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:

\(\ds \set {E_n \text { eventually} }\)

\(=\)

\(\ds \set {\omega \in \Omega : \text {there exists } N \in \N \text { such that } \omega \in E_n \text { for } n \ge N}\)

\(\ds \)

\(=\)

\(\ds \liminf_{n \mathop \to \infty} E_n\)

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:

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

Sources

• 1991: David Williams: Probability with Martingales ... (previous) ... (next): $2.8$

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