[* To show , one can either use Jensen, or else apply Cauchy-Schwarz to the functions and . -T*]

(http://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/#as-rem), which is for the discrete random variables. Are they essentially the same? ]]>

I think you want:

*[Corrected, thanks - T.]*

(b) Not directly, though in practice one can often derive recurrence results for non-invertible ergodic systems from the invertible case by a lifting trick, see e.g. Exercise 9 of http://terrytao.wordpress.com/2008/01/15/254a-lecture-4-multiple-recurrence/ for an instance of this.

]]>In the definition of measure-preserving system, when you say “ is invertible”, you mean that the map is a (measurable) bijection, so that is defined on the level of sets?

In this case, we are not considering examples such as the doubling map on the circle? I think some other texts only require that satisfy for all measurable .

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