Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a
D1159: Probability space such that
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}