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}) : \int_{\Omega} f \, d \mathbb{P} = \int_{\Omega} f \circ T \, d \mathbb{P} \end{equation} |
| (3) | \begin{equation} \forall \, f \in \mathfrak{L}^2(P \to \mathbb{C}) : \int_{\Omega} f \, d \mathbb{P} = \int_{\Omega} f \circ T \, d \mathbb{P} \end{equation} |