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](http://rads.stackoverflow.com/amzn/click/3540345132). Moreover, such random functions have been used as a [definition of mixed strategies](http://www.ma.huji.ac.il/raumann/pdf/Mixed%20and%20Behavior.pdf) in game theory and I think it would be interesting to understand them in terms of transition probabilities in a general setting.



Random Functions and Transition Probabilities
I've found a rather simple solution. No additional assumptions are necessary. There is a natural method of composing transition probabilities that gives rise to a new transition probability (see for example (2) [here](http://arxiv.org/pdf/1205.1488.pdf)). 


We can identify probability measures with transition probabilities constant in the first argument. Also, we can identify $f$ with the transition probability $\kappa:\Omega\times S\times\mathcal{T}\to[0,1]$ given by $\kappa(\omega,s,B)=1$ if $f(\omega,s)\in B$ and $\kappa(\omega,s,B)=0$ if $f(\omega,s)\notin B$. 


Also, the pointwise product of transition probabilities [is a transition probability](https://math.stackexchange.com/questions/84078/product-of-two-probability-kernel-is-a-probability-kernel/94692#94692).
So let $\kappa\_\mu:S\times\Sigma\to[0,1]$ the representation of $\mu$ as a tranition probability. Also, let $\kappa\_1:S\times\mathcal{S}\to[0,1]$ be the identity transition probability on $S$. Let $\pi:S\times\mathcal{S}\otimes\mathcal{T}\to[0,1]$ be the pointwise product of $\kappa\_\mu$ and $\kappa\_1$. 


Then $\pi\_f=\kappa\circ\pi$ and is therefore a transition probability as the composition of transition probabilities. 


