Definition and derivation of conditional expectation/probability

I read quite a few books introducing the notion of conditional probabilities/expectation by putting a formula out there coming from what they call "intuition".

Can someone provide me a good measure theoretic derivation of how to find the conditional expectation or probability of a random variable ? Or even a link or something.

Basically knowing the basic in measure theory,let $(\Omega, Z,P)$ be a probability space.

How do you derive finally that any function g s.t :

\begin{align} \int_A YdP = \int_A gdP_{|G} \end{align}

is the expectation of $Y$ knowing $G$. And also how do we go from there to find : \begin{align} P(A|B) = \frac{P(A,B)}{P(B)} \end{align}

• – BCLC
Aug 2 '18 at 12:35