Given a set Ω and a σ-algebra F of subsets, is there some natural way to assign something like a "uniform" measure on the space of all measurable functions on this space? (I suppose first it would be useful to know if there's even a natural σ-algebra to use on this space.)
The reason I'm asking is because I'd like to do the following. Let Ω be the (2-dimensional) surface of a sphere, with the uniform probability distribution. Let F be the Borel σ-algebra, and let G be the sub-algebra consisting of all measurable sets composed of lines of longitude. (That is, S is in G iff S is measurable and for all x in S, S contains all points with the same longitude as x.) Let A be the set of all points with latitude 60 degrees north or higher (a disc around the north pole).
Let f be a G-measurable function defined on Ω such that the integral of f over any G-measurable set B equals the measure of (A\cap B). (This is a standard tool in defining the conditional probability of A given G-measurable sets.) It's not hard to show that for any such function f, for almost-all x, f(x) will equal the unconditional measure of A.
What I'd like to be able to say is that for any x, for almost-all such functions f, f(x) will equal the unconditional measure of A. However, I can't say "almost-all" on the functions unless I have some measure on the space of functions.
Clearly I can do this by concentrating all the measure on the single constant function in this set. But I'd like to be able to pick out this most "generic" such function even in cases where A isn't so nice and symmetric.
Maybe there's some other, simpler question I should be asking first?