Let $P = (\Omega, \mathcal{F}, \mathbb{P}, T)$ be a D2839: Probability-preserving system such that

(i)	$E \in \mathcal{F}$ is an D1716: Event in $P$
(ii)	\begin{equation} \mathbb{P} \left( E \triangle T^{-1} E \right) = 0 \end{equation}
(iii)	\begin{equation} F : = \bigcap_{n = 0}^{\infty} \bigcup_{m = n}^{\infty} T^{-m} E \end{equation}

Then

(1)	\begin{equation} \mathbb{P}(E) = \mathbb{P}(F) \end{equation}
(2)	\begin{equation} T^{-1} F = F \end{equation}