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Let $(X,\mathscr{F})$ be a measurable space, and let $M$ be the set all probability measures $\mu: \mathscr{F} \to [0,1]$. Let us denote with $\mathscr{M}$ the $\sigma$-algebra on $M$ generated by the mappings $\mu \mapsto \mu(F)$, with $F \in \mathscr{F}$.

Now fix a Borel set $B$ of $\mathbb{R}$ and $A \in \mathscr{F} \otimes \mathscr{F}$. How can we prove that $$ \{\mu \in M: (\mu \times \mu)(A) \in B\} \in \mathscr{M}? $$

[Here $(\mu\times \mu)$ stands for the product measure and $\mathscr{F}\otimes \mathscr{F}$ for the product $\sigma$-algebra]

[Linked thread on MSE: here]

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    $\begingroup$ Is this a homework exercise? $\endgroup$
    – Anthony Quas
    Commented Mar 1, 2016 at 19:30
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    $\begingroup$ Not really, just a curiosity which I would like to solve (to be precise, I asked myself why a variant of deFinetti representation theorem was meaningful); I posted this morning the same problem on MSE, but it didn't receive much attention.. $\endgroup$
    – Nduccio
    Commented Mar 1, 2016 at 20:02
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    $\begingroup$ So in general it's considered impolite to simultaneously post to MO and MSE (it can lead to people duplicating work). The preferred strategy if you're not sure is to post to MSE; wait for a day or two before posting to MO. Also, when you do cross-post, make sure that each posting refers to the other. $\endgroup$
    – Anthony Quas
    Commented Mar 1, 2016 at 21:39
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    $\begingroup$ I am sorry Anthony, I will take it into account next time.. I added the links to both threads, let me know if I forgot something.. $\endgroup$
    – Nduccio
    Commented Mar 1, 2016 at 22:44
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    $\begingroup$ Answered on Math.SE. $\endgroup$
    – Nate Eldredge
    Commented Mar 1, 2016 at 23:34

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