Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and let $X:(\Omega, \mathcal{A}) \rightarrow (\mathcal{X}, \mathcal{F})$ and $Y:(\Omega, \mathcal{A}) \rightarrow (\mathcal{Y}, \mathcal{G})$ be random variables.
Now I am interested in the differences between the differences between the following two objects:
i.) Regular conditional distribution of X under Y:
A Markov kernel from $(\Omega, \sigma(Y))$ into $(\mathcal{X}, \mathcal{F})$ is called regular conditional distribution of X under Y, $K:= \mathbb{P}_{X\mid Y}$ iff: $$K(\cdot, A)=\mathbb{P}(X\in A\mid \sigma(Y)), \quad \text{for all } A\in \mathcal{F} $$
ii.) Regular conditional distribution of X under Y=y:
A Markov kernel from $(\mathcal{Y}, \mathcal{G})$ into $(\mathcal{X}, \mathcal{F})$ is called a factorized regular conditional distribution of X under Y=y, $K(y, \cdot) = \mathbb{P}_{X\mid Y=y}$, iff: $$K(y, A) = \mathbb{P}(X\in A\mid Y=y), \quad\mathbb{P}\text{-almost surely, } \quad A\in \mathcal{F}$$
My questions are now the following:
1.) Are both $\mathbb{P}_{X\mid Y}$ and $\mathbb{P}_{X\mid Y=y}$ pushforward measures?
2.) why is (ii) only defined up to $\mathbb{P}$-almost surely ( (and (i) not) ?
3.) So $\mathbb{P}_{X\mid Y}$ should accept to inputs, right? One $A\in\mathcal{F}$ and the other a number. The question is though from which set? Does it get an $\omega \in \Omega$ or a $y\in\mathcal{Y}$ which the notation suggests? Which one does it use as input and why?
4.) And how are the two definitions generally related? The notation suggests that $\mathbb{P}_{X\mid Y}$ should accept two arguments, $y\in \mathcal{Y}$ and $A\in\mathcal{F}$, while $\mathbb{P}_{X\mid Y=y}$ should only accept an $A\in\mathcal{F}$ (assuming here that the answer to Question 1.) was yes that indeed both are pushforward measures). So the notation also seems to suggest that $\mathbb{P}_{X\mid Y=y}$ is somehow a paremetric version of $\mathbb{P}_{X\mid Y}$ where one variable was just simply fixed. Is that indeed the case?