Does the use of metric spaces to describe the support of a random variable provide any greater illumination? I ask this after reading about how metrics spaces have been used to unify the mathematical measure theoretic nature of probability and the physical intuition that most associate with probability. You can read my inspiration here: http://www.arsmathematica.net/archives/2009/02/14/complete-metric-spaces-and-the-interpretation-of-probability/
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Here are some technical conveniences of separable metric spaces (a) If $X$ and $X'$ take values in a separable metric space $(E,d)$ then the event $\{X=X'\}$ is measurable, and this allows to define random variables in the elegant way: a random variable is the equivalence class of $X$ for the "almost surely equals" relation (note that the normed vector space $L^p$ is a set of equivalence class) (b) The distance $d(X,X')$ between the two $E$-valued r.v. $X, X'$ is measurable; in passing this allows to define the space $L^0$ of random variables equipped with the topology of convergence in probability (c) Simple r.v. (those taking only finitely many values) are dense in $L^0$
And some techical conveniences of complete separable (Polish) metric spaces : (d) Existence of the conditional law of a Polish-valued r.v. (e) Given a morphism between probability spaces, a Polish-valued r.v. on the first probability space always has a copy in the second one
answered Jan 8, 2012 at 17:14
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Interesting reference. Its value for me lies in questioning the ability of measure theoretic probability to capture an "intuition" about probability (whatever that might mean) and going on to propose an intriguing distinction; namely, between a set of measure zero having a measure zero neighborhood and a set of measure zero all of whose proper neighborhoods have positive measure.
It is not apparent that separable metric spaces are the "right" way to capture this idea, though, as the comment by Matt Heath points out. It sounds like we only need a predefined subcollection of measurable sets (not necessarily even satisfying the axioms of a topology). Such a collection is conveniently obtained in a separable metric space but there are other ways to create such collections, too. Thus it appears that the idea presented here illuminates the connection between abstract measure theory and using random variables in models, but the use of metric spaces may be a bit of a red herring.
