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}

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}