Ergodic decomposition

For a measure-preserving system on a standard probability space, an invariant probability measure is an integral of ergodic invariant probability measures. Its conditional measures given the invariant sigma-algebra are the ergodic components.

If $\mu =\int \nu d\tau (\nu )$ is an ergodic decomposition, then the Kolmogorov-Sinai entropy satisfies $h_{\mu }(T)=\int h_{\nu }(T)d\tau (\nu )$.