Theorem
Kolmogorov–Sinai generator theorem
A finite generating partition computes the entire measure-theoretic entropy.
Statement
If is a finite generating partition of an invertible probability-preserving system, then
The analogous statement holds for a finite one-sided generator of a noninvertible system. This is the Kolmogorov–Sinai generator theorem.
Why it helps
The entropy definition takes a supremum over all finite partitions. A generator contains all measurable information through its iterates, so one partition suffices. For Bernoulli shifts this reduces entropy to the Shannon entropy of one coordinate; for circle doubling, the two half-circle cells give entropy .
References
- Karma Dajani, Introduction to Ergodic Theory and its Applications to Number Theory, 2014, Theorem 5.3.1 and Exercise 5.3.2(c).