Let $(\Omega_i, \mathcal{F}_i), i=1,2$ be measurable spaces. Their product measurable space is $(\Omega, \mathcal{F})$.
Let $\mu_1$ be a probability measure on $(\Omega_1, \mathcal{F}_1)$, and let $(\mu_{ω_1})_{ω_1∈\Omega_1}$ be a transition probability from $\Omega_1$ to $\Omega_2$. Then there exists a probability measure $\mu$ , defined by $$\mu(A)= \int_{\Omega_1} \mu_{ω_1}(A_{ω_1})\mu_1(dω_1), \quad \forall A \in \mathcal{F}$$ where $A_{ω_1}:= \{\omega_2 \in \Omega_2: (\omega_1, \omega_2) \in A\}$.
My questions are:
Conversely, given any probability measure $\mu$ on $(\Omega, \mathcal{F})$, do there exist a probability measure $\mu_1$ on $(\Omega_1, \mathcal{F}_1)$, and a transition probability $(\mu_{ω_1})_{ω_1∈\Omega_1}$ from $\Omega_1$ to $\Omega_2$, such that $$\mu(A)= \int_{\Omega_1} \mu_{ω_1}(A_{ω_1})\mu_1(dω_1), \quad \forall A \in \mathcal{F} ?$$ Can they be explicitly or implicitly determined?
Are such probability measure $\mu_1$, and transition probability $(\mu_{ω_1})_{ω_1∈\Omega_1}$ unique?
- What if considering general measures instead of probability measures? Are the answers yes only up to scaling of measures?
Thanks and regards!