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

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

Definition D2940

Measure-preserving endomorphism

Formulation 0

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

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}

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

Formulation 1

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

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

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

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 |