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Let $(\Omega,\Sigma,\mu)$ be a probability space, $(X,\mathcal{X})$ be a measurable space and $R(\Omega,X)$ be the set of equivalence classes of measurable functions from $\Omega$ to $X$ under almost everywhere equality, they are random elements. Let $(\mathcal{A},\mu_A)$ be the measure algebra of $(\Omega,\Sigma,\mu)$, that is $\mathcal{A}$ identifies elements of $\Sigma$ if their symmetric difference has outer measure zero and $\mu_A$ is defined in the natural way in terms of its representatives.

I would like to know if one can identify the elements of $R(\Omega,X)$ with something that can be canonically be constructed in terms of $(\mathcal{A},\mu_A)$ and $(X,\mathcal{X})$.

The motivation behind the question is the following: I work with certain random elements that are defined on a countably generated probability space. By Maharam's theorem, this amounts to the measure algebra being isomorphic to one that consists of a convex combination of Lebesgue measure on $[0,1]$ and a discrete probability space. I would like to know whether it makes sense for me to say that I'm essentially working with such a probability space.

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Perhaps this calls for consideration of the Image measure catastrophe ... – Gerald Edgar Mar 10 '13 at 16:27
@Gerald: could you please summarize the image measure catastrophe in a comment? – Tom LaGatta Mar 11 '13 at 0:34

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