Where the separability assumption were used in the first proof of Theorem 2?

*[The first proof is in fact valid for inseparable Hilbert spaces also. -T]*

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

(https://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 https://terrytao.wordpress.com/2008/01/15/254a-lecture-4-multiple-recurrence/ for an instance of this.

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