# Prove that $\frac{d \lambda}{d \mu} = \frac{d \lambda}{d \nu} \frac{d \nu}{d \mu}$ for $\sigma$-finite measures $\mu,\nu, \lambda$.

Suppose that $\mu,\nu, \lambda$ are $\sigma$-finite measures on a measurable space $(X,\mathcal{M})$ and that $\lambda \ll \nu \textrm{ and } \nu \ll \mu .$ Prove that $\frac{d \lambda}{d \mu} = \frac{d \lambda}{d \nu} \frac{d \nu}{d \mu} \;\; \mu\textrm{-a.e.}.$