Metric spaces and the support of a random variable

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

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