Fibered product of probability spaces

Question

Let $(\Omega_1, \mathcal{F}_1, \mu_1)$, $(\Omega_2, \mathcal{F}_2, \mu_2)$ and $(T, \mathcal{G}, \nu)$ be probability spaces. A probability space morphism is a measurable function that preserves the probability measure, i.e., $\Omega_1 \xrightarrow{f} T \xleftarrow{g} \Omega_2$ means that $f$, $g$ are measurable and $\mu_1 \circ f^{-1} = \nu = \mu_2 \circ g^{-1}$. The fibered product of $\Omega_1$, $\Omega_2$ along $f$, $g$ is $U = \Omega_1 \times_{T} \Omega_2 = \{ (\omega_1, \omega_2) \mid \omega_1 \in \Omega_1, \omega_2 \in \Omega_2, f(\omega_1) = g(\omega_2) \}$. We can equip $U$ with the smallest $\sigma$-algebra $\mathcal{H}$ such that the functions $p_1((\omega_1, \omega_2)) = \omega_1$ and $p_2((\omega_1, \omega_2)) = \omega_2$ are measurable to form a measurable space $(U, \mathcal{H})$.

Under what conditions does a probability measure $\rho$ exist such that $p_1$ and $p_2$ are probability space morphisms from $(U, \mathcal{H}, \rho)$?

Answer 1

According to a construction from Lewis Bowen, Robin Tucker-Drob (2021). Superrigidity, measure equivalence, and weak Pinsker entropy, where they call the resulting product the relatively independent product, it is sufficient for $(\Omega_1, \mathcal{F}_1, \mu_1)$, $(\Omega_2, \mathcal{F}_2, \mu_2)$, $(T, \mathcal{G}, \nu)$ to be standard Borel.

According to the disintegration theorem, we can decompose the measures $\mu_1$ and $\mu_2$ as

\begin{align*} \mu_1(F) &= \int_T (\mu_1)_\tau(F_\tau) \;d\nu(\tau)\text, &\mu_2(F) &= \int_T (\mu_2)_\tau(F_\tau) \;d\nu(\tau)\text, \end{align*} where $F_\tau = \{ \omega_1 \in F \mid f(\omega_1) = \tau \}$ (resp. $F_\tau = \{ \omega_2 \in F \mid g(\omega_2) = \tau \}$) are the fibers of $F$.

Then we can define $\rho$ as \begin{equation*} \rho(H) = \int_T \bigl[ (\mu_1)_\tau \otimes (\mu_2)_\tau \bigr](H_\tau) \;d\nu(\tau)\text, \end{equation*} where $(\mu_1)_\tau \otimes (\mu_2)_\tau$ is the product measure of the disintegrated measures at $\tau$, and $H_\tau = \{ (\omega_1, \omega_2) \in H \mid f(\omega_1) = g(\omega_2) = \tau\}$ are the fibers of $H$.