Suppose $X,Y$ are two real valued random variables on $(\Omega,F,P)$, i.e. measurable maps $X:\Omega\to\mathbb R$ and $Y:\Omega\to\mathbb R$.
I am interested in defining a conditional distribution function $F_{Y\mid X=x}(y)$
Since $\mathbb R$ is polish, we can define the regular conditional distribution of $Y$ given $X=x$ by a Markov kernel $k_{Y,X}(x,A)$ where $P(Y\in A\mid X=x)=k_{Y,X}(x,A)$ for almost all $x\in\mathbb R$ and for all $A\in \mathbb R$. Now I would like to define $F_{Y\mid X=x}(y):=k_{Y,X}(x,(-\infty,y])$ and call this a conditional distribution function. But I am curious if this makes any sense since $P(Y\in A\mid X=x)=k_{Y,X}(x,A)$ is just defined almost everywhere.
So my question is, if it is possible to define a conditional distribution function in this setting? Does my definition makes sense? If so could someone elaborate on this?
The Markov kernel $k=k_{Y,X}$ has two key properties: (1) $x\mapsto k(x,A)$ is Borel measurable for each Borel set $A\subset \Bbb R$; (2) $A\mapsto k(x,A)$ is a probability measure on $(\Bbb R,\mathcal B(\Bbb R))$ for each $x\in\Bbb R$. And moreover, $$ P[Y\in A, X\in B] =\int_B k(x,A) P_X(dx), $$ for Borel sets $A,B\subset\Bbb R$. [$P_X$ is the distribution measure of $X$.] Because of (2), $F_{Y|X=x}(y):= k(x,(-\infty,y])$ is a cumulative distribution function on $\Bbb R$ for each $x\in\Bbb R$, and $$ P[Y\le y, X\in B]=\int_B F_{Y|X=x}(y) P_X(dx), $$ for each $Y$. This justifies calling $F_{Y|X=x}(y)$ a conditional distribution function.