I was reading a book on information theory and entropy by Robert Gray, when I saw the following definition of conditional probability:
Given a probability space $(\Omega,\mathcal{B}, P)$ and a sub-$\sigma$-field $\mathcal{G}$, for any event $H\in\mathcal{B}$ the conditional probability $m(H\text{ }|\text{ }\mathcal{G})$ is defined as any function , say $g$, which satisfies the two properties:
(1) $g$ is measurable with respect to $\mathcal{G}$
(2) $\displaystyle\int_{G}ghdP=m(G\bigcap{}H)$; all $G\in\mathcal{G}$
I am quite confused with this definition since it is very different from the definition through joint probability of events.
I understand what measurable function, sub-$\sigma$-field and probability space are, and I'm guessing that the author is trying to definie the measure $m$ through the measurable function $g$, but I don't quite understand what the second requirement is saying. Especially, what does that h
in $\displaystyle\int_{G}ghdP$ refer to? it just jumped out of nowhere in the book, so I'm suspecting that it may have some conventional meaning?
I'd appreciate it a lot if someone can help. Thank you!!