> [!definition] > > Let $(\Omega, \cf, \bp)$ be a [[Probability|probability space]]. A [[Measurable Function|measurable function]] $T: \Omega \to \Omega$ is **measure preserving** if $\bp$ is invariant under [[Pushforward of Vector Field|pushforward]] by $T$, with the pairing $(\Omega, \cf, \bp, T)$ known as a **measure system**. > [!definition] > > Let $(X, S), (Y, T)$ be measure systems, then they are isomorphic if there exists a measure preserving bijection $f: X \to Y$ such that $S = f^{-1} \circ T \circ f$. > [!definition] > > The **orbit equivalence relation** $\mathbf E_T \subset X^2$ is defined as > $ > (x, y) \in \mathbf{E}_T \Leftrightarrow \exists m, n \in \nat: T^m(x) = T^n(y) > $ > In particular, if $T^{-1}(x)$ is countable or finite for each $x \in X$, then each orbit is countable. > [!definition] > > Let $E \subset X$, then $E$ is $\mathbf{E}_T$-**invariant** if and only if it is a union of $\mathbf{E}_T$-orbits. If $f: X \to Y$ is a function, then $f$ is $\mathbf{E}_T$-invariant if and only if $f$ is constant on the $\mathbf{E}_t$-orbits.