ThmDex – An index of mathematical definitions, results, and conjectures.
Formulation F10590 on R4550:
F10590
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P}, T)$ be a D2839: Probability-preserving system such that
(i) $P$ is an D4492: Ergodic probability-preserving system
(ii) $E \in \mathcal{F}$ is an D1716: Event in $P$
Then \begin{equation} \lim_{N \to \infty} \frac{ \# \left\{ n \in \{ 0, \ldots, N - 1 \} : T^n \in E \right\} }{N} \overset{a.s.}{=} \mathbb{P}(E) \end{equation}