Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that
(i) | $T : X \to X$ is a D201: Measurable map on $M$ |
(ii) | $T$ is an D976: Invertible map with an D216: Inverse map $T^{-1} : X \to X$ |
(iii) | $T^{-1}$ is a D201: Measurable map on $M$ |
Then $T$ is a D2940: Measure-preserving endomorphism on $M$ if and only if
\begin{equation}
\forall \, E \in \mathcal{F} :
\mathbb{P}(T E) = \mathbb{P}(E)
\end{equation}