Ergodic decomposition theorem

Statement

For a Borel probability-preserving transformation $T$ on a standard Borel space, there is a measurable family $y\mapsto\mu_y$ of invariant ergodic Borel probability measures, with a probability $\nu$ on an indexing space, such that

\mu(E)=\int\mu_y(E)\,d\nu(y) \qquad(E\text{ Borel}).

The family is the disintegration over the invariant factor, unique modulo null sets and equivalence of indexing. It is the ergodic decomposition of the system.

Time averages within components

The component average of $f$ supplies $\mathbb E[f\mid\mathcal I_T]$ . Different components can therefore give different long-time statistics even though the overall measure is preserved.

Finite example

For a permutation with cycles $C_j$ , let $\mu_j$ be uniform on $C_j$ . The uniform measure on all $n$ points decomposes as $\sum_j(|C_j|/n)\mu_j$ . Each cycle system is ergodic. On an atomless system the components can each have total $\mu$ -measure zero, so this is a measure mixture, not necessarily a countable partition into positive-measure pieces.

References

Terence Tao, “254A, Lecture 9: Ergodicity”, Proposition 4 (Ergodic decomposition).