Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that

(i)	$T : X \to X$ is a D201: Measurable map on $M$

Then $T$ is a measure-preserving endomorphism on $M$ if and only if \begin{equation} \forall \, E \in \mathcal{F} : \mu(T^{-1} E) = \mu(E) \end{equation}

Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that

(i)	$T : X \to X$ is a D201: Measurable map on $M$

Then $T$ is a measure-preserving endomorphism on $M$ if and only if \begin{equation} \mu \circ T^{-1} = \mu \end{equation}