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}