Probabilist often work on Polish spaces. Does somebody know an ("nonexotic") 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 nonexotic example I mean something like a stochastic process, which is really used in applications, and cannot be defined on a Polish space...(I posted the this question also here).

This answer is identical to the one I gave when the same question was posted at M.SE.: There are a number of constructions that do not work for Polish spaces, but a certain class of probability spaces, variously known as superatomless, 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 superatomless if there is no $A\in\Sigma$ satisfying $\mu(A)>0$, such that the pseudometric space obtained by endowing the trace $\sigma$algebra on $A$ with the pseudometric $d(A,B)=\mu(A\triangle B)$ is separable. 

