continuity theorem for monotone events, $P(A^\varepsilon)=\lim\limits_{n \to \infty} P(A_n^\varepsilon)$.)

1. Suppose $P(A)=1$. Note that $A\subseteq A^\varepsilon$. Indeed, in detail, $A^\varepsilon$ is the following set of outcomes: \begin{align*} \quad \ A^\varepsilon & = \{\omega \in \Omega: \exists \,n_\varepsilon(\omega) \text{ such that } \omega \in A_k^\varepsilon, \ \forall k\, \ge n_\varepsilon(\omega)\}&\\ & =\{\omega \in \Omega: \exists \,n_\varepsilon(\omega) \text{ such that } |X_k(\omega)-X(\omega)| \le \varepsilon, \ \forall \, k \ge n_{\varepsilon}(\omega)\}& %& = \{\omega \in \Omega: \exists n_\varepsilon(\omega) \text{ such that } \omega \in A_k^\varepsilon, \ \forall k\ge n_\varepsilon(\omega)\}& \end{align*} Let $\omega \in A$. Then for a chosen $\varepsilon>0$, $\exists \, n_\varepsilon(\omega)$ such that $\forall \,(k \ge n_\varepsilon(\omega)) \ |X_k(\omega)-X(\omega)|\le\varepsilon$. Clearly then, $\omega \in A^\varepsilon$.
   The fact that $A\subseteq A^\varepsilon$ implies that $P(A^\varepsilon) \ge P(A)=1$ and, thus, because probability cannot be greater than $1$, $P(A^\varepsilon) = 1$.

2. Suppose now that $P(A^\varepsilon)=1$ for any $\varepsilon>0$. Take $\varepsilon=1,\frac{1}{2},\frac{1}{3},\ldots$ Then $A^1\supset A^\frac{1}{2} \supset A^\frac{1}{3}\supset \ldots$ $-$ a decreasing sequence of events, for which $\lim\limits_{m \to \infty}A^\frac{1}{m}=\bigcap\limits_{m=1}^{\infty}A^\frac{1}{m}$. By the continuity theorem, $$P\left(\bigcap\limits_{m=1}^{\infty}A^\frac{1}{m}\right) = P\left(\lim\limits_{m \to \infty}A^\frac{1}{m}\right)=\lim\limits_{m \to \infty}P\left(A^\frac{1}{m}\right)=1.$$ Now notice that $A=\bigcap\limits_{m=1}^{\infty}A^\frac{1}{m}$ and, thus, $P(A)=1$.