Alternate definitions of compact and weak mixing extensions

Question

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!

Answer 1

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.