In the Wikipedia article for conditional expectation, conditional probability is defined in terms of conditional expectation.
Given a sub sigma algebra of the one on a probability space.
Given a probability space $(\Omega, \mathcal{F}, P)$, a conditional probability $P(A \mid \mathcal{B})$ of a measurable subset $A \in \mathcal{F}$ given a sub sigma algebra $\mathcal{B}$ of $\mathcal{F}$, is defined as the conditional expectation $E(A \mid \mathcal{B})$ of indicator function $i_A$ of $A$ given $\mathcal{B}$, i.e. $ P(A \mid \mathcal{B}): = E(A \mid \mathcal{B}), \forall A \in \mathcal{F}.$
So actually the conditional probability $P(\cdot \mid \mathcal{B})$ is a mapping $: \Omega \times \mathcal{F} \rightarrow \mathbb{R}$.
A conditional probability $P(\cdot \mid \mathcal{B})$ is called regular if $P(\cdot|\mathcal{B})(\omega), \forall \omega \in \Omega$ is also a probability measure.
Question:
what are some necessary and/or sufficient conditions for a conditional probability $P(\cdot \mid \mathcal{B})$ to be regular?
Given a r.v. on a probability space.
Suppose $(\Omega, \mathcal{F}, P)$ is a probability space and $(U, \mathcal{\Sigma})$ is a measurable space. There seem to be two ways of defining the conditional expectation $E(X\mid Y)$ of a r.v. $X: \Omega \rightarrow \mathbb{R}$ given another r.v. $Y: \Omega \rightarrow U$, either as a $\sigma(Y)$-measurable mapping $: \Omega \rightarrow \mathbb{R}$, or as a $\Sigma$-measurable mapping $: U \rightarrow \mathbb{R}$, as in my previous post.
If one let $X$ to be the indicator function $1_A$ for some $A \in \mathcal{F}$, one can similarly define $E(1_A \mid Y)$ to be conditional probability of $A$ given $Y$, denoted as $P(A\mid Y)$. Therefore $P(\cdot \mid Y)$ is a mapping $: \Omega \times \mathcal{F} \rightarrow \mathbb{R}$ or a mapping $: U \times \mathcal{F} \rightarrow \mathbb{R}$.
Questions:
(1). What are some necessary and/or sufficient conditions for $P(\cdot \mid Y)$ to be regular, i.e. to be a mapping $: \Omega \rightarrow \{ \text{probability measures on }(\Omega, \mathcal{F}) \}$ or a mapping $: U \rightarrow \{ \text{probability measures on }(\Omega, \mathcal{F}) \}$?
(2). Under what kinds of conditions, will $P(X \mid Y)$ defined as above be equal to the ratio $\frac{P(X, Y)}{P(Y)}$, the definition used in elementary probability?
Thanks and regards! References (links or books) will also be appreciated!