I have a question regarding defining conditional probabilities and the concept of regular conditional probabilities. I have read other posts but I am still struggling with certain issues.
Imagine we have a measurable map $X$ from a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ to a measure space $(\tilde{\Omega}, \mathcal{G})$. Let $\mathcal{F}_0 \in \mathcal{F}$. Let $A \in \mathcal{G}$.
I want to define $\mathbb{P}(X \in A| \mathcal{F}_0)$ as $\mathbb{E}(\mathbb{1}_A(X)|\mathcal{F}_0)$. $\:$(*)
1) Is this an acceptable/standard definition?
2) If not, is the reason for regular conditional probabilities to deal with issues that may arise from the above definition?
3) I know conditional expectation is unique only up to null sets. Is a possible issue with the above definition that for another $A' \in \mathcal{G}$ it may be the case that $\mathbb{P}(X \in A| \mathcal{F}_0)$ and $\mathbb{P}(X \in A'| \mathcal{F}_0)$ do not "align" in the way we would want them to for certain $\omega$.
4) If the spaces are sufficiently "nice" is (*) a reasonable definition?
Best Answer
In the definition of regular conditional probability we want $P(X \in A|\mathcal F_0)$ to be a measure in $A$ and a measurable fuction for fixed $A$. In view of the fact that conditional probabilites are defined only up to null sets (those null sets varying with $A$) it is impossible to get regular conditional probabilities directly from the definition of conditional probabilities, even in nice spaces. Considerable amount of work is needed to show that regular conditional probabilities exist in nice spaces (like Polish spaces).