Finding a distribution satisfying uncountably many constraints. Any relevant references?

Question

The problem I'm dealing with has the following form. Let $X$ be some uncountable set, and $Y$ be some finite set. Suppose $f: X \times Y \to [0,1]$, and given $\mathcal{H} \subseteq Y^X$, I'm looking for some $P$, a distribution with support on $\mathcal{H}$, such that

$$ \forall x, \ f(x, y) = \int_{h\in \mathcal{H}: \ h(x) = y}dP(h), $$

assuming it exists. In the case that $X$ is finite, this boils down to a linear system, but given my background, I'm totally at a loss in approaching the general case. It seems to me that this is something of a infinite dimensional feasibility problem. Maybe some optimization literature review is in order? Any other literature I should be looking at?

Answer 1

It seems that in general this is an almost arbitrarily hard problem.

Consider the simpler countably infinite case $X=\mathbb N$, $Y=\{0,1\}$. Thus $H=\{x:h(x)=1\}$ is a "random set".

Fix $g:X\to\{0,1\}$ and let $f(x,g(x))=1$ (which forces $f(x,1-g(x))=0$). Thus $G=\{x:g(x)=1\}$ is an arbitrary subset of $\mathbb N$ and by $\sigma$-additivity we get $P(H=G)=1$.

Since $P$ is supposed to be concentrated on $\mathcal H$, $P$ as required exists iff $G\in\mathcal H$.

So determining, given $f$ and $\mathcal H$, whether $P$ exists is at least as hard as determining whether a given real number (or equivalently subset of $\mathbb N$) belongs to a given set of reals.