If $\nu$ and $\mu$ are $\sigma$-finite measures on $(X,F)$ such that $\mu \ll \nu$ we have a $\nu$-unique random variable $X \in F$ such that $\mu(A) = \int_A Xd\nu \space \forall A \in F$
I so not quite understand how we show that $X = \frac{d\mu}{d\nu}$. We can also express this as: $d\mu = X d\nu$
Moreover, what does it mean for a random variable Y to be distributed $dP$ so: $Y \sim dP$ where P is a probability measure?
Thank you for insight!!!