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For a measure-preserving (finite) system $(X,\mathcal{B},\mu,T)$, is it correct that the following are equivalent?

  1. For every $A,B\in\mathcal{B}$ , $\displaystyle\lim_{n\rightarrow\infty}\mu(A\cap T^{-n}B)=\mu(A)\mu(B)$.

  2. For every $A,B\in\mathcal{B}$ of positive measure, there is some $n_0\in\mathbb{N}$ such that for every $n>n_0$, $\mu(A\cap T^{-n}B)>0$.

Clearly 1 implies 2. Is the opposite direction also correct?

  • 0
    Is every invariant measure a mixture of ergodix measures ? I think this is Choquet's thm but ..., are all ergodic measure mutually singular ? Can you use this to show the mixture must be trivial ?2012-12-10
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    @mike: The ergodicity of $\mu$ follows from "2" immediately.2012-12-10
  • 4
    This exact question has already been asked in 1968 : it is proposed as a conjecture at the end of the paper “On weak mixing automorphisms” by James W. England and N.F.G. Martin, Bulletin of the AMS, 74, pp.505-507.2012-12-13
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    Ergodicity implies convergence in the Cesàro sense, thus, should it exist, the limit in (1) is $\mu(A)\mu(B)$.2012-12-19
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    The answer seems to be No. The question is answered here: http://mathoverflow.net/questions/125245/silly-question-about-mixing2013-04-11

2 Answers 2