In Folland's Real Analysis book on page 91, proposition 3.9 is referred as Chain rule,
Let $\nu$ be sigma finite signed measure and $\mu,\lambda $ be a sigma finite measure on $(X,\mathscr{M})$, such that $\nu \ll \mu$ and $\mu \ll \lambda$, then (a)For all $g\in L^1(\nu)$ then $g(\frac{d\nu}{d\mu})\in L^1(\mu)$ and $$\int gd\nu=\int g \frac{d\nu}{d\mu}d\mu$$ (b)We have $\nu \ll \lambda$ and $$\frac{d\nu}{d\lambda}=\frac{d\nu d\mu}{d\mu d\lambda}\text{ a.e.}$$
Should I think of $g(\frac{d\nu}{d\mu})$ as a composite of a $\nu$-integrable function with a $\mu$-integrable function? but then later in the integral $\int g \frac{d\nu}{d\mu}d\mu$ the parenthesis is taken away, so is this a product instead? Can someone help me understand? Thanks!
No. It’s a product of functions.
The brackets are because Folland’s type-setting is such that he doesn’t like to use inline fractions, so he writes $g(d\nu/d\mu)$.