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:

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

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


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