Let $\displaystyle \mu,\nu$ be measures on $\displaystyle (X,F)$ such that $\displaystyle \nu << \mu$ and $\displaystyle \lambda = \mu+\nu$. Show that $\displaystyle f= \frac{d\nu}{d\lambda}$ exists and satisfies $\displaystyle 0 \leq f \leq 1$ a.e $\displaystyle \lambda$.

(Hint: $\displaystyle \mu_1 << \mu_2 $ and $\displaystyle \mu_2 << \mu_1$ implies $\displaystyle \frac{d\mu_1}{d\mu_2}=(\frac{d\mu_2}{d\mu_1})^{-1}$.

i do not have any idea how to do this. any help would be appreciated.