ThmDex – An index of mathematical definitions, results, and conjectures.
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Deduction system
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Zermelo-Fraenkel set theory
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Measure-preserving endomorphism
Definition D3120
Probability-preserving endomorphism
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $T : \Omega \to \Omega$ is a D201: Measurable map on $P$
Then $T$ is a probability-preserving endomorphism on $P$ if and only if \begin{equation} \forall \, E \in \mathcal{F} : \mathbb{P}(T^{-1} E) = \mathbb{P}(E) \end{equation}
Formulation 1
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $T : \Omega \to \Omega$ is a D201: Measurable map on $P$
Then $T$ is a probability-preserving endomorphism on $P$ if and only if \begin{equation} \mathbb{P} \circ T^{-1} = \mathbb{P} \end{equation}