Invariant sigma-algebra

The invariant sigma-algebra of a probability-preserving system is

\mathcal I_T=\{E\in\Sigma:\mu(T^{-1}E\triangle E)=0\},

where $\triangle$ is symmetric difference. Its events are invariant modulo null sets. For an action of $G$ , require this equality for every $T_g$ , and write $\mathcal I_G$ .

Why it is a sigma-algebra

Preimages commute with complements and countable unions, and a countable union of null sets is null. These facts verify closure. Every measurable null set belongs to $\mathcal I_T$ .

Statistical meaning

The limit of time averages is conditional expectation onto this sigma-algebra. It keeps the information that does not change under the dynamics.