Question 1.1)
Let $X_1 = 1_{A}$ for $A\in \mathbb{F}$ and $X_2$ be a R.V. Where $X_2$ takes value in an arbitrary Borel Space. Let $X_1,X_2$ take values in $(A_1, \mathbb{A}_1)$, $(A_2, \mathbb{A}_2)$ Find an expression for the regular conditional distribution (RCD) of $X_2$ given $X_1 = x$.
I know that since $X_2$ takes value in a Borel space, the RCD exists. I know it should be on the form $p: A_1\times \mathbb{A} \to [0,1]$ where $B_2 \to p(x_1, B_2)$ should be a p-measure on $(A_2, \mathbb{A}_2)$.
I have tried to choose $$p(x_1, B_2) = 1_{A}(x_1) P(X_2 \in B_2),$$ but obviously this isn't a p-measure if $x_1\notin A$ and the construction also gives a feeling of independence.
I did also think about $$ p(x_1, B_2) = P(X_2\in B_2, x_1 \in A),$$ but I feel like that is too obvious? I feel like I have to use the indicator in some way? EDIT: And an issue is, that $x_1 \in A_1$, and we have defined A to be a subset of $\mathbb{F}$.
I struggle construction that RCP function, are there any obvious explaination I am missing out on?
What you wrote is kind of confusing. If $X_1$ is an indicator function, then it takes values in $\{0,1\}$, so why mention an abstract space of values $(A_1, \mathbb A_1)$? Also there needs to be some underlying probability space. Is $\mathbb F$ the sigma-algebra on that space? Moreover, what is $\mathbb A$?
If we assume that $(\Omega, \mathbb F, P)$ is a probability space, $A \in \mathbb F$, and $X_2$ is a r.v. taking values in the Borel space $(A_2, \mathbb A_2)$, then the problem has a simple answer.
The sigma-algebra (of subsets of $\Omega$) generated by $X_1=1_A$ is just $\{\emptyset, A, A^c, \Omega\}$. Let $\mathcal A(\omega)=A$ if $\omega \in A$ and $= A^c$ if $\omega \in A^c$. Then,
$$p(\omega, B) = \frac{P(\{X_2 \in B\} \cap \mathcal A(\omega))}{P(\mathcal A(\omega))}$$
defines a rcd (if $P(\mathcal A(\omega))=0$ for some $\omega$, then let $p(\omega, \cdot)=p$ or any other probability measure).