Assume we have random variables $X,Y$ mapping from a probability space $(\Omega,\mathcal{F},\mathbb{P})$ to some spaces $\mathcal{X},\mathcal{Y}$.
We assume there is some measure $\mu$ on the product space $\mathcal{X} \times \mathcal{Y} $ such that there exists a radon-nikodyn derivative of the joint distribution with respect to this measure, that is $$ \frac{d \mathbb{P}_{(X,Y)}}{d\mu} = p(x,y).$$
Now in the case that $\mu$ factorizes we can obtain conditional densities via
$$p(x|y) = \frac{p(x,y)}{p(y)} = \frac{\frac{d \mathbb{P}_{(X,Y)}} {d\mu}}{ \frac{d\mathbb{P}_X} {d\mu_{\mathcal{X}} }}.$$
I was wondering what happens in the general case, i.e. where $\mu$ does not necessarily behave like e.g. the Lebesgue-measure. I suspect that I have to use the disintegration theorem in some way, but I cannot wrap my head around this. Maybe I would need something like $$ \frac{\frac{d \mathbb{P}_{(X,Y)}} {d\mu}}{ \frac{d\mathbb{P}_X} {d\mu(dx|y) }}$$
but I can not really prove it with my basic knowlege of measure theory.