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In Furstenberg's proof of the multiple recurrence theorem in ergodic theory, one makes use of the concept of compact and weak mixing extensions of a measure preserving system. The following definition is taken from the book Ergodic Theory with a view toward Number Theory by Ward:

Definition 1 (Weak mixing extension): An extension $(X, \mathcal B_X, \mu, T) \to (Y, \mathcal B_Y, \nu, S)$ of measure preserving systems is said to be be relatively weak mixing if the system $(X \times X, \mu \times_Y \mu, T \times T)$ is ergodic, where $\mu \times_Y \mu$ is the relatively independent joining over $Y$ (see, e.g. here for a definition).

The motivation given for the definition is that whenever $Y$ is trivial, that is, measurably isomorphic to a one point space, then the relatively independent joining over $Y$ reduces to the product measure, and hence the extension $X \to Y$ is relatively weak mixing if and only if $X$ is weak mixing in the classical sense.

While the definition is in a very useful form, as evidenced by its use in the proof, it is not immediately intuitive to me. Naively, I would expect relative weak mixing to look something like the following (in analogy to relative independence in probability theory):

Definition 2 (Weak mixing extension?): An extension $(X, \mathcal B_X, \mu, T) \to (Y, \mathcal B_Y, \nu, S)$ of measure preserving systems is said to be be relatively weak mixing if the system $(X, \mathcal B_X, \mu_y^Y, T)$ is classically weak mixing for $\nu$-a.e. $y \in Y$, where $\mu_y^Y$ are the conditional measures of $\mu$ given $Y$.

Hence the above naive definition just asks for $X$ to be weak mixing above every fibre of $Y$.

Question: Are definitions 1 and 2 equivalent? If not, is there any equivalent formulation of relative weak mixing that is more immediately intuitive? Ideally I would like something that works directly with the conditional measures/disintegration over $Y$, instead of transferring to the $L^2$ setting.

I seek also a similar (equivalent) reformulation of compact extensions to a potentially less tractable/useful but more immediately intuitive form. The definition, also given in Ward is:

Definition 3 (Compact extension): An extension $(X, \mathcal B_X, \mu, T) \to (Y, \mathcal B_Y, \nu, S)$ of measure preserving systems is said to be be relatively compact if the set of functions satisfying the following almost periodic property is dense in $L^2_\mu (X)$.

A function $f \in L^2_\mu (X)$ is said to be almost periodic with respect to $Y$ if every $\varepsilon > 0$, there exist functions $g_1, \dots, g_r \in L_\mu^2 (X)$ such that

$$\min_{s = 1, \dots, r} \|T^n f - g_s\|_{L^2_{\mu^Y_y}} < \varepsilon$$

for all $n \geq 1$ and for $\nu$-almost every $y \in Y$.

Again I would prefer a definition that works directly with the measures instead of function space.

Thanks in advance!

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    $\begingroup$ The measures $\mu^Y_y$ are almost never $T$-invariant (this basically only occurs when $S$ is trivial). $\endgroup$ Commented Oct 6, 2023 at 6:07
  • $\begingroup$ @TerryTao Oh true... maybe the fibres need to be moving. According to Corollary 5.24 in Ward, we have $T_* \mu^Y_{y} = \mu^Y_{Sy}$ for $\nu$-a.e. $y$, so we can try to define relative weak mixing to be that the limit $$\lim_{N \to \infty} \frac{1}{N} \sum_{n = 1}^n |\mu^Y_{S^n y} (T^{-n} A \cap B) - \mu^Y_{S^n y} (A) \mu^Y_{S^n y} (B)|$$ exists and equals $0$ for $\nu$-almost every $y \in Y$ and $A, B \in \mathcal B_X$. Does this definition work? $\endgroup$ Commented Oct 6, 2023 at 7:43
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    $\begingroup$ You should measure $A$ in $\mu^Y_y$ rather than $\mu^Y_{S^n y}$, and integrate in $y$ rather than ask for pointwise ae convergence, then it will be an equivalent form of relative weak mixing. $\endgroup$ Commented Oct 6, 2023 at 14:25
  • $\begingroup$ @TerryTao Thank you very much, I will work out the equivalence. $\endgroup$ Commented Oct 6, 2023 at 14:34

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As Terry Tao mentioned in a comment, definition 2 is problematic and there is no point in answering the stated question. Instead, I will use this opportunity to give equivalent definitions to weak mixing and compact extensions, which I find very natural.

Let me start, for the sake of motivation, with the absolute version. Let $G$ be a group and $X$ be a probability measure preserving Lebesgue $G$-space. By a separable $G$-semimetric on $X$ I mean an a.e defined, measurable $G$-invariant function $X\times X \to [0,\infty)$ which satisfies the usual axioms of a semimetric (a metric, with the possibility of distance 0 between distinct points) such that the natural metric quotient is separable (has a countable dense subset). Obvious extremal examples are the trivial semimetric (the function 0) and actual metrics. It is a nice exercise to check that $X$ is compact (in the sense used in ergodic theory) iff it admits a separable $G$-metric and it is weakly mixing iff the only separable $G$-semimetric it admits is the trivial one. The last property is also denoted metric ergodicity.

Consider now a $G$-extension $X\to Y$ of pmp spaces. By a separable $G$-semimetric on $X$ relative to $Y$ I mean an a.e defined, measurable $G$-invariant function $X\times_Y X \to [0,\infty)$ which satisfies fiberwise the usual axioms of semimetric such that natural metric quotient on a.e fiber is separable. Then one checks that the extension is compact iff it there exists a separable $G$-metric on $X$ relative to $Y$ and it is weakly mixing iff the only separable $G$-semimetric on $X$ relative to $Y$ is the trivial one. The last property is also denoted relative metric ergodicity.

Let me also note that Furstnberg's structure theorem is actually a rather easy corollary of the above-mentioned exercise.

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  • $\begingroup$ Thank you for your alternative definition which I will go through shortly. Is it really pointless to try to formulate things measure theoretically though? It seems like Definition 2 can be repaired, though it may be asking too much for something similar for compact extensions… $\endgroup$ Commented Oct 6, 2023 at 15:27
  • $\begingroup$ It is not pointless at all to formulate things measure theoretically. I would say that the usual definition of a weakly mixing extension is a measure theoretic one. However, I wouldn't call this a repair of your Definition 2... $\endgroup$ Commented Oct 6, 2023 at 15:42
  • $\begingroup$ Ah, I was thinking about the suggested repair by Terence Tao in the comments. $\endgroup$ Commented Oct 6, 2023 at 15:44
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    $\begingroup$ @NateRiver, you asked "is there any equivalent formulation of relative weak mixing that is more immediately intuitive?" I guess this is rather subjective. $\endgroup$ Commented Oct 6, 2023 at 15:47
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    $\begingroup$ Yes, this and the corollary that every pmp action is a weakly mixing extension of a tower of compact ones. $\endgroup$ Commented Oct 6, 2023 at 15:53

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