Probabilist often work on Polish spaces. Does somebody know an ("non-exotic") example, for which it is not possible to work on a Polish space, but instead one has to work on a general measurable space? By non-exotic example I mean something like a stochastic process, which is really used in applications, and cannot be defined on a Polish space...(I posted this question also here).
Probabilistic Example which might not be defined on a Polish space
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1There is a good [link](http://mathoverflow.net/questions/20919/polish-spaces-in-probability) on MO, with examples of what can go wrong when the assumption Polish is being dropped...this might help in constructing an example... – 2012-07-10
1 Answers
There re a number of constructions that do not work for Polish spaces, but a certain class of probability spaces, variously known as super-atomless, saturated, nowhere countably generated and a number of other names. A nice overview can be found here.
A probability space $(\Omega,\Sigma,\mu)$ is saturated if for every two Poilsh spaces $X$ and $Y$, every probability measure $\nu$ on $X\times Y$ and every random variable $f:\Omega\to X$ such that its distribution $\mu f^{-1}$ equals the marginal of $\nu$ on $X$, there is a random variable $g:\Omega\to Y$ such that the joint distribution of $(f,g)$ is $\nu$.
The following definition is conceptually different, but can be shown to be equivalent:
A probability space $(\Omega,\Sigma,\mu)$ is super-atomless if there is no $A\in\Sigma$ satisfying $\mu(A)>0$, such that the pseudo-metric space obtained by endowing the trace $\sigma$-algebra on $A$ with the pseudo-metric $d(A,B)=\mu(A\triangle B)$ is separable.
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0@Andy: Section 4 is about the existence of Nash equilibria, which are certain functions, even though the proofs make use of set-valued functions. [Here](http://homepage.univie.ac.at/papers.econ/RePEc/vie/viennp/vie0807.pdf) is a paper that uses these kind of spaces to construct extended product spaces which are used in the modeling of ideosyncratic risk in large populations in economics. – 2012-07-11