# Fibered product of probability spaces

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 $$(U, \mathcal{H}, \rho)$$?

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 $$H$$.