# Tagged Questions

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.

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### Reference for Ergodic Theory

I am looking for a good introductory book on ergodic theory. Can someone recommend some introductory texts on that?
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### Properties of the space of $T$-invariant probability measures over a compact topological space.

Let $T: X \to X$ be a continuos map defined on the compact space (Maybe Haussdorff) $X$. Denote by $M_T$ the space of $T$-invariant probability measures defined over the Sigma Algebra of Borel sets. ...
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### Uniform convergence of finitely additive measure along a tree of partitions

Learning about Lebesgue-Rohlin spaces is prominently on my to-do-list, so I'm reading Fundamentals of measurable dynamics by Daniel Rudolph, were I'm stuck on an exercise. Framework There is a ...
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### A generic point for a non-ergodic measure

Definition: Let $X$ be a compact metric space and let $\mu$ be a Borel probability measure on $X$, then we say that the sequence $(x_n)$ of elements of $X$ is equidistributed with respect to $\mu$ if ...
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### Weak convergence of finite measure preserving transformations

I am reading King's paper "The commutant is the weak closure of the powers, for rank-1 transformation" and I am not able to show that: (0.1) "If the $T_i$ are invertible measure preserving ...
In the definition of measure-preserving dynamical system, the crucial equation is $$\mu \left(T^{-1} \left(A\right)\right) = \mu\left(A\right) .$$ Why is it not the seemingly more natural $$\... 1answer 100 views ### Continuity in the Krylov-Bogoliubov theorem I have the following proof of the Krylov-Bogoliubov theorem, which asserts that given a compact metric space X endowed with a continuous transformation T \colon X \to X one can find a T-... 1answer 29 views ### Extension of ergodic theorem with WLLN Suppose you have a ergodic (or irreducible) Markov chain (A_t)_{t\geq0} in continuous time. Denote by \pi the invariant distribution of A. If f is a function of A_s which is integrable w.r.t.... 1answer 58 views ### If x=[a_0,a_1,\dots] show that \mu-almost every x \in (0,1/N] is infinitely recurrent Let G be the Gauss map,$$G(x)= \begin{cases} 0 & \text{if} \ x=0 \\ \{\frac{1}{x} \}=\frac{1}{x} \ \mathrm{mod} \ 1 & \text{if $0<x\leq 1$}\end{cases}$$and \mu be the Gauss ... 1answer 48 views ### Ergodic system, show an implication Let (\Omega,\mathfrak{A},\mu,T) be a dynamic system in measure theory and p\geq 1. Show the implication (1)\implies (2), whereat$$ (1)~~~~~\forall f\in L_{\mu}^p: f= f\circ T\implies f=\...
Let $X$ be a compact metric space and $\mathcal{B}$ its Borel $\sigma$-algebra. Suppose that $(X,\mathcal{B},\mu,T)$ is an invertible ergodic system ($T$ is only a measurable isomorphism, not ...