Let $(S,\mathcal{S})$ and $(T,\mathcal{T})$ be measurable spaces. A *transition probability* from $S$ to $T$ is a function $\pi:S\times\mathcal{T}\to [0,1]$ such that $\pi(s,\cdot)$ is a probability measure for all $s\in S$ and $\pi(\cdot,B)$ is measurable for all $B\in\mathcal{T}$.

Now let $(\Omega,\Sigma,\mu)$ be a probability space and let $f:\Omega\times S\to T$ be a jointly measurable function.

Under what conditions is $\pi_f:S\times\mathcal{T}\to [0,1]$ given by $\pi_f(s,B)=\mu > \{\omega:f(s,\omega)\in B\}$ a transition probability?

Clearly, the issue is measurability.

Motivation: If $(T,\mathcal{T})$ is standard Borel, there always exists a function such that $\pi=\pi_f$ when $\Omega$ is atomless, this is Proposition 10.7.6. in Bogachev. Moreover, such random functions have been used as a definition of mixed strategies in game theory and I think it would be interesting to understand them in terms of transition probabilities in a general setting.