Pinsker sigma-algebra

The Pinsker sigma-algebra consists of the measurable sets $A$ whose binary partition $\{A,X\setminus A\}$ has entropy rate zero. It is a sigma-algebra and is the largest invariant factor with zero Kolmogorov-Sinai entropy.

A set belongs to the Pinsker sigma-algebra exactly when, modulo a null set, it belongs to $\mathcal{T}(\xi )$ for some finite measurable partition $\xi$. For a zero-entropy binary partition one may take that partition itself: zero conditional entropy makes its present atom measurable from every remote future.