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 the following statements are equivalent
(1) | $T$ is a D3120: Probability-preserving endomorphism on $P$ |
(2) | \begin{equation} \forall \, f \in \mathfrak{L}^1(P \to \mathbb{C}) : \mathbb{E}(f) = \mathbb{E}(f \circ T) \end{equation} |
(3) | \begin{equation} \forall \, f \in \mathfrak{L}^2(P \to \mathbb{C}) : \mathbb{E}(f) = \mathbb{E}(f \circ T) \end{equation} |