Let $(X,A,\nu)$ be a probability space and $T:X\to X$ a measure-preserving transformation.
The Pinsker sigma algebra is defined as the lower sigma algebra that contains all partition P of measurable sets such that $h(T,P)=0$ ( entropy of T with respect to P).
How calculate the Pinsker sigma algebra of shift Bernoulli $\left(\dfrac{1}{2},\dfrac{1}{2}\right)$?
I think that the Pinsker sigma algebra is the sigma algebra of all measurable sets of measure $0$ or $1$.
And another question is the why is the (SA) Pinsker important for ergodic theory?