I have trouble understanding the connection of conditional densities for two (continuous) random variables and conditional distributions for measurable sets.
Let $\nu$ be a probability measure on $\mathbb{R}^N$ with Radon-Nikodym-derivative $f$ (w.r.t. the Lebesgue-measure $\lambda^N$). Now, for two measurable sets $A,C \subset \mathbb{R}^N$ with $\nu (C) \neq 0$ the conditional probability is given via $$\nu (A|C) := \int_{\Omega}\frac{\chi_C}{\nu (C)} d\nu = \frac{\nu (A \cap C)}{\nu (C)}.$$
So we could write the Radon-Nikodym-derivative w.r.t. $\nu$ as $$\frac{d \nu (\cdot | C)}{d \nu} = \frac{\chi_C}{\nu (C)} $$ and with the "chain rule" we get $$\frac{d \nu (\cdot | C)}{d \lambda} =\frac{f \chi_C}{\nu (C)} . $$
So far, so good. No on to random variables. Let $\Omega$ be a probability space and $X,Y: \Omega \rightarrow \mathbb{R}$ two random variable with the joint probability measure $\mu_{X,Y}$ with joint density $f_{X,Y}$ and marginal densities $f_X(x)$, $f_Y(y)$, that is $$\mu_{X,Y} (A \times B ) = \mu_{X,Y} (X\in A \land Y \in B ) = \int_{A \times B} f(x,y) \,d\lambda(x) \, d\lambda(y).$$ $$\mu_X(A) := \mu (A \times \mathbb{R}) = \int_A \underbrace{\int_{\mathbb{R}} f(x,y) \,d\lambda (y)}_{f_X(x)} \, d\lambda (x) = \int_A f_X(x) d\lambda (x) $$
Now, in literature, the conditional density is given via $$f_{Y|X}(y|x) = \frac{f_{X,Y}(x,y)}{f_X(x)}.$$ And this is what I don't understand. Is this a mere definiton, or does this result from all the above definitions? If the latter, how exactly? How would even the conditioned probability measure for random variables look like? Something like $\mu_{Y|X} (B|A)$? If so, then $A$ should at least show up in the density definiton, right?