Hi Isobel, what is exactly your idea?
As I see it, you might mean you have a measurable space A and a `random variable' X from A to another space B which is given a measurable structure by a family of functions f:B\to R. Then you would want to use the values E[f(X)], for f ranging over the family, in order to identify an element b\in B for which you would define
E[X]:=b.
Is that right? And also: is it related to your previous question about Riesz spaces?
This seems related to the ideas of Irving Segal about abstract probability spaces.
If you have a sigma-algebra generated by functions from a set S you can focus on the functions and forget about the sigma-algebra. The functions generate an abelian complex *-algebra (you can add and multiply functions, and take their complex conjugates), denote it by A. Segal defines an expectation operator as a complex-linear functional E on the algebra, which is positive and normalised. This means
E[aa*] > 0 for all a in the algebra
E[1] = 1
That is enough to represent the abstract algebra and the state as a concrete algebra of continuous functions on a concrete topological space, with E represented as the integral with respect to a Borel probability measure.
See: http://www.jstor.org/stable/2372714
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