Transition, marginal probability measures and probability measure on product space 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!
 A: *

*The answer is in general no. The first such an example was constructed by Dieudonne and can be found in many advanced books on probability. Existence is guaranteed under various assumptions that are mostly topological or emulate topological regularity conditions. The strongest results obtained this way are found in a paper by Pachl. The simplest case where existence is guaranteed is when one deal with (the product of) the real line. An accessible proof for that case can be found, for example in the book by Lehmann and Romano (I like their exposition, but the result can be found in many books). Computation of these probabilites is in general not possible. Rao has written a book that carefully looks at the challenges of calculating these conditional probabilities- and pretty much everything else about the topic. The book makes for hard reading.

*They are unique up to a measure zero subset of $\Omega_1$ for countable generated probability or measure spaces $\Omega$ (10.4.3. in Bogachev). Otherwise, not necessarily (10.10.44 in Bogachev).

*One can certainly do this for finite measures and to some degree also for infinite measures. The simple proof in Lehmann and Romano alluded to above does not work for general spaces, however (at least not without major adaptions).  
Generally, the whole area, known as regular conditional probabilities and, relatedly, disintegrations, is fairly technical and advanced. A good guide is given by Chapter 10 in Bogachev.
