Cfrgtkky

This question comes from my attempt to understand theorem $7.21$ in E-W. This concerns the dichotomy between relatively weak-mixing extensions and compact extensions. I cannot understand the proof as I do not understand these concepts. I will sketch my understanding of the proof until the part I'm certain I don't understand.

All systems are taken to be ergodic, invertible, and Borel.

Definition 1: An extension $psi:(X,mathscr{B},mu,T) to (Y,mathscr{A},nu,S)$ is said to be relatively weak-mixing if the system $(Xtimes X,mathscr{B}otimesmathscr{B},bar{mu},Ttimes T)$ is ergodic. Here $bar{mu}$ is the relatively independent join over $Y$(See E-W definition $6.15$).

Definition 2: Again, let $psi:(X,mathscr{B},mu,T) to (Y,mathscr{A},nu,S)$ be an extension. Let $mu^{psi^{-1}mathscr{A}}_x$ be the conditional measures on $(X,mathscr{B})$. A function $fin L^2(X,mathscr{B},mu)$ is almost-periodic $(AP)$ with respect to $Y$ if given $varepsilon>0$, there exist $g_1,dots,g_rin L^2(X,mathscr{B},mu)$ such that $forall n in mathbb{Z}$, $nu$-a.e $y$, $$min_{i}|U_T^nf-g_i|_{L^2(mu_y^{mathscr{A}})}<varepsilon.$$
Here $U_Tf:=fcirc T$ and $mu_y^{mathscr{A}}$ are the measures on $(X,mathscr{B})$ obtained by completing the conditional measure diagram:
$require{AMScd}$
begin{CD}
X @>mu^{psi^{-1}mathscr{A}}_.>> Mleft(bar{X}right)\
@V{psi}VV @| \
Y @>mu^{mathscr{A}}_.>> Mleft(bar{X}right)
end{CD}

Definition 3: The extension $psi:(X,mathscr{B},mu,T) to (Y,mathscr{A},nu,S)$ is said to be a compact extension if the set of $AP$ (with respect to $Y$) functions in $L^2(X,mathscr{B},mu)$ is dense.

Theorem 7.21: If the extension $psi:(X,mathscr{B},mu,T) to (Y,mathscr{A},nu,S)$ is not relatively weak-mixing, there exists a non-trivial intermediate factor $Xto X^* to Y$ with the property that $X^* to Y$ is a compact extension.

Sketch of proof: (For details see E-W page $200$ onwards) By the hypothesis, there exists a non-constant $H in mathscr{L}^infty(Xtimes X,bar{mu})$ invariant under $Ttimes T$ ($mathscr{L}$ denotes the fact that $H$ is bounded and not just essentially-bounded). For $phi in L^2(X,mathscr{B},mu)$, define $$H*phi(x) = int H(x,x')phi(x')dmu^{psi^{-1}mathscr{A}}_x(x').$$

One can show this formula defines a bounded operator $L^2(X,mathscr{B},mu)to L^2(X,mathscr{B},mu).$ And, a priori, this formula defines a compact operator $L^2(X,mathscr{B},mu^{psi^{-1}mathscr{A}}_x) to L^2(X,mathscr{B},mu^{psi^{-1}mathscr{A}}_x)$.

Using this and the additional fact that $U_T(H*phi)=H*(U_Tphi)$, one shows that $lbrace H*phi : phi in mathscr{L}^{infty}(X,mathscr{B},mu)rbrace$ is an $AP$ (wrt. $Y$) family in $L^{infty}(X,mathscr{B},mu)$. Moreover, one can show that this family contains functions which are not $psi^{-1}mathscr{A}$-measureable.

Then if we define $mathscr{F}=lbrace fin L^{infty}(X,mathscr{B},mu): f text{ is } AP text{ wrt. } Yrbrace $ and $mathscr{B}^*$ to be the smallest $sigma$-algebra making the functions in $mathscr{F}$ measurable, one can show that $mathscr{F} subset L^2(X,mathscr{B}^*,mu)$ is dense.

Here's the part I don't understand: It is then claimed that the $T$-invariant $sigma$-algebra $mathscr{B}^* (subset mathscr{B}$) gives rise to the intermediate, non-trivial, compact extension. I assume that this is done by completing the following system of extensions from $Mleft(bar{X}right)$ to $Y$:
begin{CD}
X @>psi>> Y\
@Vmu^{mathscr{B}^*}_.VV \
Mleft(bar{X}right)
end{CD}

The $T$-invariance of $mathscr{B}^*$ shows that $mu^{mathscr{B}^*}_.$ gives an extension. And I guess the density of $mathscr{F}$ will be used to show $Mleft( bar{X}right)$ is a compact extension of $Y$.

Question: To complete the above diagram, don't we need to show that $psi^{-1}mathscr{A}subset mathscr{B}^*$? Is this obvious? It is not explicitly addressed in the text.

Thanks for reading, all help is appreciated.

Edit: As John Griesmer points out, $psi^{-1}mathscr{A}subsetmathscr{B}^*$ would follow if one could show every characteristic function $1_{psi^{-1}A}$ on $X$ was AP with respect to Y. I don't quite see why such a statement should be true though.

Take the function $1_{A}circpsi$. I want to find functions $g_iin L^2(X,mathscr{B},mu)$ and estimate
$$int|1_{A}(psi(T^nx))-g_i(x)|^2dmu_{x'}^{psi^{-1}mathscr{A}}(x) = int|1_{A}(S^n(psi x))-g_i(x)|^2dmu_{x'}^{psi^{-1}mathscr{A}}(x)$$

as $x'$ varies in $X$. Now if the $g_i$'s happened to be functions on $Y$, I could rewrite the last integral as $$int|1_{A}(S^ny)-g_i(y)|^2dnu_{psi x'}^{mathscr{A}}(x) = Eleft(|U_S^n1_A-g_i|^2:mathscr{A},nuright)(psi(x')) = |U^n_S1_A - g_i|^2(psi(x'))$$
by the properties of the conditional expectation operator. So I guess, I'm left trying to play around with the equation
$$min_i|U_s^n1_A(y) - g_i(y)|^2 < varepsilon, $$
and hope that it holds for almost every $y$ and for all $n$. Am I on the right track? what would be sensible choices for the $g_i$? Apologies if I'm missing something obvious; I only understand these things formally and have no intuition for what's actually going on.

You are correct: you need to show $psi^{-1}mathscr A subset mathscr B^*$ (modulo null sets). Brief outline: if $A$ is a measurable subset of $Y$, then it should follow quickly from the definition that the characteristic function of $psi^{−1}(A)$ is $AP$ wrt $Y$ (I don't have their exact definition in front of me, so I won't try to fill in the details here). Then the definition of $mathscr F$ and $mathscr B^*$ imply $Ain mathscr B^*$.