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 D2940: Measure-preserving endomorphism on $M$ if and only if \begin{equation} \forall \, f \in \mathfrak{L}^1(M \to \mathbb{C}) : \int_X f \, d \mu = \int_X (f \circ T) \, d \mu \end{equation}