How are these statements of the Radon-Nikodym theorem the same

Question

I am reading An Informal Introduction to Stochastic Calculus with Applications and I've come across this statement of the Radon-Nikodym theorem.

"Consider the probability space $(\Omega ,\mathcal {F},P)$, and let $\mathcal {G}$ be a $\sigma$ field in $\mathcal {F}$. The for any random variable $X$ there is a $\mathcal {G}$ measurable random variable such that $Y$ such that

$\int_{A}XdP=\int_{A}YdP$ "

this occurs after some discussion of conditional probabilities.

How does this statement correspond to the more familiar

${\displaystyle \nu (A)=\int _{A}f\,d\mu ,}$

Answer 1

Assuming that $X$ has finite mean, $\nu (A)=\int_AXdP$ define a (real) measure $\nu$ and $\nu (A)=0$ whenever $P(A)=0$. So there exists $Y$ such that $\nu(A)=\int_A YdP$. [RNT applies to real (and complex) measures also].