Trying to obtain some exchangeability-related results, I ended up with the following questions, which I couldn't answer (at least, not in the negative); this is also related to this [MO thread](https://mathoverflow.net/questions/152907/convex-hulls-of-families-of-probability-measures) (edit: actually, not really)


Given a measurable space $(S,\mathscr{S})$, let $\pi$ be a probability measure on the product space $(S^2,\mathscr{S}^2)$. Also, endow the set $\mathcal{P}(S)$ of probability measures over $(S,\mathscr{S})$ with the $\sigma$-algebra generated by the measurable maps $\nu \mapsto \nu(F)$, with $F \in \mathscr{S}$ (and similarly for $\mathcal{P}(S)^2$).



> 
> **Question.** Is it true that we can find *always* a probability measure $\mu$ on $\mathcal{P}(S)^2$ such that
>  $$
> \pi(A)=\int\_{\mathcal{P}(S)^2} (\alpha \otimes \beta)(A) \,\mu(\mathrm{d}(\alpha\otimes \beta))\,\,\,\,\,?
> $$
>  for all $A \in \mathscr{S}^2$.
> 
> 
> 


[Here, given probability measures $\alpha,\beta$ on $(S,\mathscr{S})$, we let $\alpha \otimes \beta$ denote their product measure]



Representation of probability measure over product spaces
It's true; we can use the following rather trivial construction.


Notation: for measurable $A \subset S^2$, let $F\_A : \mathcal{P}(S)^2 \to [0,1]$ be the "evaluation map" $F\_A(\alpha, \beta) = (\alpha \otimes \beta)(A)$.


Consider the map $T : S^2 \to \mathcal{P}(S)^2$ defined by $T(x,y) = (\delta\_x, \delta\_y)$. This is the natural way to embed a space into its space of probability measures: map a point to a point mass. We note that $T$ is measurable, since for any $F\_A$ we have $F\_A(T(x,y)) = (\delta\_x \otimes \delta\_y)(A) = 1\_A(x,y)$. So $F\_A \circ T = 1\_A$ is a measurable map from $S^2$ to $[0,1]$. Since the $F\_A$ generate the $\sigma$-algebra on $\mathcal{P}(S^2)$, the measurability of $T$ follows.


Now for a given measure $\pi$, simply let $\mu = \pi \circ T^{-1}$ be the pushforward of $\pi$ under $T$. Then by change of variables
$$\begin{align\*}\int\_{\mathcal{P}(S)^2} F\_A(\alpha, \beta) \,\mu(d\alpha, d\beta) &= \int\_{S^2} F\_A(T(x,y))\,\pi(dx, dy) \\ &= \int\_{S^2} 1\_A(x,y)\,\pi(dx,dy) \\ &= \pi(A).\end{align\*}$$


