Definition:Eventually (Probability Theory)

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

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

We define:

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

Notation
We may abbreviate eventually as ev and write:


 * $\set {E_n \text { eventually} } = \set {E_n \text { ev} }$