Let $X,U,V$ be three subsets of $\mathbb{R}^n,\mathbb{R}^a,\mathbb{R}^b$ respectively and $\mu$ a probability measure supported on $X\times U\times V$.
The marginal $\mu_X$ of $\mu$ w.r.t $x$ is the Lebesgue measure: $$ dx = d\mu_X(x) = \int_{U\times V} d\mu(x,u,v) \qquad (\star) $$
We disintegrate $\mu$ as
$$ d \mu (x,u,v) = \Psi((u,v)\mid x) dx $$
- Is the condition $(\star)$ is necessary to have such disintegration ?
I would like to force $\Psi((u,v)\mid x)$ to be as singular as possible, that is $$ \Psi((u,v)\mid x) = \delta_{u(x),v(x)} $$ where $\delta$ is the Dirac measure on $U\times V$.
- Can we impose conditions on the moments of the measure $\mu$ such that $\Psi$ is this Dirac ?