Characterization of a joining over a common subsystem. Given a joining measure $\lambda$ on $X\times Y$, where $(X,\mu)$ and $(Y,\nu)$ are two probability measure space, let $\lambda=\int (\lambda_y \times \delta_y)\,d\nu(y)$ be the disintegration of $\lambda$ over $\nu$. We denote by $P_{\lambda}:L^2(X,\mu)\rightarrow L^2(Y,\nu)$ the conditional expectation operator given by $$(P_{\lambda}f)(y)=\mathbb{E}_{\lambda}^Y(f)(y)=\int_xf(x)\,d\lambda_y(x) \text{ for } \nu\text{-a.e.} y.$$
In Theorem $6.8$ of the book 'Ergodic theory via joining' by Glasner, the author says that if $\lambda=\mu\times_{\eta}\nu$, where $(Z,\eta)$ is the common factor of $(X,\mu)$ and $(Y,\nu)$ determined by $\lambda$, then the operator $P_{\lambda}:L^2(X,\mu)\rightarrow L^2(Y,\nu)$ is the projection onto $L^2(Z,\eta)$. My doubts are the following:

*

*What does the author mean by $(Z,\eta)$ to be the common factor of $(X,\mu)$ and $(Y,\nu)$ determined by $\lambda$?

*How can we show that if $\lambda=\mu\times_{\eta}\nu$, then $P_{\lambda}=P_Z$, i.e. the projection onto $L^2(Z,\eta)$? $\big($Note that, if $\mu=\int_z\mu_z\,d\eta (z)$ and $\nu=\int_z \nu_z\,d\eta (z)$ be the disintegrations of $\mu$ and $\nu$, respectively with respect to $\eta$, then $\mu\times_{\eta}\nu:=\int_z(\mu_z\times\nu_z)d\eta(z)$$\big)$.

Thanks in advance for any help.
 A: Below I won't get into complete details. One important point is that since the spaces on which the group acts is "standard" from the measure theoretical point of view, morphisms between $\sigma$-algebras (or measure algebras) determine a.e.-unique morphisms of measure spaces, and conditional expectations can be upgraded to conditional measures. Thus the standardness assumption allows one to transition between multiple categories with minimal hassle.


*

*According to Thm.6.6 on p.129, any joining $\lambda\in \mathbb{J}(\mathfrak{X},\mathfrak{Y})$ of two systems $\mathfrak{X}=(X,\mathcal{X},\mu,\Gamma)$ and $\mathfrak{Y}=(Y,\mathcal{Y},\nu,\Gamma)$ determines two invariant sub-$\sigma$-algebras $\mathcal{L}\leq \mathcal{X}$ and $\mathcal{R}\leq \mathcal{Y}$ such that $\mathcal{L}\times Y=_\lambda X\times \mathcal{R}$ (as Boolean sub-$\sigma$-algebras of $\mathcal{X}\otimes\mathcal{Y}$). More precisely, $\mathcal{L}$ and $\mathcal{R}$ are defined like so:

$$\mathcal{L}=\{A\in\mathcal{X}\,|\, \exists B\in\mathcal{Y}: \mu((A\times Y)\triangle (X\times B))=0\},$$
$$\mathcal{R}=\{B\in\mathcal{Y}\,|\, \exists A\in\mathcal{X}: \mu((A\times Y)\triangle (X\times B))=0\}.$$
(In my opinion it's useful to draw a caricature here. A square represents $X\times Y$, and by way of a joining $\lambda$ we can replace vertical strips (i.e. elements of $\mathcal{L}\times Y$) with the horizontal strips (i.e. elements of $X\times \mathcal{R}$).)
This gives an isomorphism of Boolean $\sigma$-algebras $\mathcal{L}\to\mathcal{R}$. Since the all the spaces involved are "standard" $\mathcal{L}$ determines a factor $\mathfrak{L}$ of $\mathfrak{X}$ and likewise $\mathcal{R}$ determines a factor $\mathfrak{R}$ of $\mathfrak{Y}$, and the isomorphism $\mathcal{L}\to\mathcal{R}$ gives an isomorphism $\mathfrak{L}\to\mathfrak{R}$ of systems. $\mathfrak{Z}_\lambda=(Z_\lambda,\mathcal{Z}_\lambda,\xi_\lambda,\Gamma)$ is an anonymous representative from the isomorphism class of $\mathfrak{L}$ (which is the same as the isomorphism class of $\mathfrak{R}$). $\mathfrak{Z}_\lambda$ is called the common factor of $\mathfrak{X}$ and $\mathfrak{Y}$ determined by the joining $\lambda\in\mathbb{J}(\mathfrak{X},\mathfrak{Y})$.
Note that Thm.6.5 gives the analogous statement with $\mathcal{L}$ replaced with the whole $\mathcal{X}$ and $\mathcal{R}$ replaced with the  whole $\mathcal{Y}$; in this case the systems themselves are isomorphic, instead of having isomorphic factors.



*Let us consider the category of systems (or the standard probability spaces, resp.), with the arrows equivariant measurable measure-preserving maps (or arrows measurable measure-preserving maps, resp.). For readability let us suppress everything but the space (= point set = largest element of the associated $\sigma$-algebra = spatial model) and the associated measure, unless it is necessary to be explicit. Let us also denote by $L^0$ the space of real-valued measurable functions identified a.e. according to the prescribed measure. If $\phi: (X,\mu)\to (Y,\nu)$ is an arrow, on function spaces we have two maps: one is the categorical pullback ( = Koopman operator) $\overleftarrow{\phi}: L^0(Y,\nu)\to L^0(X,\mu)$. The other is the conditional expectation $\phi_!: L^0(X,\mu)\to L^0(Y,\nu)$. More explicitly $\phi: (X,\mu)\to (Y,\nu)$ determines a sub-$\sigma$-algebra $\overleftarrow{\phi}(\mathcal{Y})\leq \mathcal{X}$, which in turn determines a conditional expectation operator $\mathbb{E}_\mu\left(\cdot\,\left|\, \overleftarrow{\phi}(\mathcal{Y})\right.\right): L^0(X,\mathcal{X},\mu)\to L^0\left(X,\overleftarrow{\phi}(\mathcal{Y}),\left.\mu\right\vert_{\overleftarrow{\phi}(\mathcal{Y})}\right)$. Then the following diagram defines $\phi_!$ unambiguously:

What is more, $\phi_!$ is the left inverse to $\overleftarrow{\phi}$, i.e. $\phi_!\circ \overleftarrow{\phi}=\operatorname{id}_{L^0(Y,\nu)}$.
Let us note the relationship between $\phi_!$ and the disintegration of $\mu$ along $\phi$. If $f\in L^0(X,\mu)$, then
$$\int_X f(x)d\mu(x) = \int_Y \int_X f(x)\, d\mu_y(x) \, d\nu(y) = \int_Y \phi_!(f)(y)\, d\nu(y),$$
that is, $\phi_!(f)(y)=\int_X f(x)\,d\mu_y(x)$ for $y\in Y$ a typical point.
Just as $\overleftarrow{\phi\circ \psi}=\overleftarrow{\psi}\circ\overleftarrow{\phi}$, we have $(\phi\circ\psi)_!=\phi_!\circ \psi_!$. Still, from the categorical point of view conditional expectation is a wrong-way map.
Next let us consider an anonymous square in the category of systems (or in the category of standard probability spaces):

Applying the pullbacks and conditional expectations we get the following squares:

To go from $L^0(X,\mu)$ to $L^0(Y,\nu)$ we can use pullbacks and conditional expectations in tandem, and indeed, Glasner's $P_\lambda$ is exactly (the $L^2$ analog of) $\chi_!\circ \overleftarrow{\phi}$, and the statement that $P_\lambda=P_Z$ is exactly (the $L^2$ analog of) $\chi_!\circ \overleftarrow{\phi}=\overleftarrow{\omega}\circ \psi_!$, that is, as a diagram,

The common factor defined by a joining construction mentioned in the previous section says that if in an anonymous square the bottom right corner is missing it can be filled in:

Likewise, the relatively independent joining construction you mention says that if in an anonymous square the top left corner is missing it can be filled in:

Further, in this case by Exr.6.7.1 on p.129 we'll have that $(Z,\xi)\cong (Z_{\mu\times_\xi \nu},\xi_{\mu\times_\xi \nu})$, i.e. up to isomorphism $(Z,\xi)$ is the common factor of $(X,\mu)$ and $(Y,\nu)$ determined by the relatively independent joining $\mu\times_\xi \nu$.
With all this background we can rewrite Thm.6.8 of Glasner like so:
Theorem: Consider an anonymous square in the category of systems:

Then $(W,\lambda)\cong (X\times Y, \mu\times_\xi \nu)$ and $(Z,\xi)\cong (Z_\lambda,\xi_\lambda)$ iff $\chi_!\circ \overleftarrow{\phi}=\overleftarrow{\omega}\circ \psi_!$.
(You asked only about the necessity so I'll mention details for that only, if needed I can add details for sufficiency also, with this formalism.)
Proof of necessity: To show necessity assume $\lambda=\mu\times_\xi \nu$ and fix an $f\in L^0(X,\mu)$. We claim that $\chi_!\circ \overleftarrow{\phi}(f)=_\nu\overleftarrow{\omega}\circ \psi_!(f)$. At a typical $y\in Y$, starting from the LHS we have:
\begin{align*}
\chi_!\circ \overleftarrow{\phi}(f)(y)
&=\int_{W} f\circ \phi(w) \,d(\mu\times_\xi \nu)_y(w)
=\int_X f(x)\, d\mu_{\omega(y)}(x)\\
&=\psi_!(f)\circ \omega(y)
=\overleftarrow{\omega}\circ \psi_!(f)(y).
\end{align*}
Here the commutativity of the diagram is used in the second equality.
