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Question about probability spaces $(X,\mathcal B,\mu)$ with a measurable map $T\colon X\to X$ preserving the measure, that is $\mu(T^{—1}A)=\mu(A)$ for all $A$ measurable.

Given a measure space $(X,\mathcal B,\mu)$, a measure-preserving transformation on $X$ is a measurable map $T\colon X\to X$ preserving the measure $\mu$. This means that $\mu(T^{—1}A)=\mu(A)$ for all measurable sets $A\subset X$.

Pre-requisites are measure theory and integration theory.

Topics include, but are not restricted to: Poincaré's recurrence theorem, Birkhoff's ergodic theorem, ergodicity, mixing, and other ergodic properties, the relation to symbolic dynamics, including Markov measures and Bernoulli measures, metric entropy and topological entropy, Shannon-McMillan-Breiman's theorem, topological entropy, variational principles, equilibrium and Gibbs measures, relation to hyperbolic dynamics, and smooth ergodic theory.

Further reading-Wikipedia