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