Suppose that a measure on $X \times Y$ is given. In what situations can one define a measure on $\{x\} \times Y$ to imitate conditional probability? More generally, if a measure on $Z$ is given, and $C$ is a (nice, but maybe measure zero) subset of $Z$, then can one define the conditional probability measure, conditioned on lying in $C$.
A reasonable litmus test for a general procedure should be: If $G$ is a compact group, and $\mu$ is the Haar measure, then if $H$ is a compact subgroup, the measure conditioned to being in $H$ should be the Haar measure of $H$.
In "geometric" situations, like $\mathbb{R} \times \mathbb{R}$, one can imagine a procedure that takes the measure of tubular neighborhoods of subsets of $0 \times \mathbb{R}$ and renormalizes them appropriately. In the case of $G$ a topological group from before, assuming that $G$ has some metric structure, then given some $U \subset H$, one could define $\mu(U | H)$ as $\lim_{\epsilon \to 0} \frac{ \mu(U + B_{\epsilon}(e))} {\mu(B_{\epsilon(e)})}$, where $B_{\epsilon}(e)$ is an epsilon ball around the identity element of the group.
Another test for the correct notion: In the case of $G$ and $H$ above, if $T$ is a (nice) set of transversals, then there ought to be a Fubini type theorem for integrating a function on $G$ in terms of integrating it over cosets of $H$ and then over $T$, using these kind of induced measures. (And something similarly for a bundles.)
(Therefore, it should also be the case that this "undoes" the product measure construction.)
Is there a general theory for this situation that someone can recommend or describe to me?
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Additional thoughts - given a family of measures $\rho_x$ on $Y$ parametrized by the measure space $(X, \Sigma, \mu)$, so that for an open $U$ in $X \times Y$, the function $x \to \rho_x(U \cap \{x \} \times Y)$ is $\mu$-integrable, we can define a measure on $X \times Y$ by Fubini's theorem. Running over all such families, what is the image of this map into the space of all measures on $X \times Y$? (In particular, it should include the product measures, but should also be larger, since the measure on the fibers can vary in some fashion.)
I'm surprised nobody answered this question yet. Given a product measure, one can disintegrate it w.r.t. any of its marginal to get a regular conditional probability. Check out this and that. Conditions are quite similar and require some topological regularity of the image space.