I am beginning my study of measure theory and I came across the Radon-Nikodym Theorem (and derivative). I tried to play with the concept around with "elementary" objects but I am having troubles understanding. Suppose I have two random variables $X,Y$ and that they have a joint distribution $dP_{XY}$ I tried to show that $dP_{X|Y=y} \ll dP_X$ for a given $y\in supp(Y)$ and it seems to work fine when they are both continuous or discrete (unless I made some mistakes). But what happens when $X$ or $X|Y=y$ are mixed distributions?
Like, suppose that $X= \mathcal{N}(0,1)$ w.p. 1/2 and $0$ otherwise. If $X|Y$ behaves in the same way for every $y$ we should not have any issues, is that correct? When is it that I lose absolute continuity? The only counter-example I can come up with is if I take $dP_X =$ Lebesgue Measure, and at that point, if $X|Y=y$ is distributed as above (Gaussian wp 1/2 and 0 otherwise) then we lose absolute continuity.
Is it even possible to have $dP_X=\lambda$ (Lebesgue measure) and $dP_{X|Y=y}$ to be a mixed distribution? Or, in general, if $dP_X$ is continuous (and thus has measure $0$ in all $\mathbb{Q}$, for instance) can $dP_{X|Y=y}$ be discrete in some intervals, considering I have a joint $dP_{XY}$ and that implies a constraint on $dP_X$?
Observe that for some measurable set $E$ \begin{align*} P_X(E)&= \int_{\mathcal Y} P_{X|Y}(E|y) dP_Y(y) \end{align*} If the first one is $0$, then $P_{X|Y}(E|y)$ is also $0$ modulo $p_Y$ since it is positive or null. Hence absolute continuity in the general case.