Let $(\Omega, \mathcal{F}, P)$ be a probability space, $\mathcal{G} \subseteq \mathcal{F}$ a $\sigma$-field in $\mathcal{F}$. Given $A \in \mathcal{F}$, the Radon-Nikodym theorem implies that there is ${ }^{[3]}$ a $\mathcal{G}$-measurable random variable $P(A \mid \mathcal{G}): \Omega \rightarrow \mathbb{R}$, called the conditional probability, such that
$$
\int_{G} P(A \mid \mathcal{G})(\omega) d P(\omega)=P(A \cap G)
$$
for every $G \in \mathcal{G}$, and such a random variable is uniquely defined up to sets of probability zero. A conditional probability is called regular if $\mathrm{P}(\cdot \mid \mathcal{B})(\omega)$ is a probability measure on $(\Omega, \mathcal{F})$ for all $\omega \in \Omega$ a.e.
I am used to the non-measure theoretic definition of conditional probability defined for events where $P(A|B)=\frac{P(A\cap B)}{P(B)}$. Why is this function not defined like
$$\frac{1}{P(G)}\int_{G} P(A \mid \mathcal{G})(\omega) d P(\omega)=\frac{P(A \cap G)}{P(G)}
$$
which would make it in line with the definition for events? I understand that $P(G)=0$ is an issue, but what is the motivation of defining this representation for $P(A\cap G)$?
Best Answer
In the case where $\mathcal G$ is the $\sigma$-algebra generated by event $G$, $P(A \mid \mathcal G)$ is (up to sets of measure $0$) $P(A|G)$ when $G$ is true, $P(A|G^c)$ when $G$ is false. The equation then says $P(A \cap G) = P(G) P(A | G)$ and $P(A \cap G^c) = P(G^c) P(A | G^c)$, which fits with your non-measure theoretic definition.