Definition:Infinitely Often (Probability Theory)

Definition

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\sequence {E_n}_{n \mathop \in \N}$ be a sequence in $\Sigma$.

We define:

$\ds \set {E_n \text { infinitely often} }$

$=$

$\ds \set {\omega \in \Omega : \omega \in E_n \text { for infinitely many } n}$

$\ds $

$=$

$\ds \limsup_{n \mathop \to \infty} E_n$

where $\ds \limsup_{n \mathop \to \infty} E_n$ is the limit superior of $\sequence {E_n}_{n \mathop \in \N}$.

We may abbreviate "infinitely often" as "i.o." and write:

$\set {E_n \text { infinitely often} } = \set {E_n \text { i.o.} }$