Let $M = (X, \mathcal{F}, \mu, T)$ be a D2827: Measure-preserving system such that
| (i) | $(X, \mathcal{F}, \mu)$ is a D2707: Sigma-bounded measure space |
| (ii) | $f : X \to \mathbb{C}$ is a D5617: Complex Borel function on $M$ |
| (iii) | \begin{equation} \Vert f \Vert_{L^1} < \infty \end{equation} |
Then
| (1) | \begin{equation} \exists \, f^* \in \mathfrak{L}^1(M \to \mathbb{C}) : \lim_{N \to \infty} \frac{1}{N} \sum_{n = 0}^{N - 1} (f \circ T^n) \overset{a.e.}{=} f^* \end{equation} |
| (2) | \begin{equation} f^* \circ T \overset{a.e.}{=} f^* \end{equation} |
| (3) | \begin{equation} \mu(X) < \infty \quad \implies \quad \int_X f^* \, d \mu = \int_X f \, d \mu \end{equation} |