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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?

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    http://www.math.univ-toulouse.fr/~bressaud/Cours_2010_Septembre.pdf These course notes by Bressaud claim that the Pinsker $\sigma$-field is degenerate if and only if $T$ is a $K$-automorphism, and it is known that Bernouilli are $K$. They also claim that the Pinsker $\sigma$-field allows to define the bigger factor of $T$ having zero entropy when $T$ is not $K$ (I am not able to explain what does it rigorously means).2013-01-31
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    @StéphaneLaurent Thanks for Bressaud's notes (well written). So please don't delete your first comment! (or include the link in your answer).2013-02-19

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