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Definition D2940
Measure-preserving endomorphism

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 $$\forall \, E \in \mathcal{F} : \mu(T^{-1} E) = \mu(E)$$

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 $$\mu \circ T^{-1} = \mu$$
Children
 ▶ Measure-preserving system ▶ Probability-preserving endomorphism
Results
 ▶ Measure of set in backward orbit under measure-preserving endomorphism ▶ Probability of event in backward orbit under probability-preserving endomorphism