Total variation distance for probability measures

Total variation distance for probability measures

I'm having troubles solving this problem, any help will be appreciate :)
Let $(\Omega,\mathcal{F})$ a measurable space, and let $\mu, \ \nu$
probability measures on $\mathcal{F}$. It's well know that
$d(\mu,\nu)=2\sup_{A\in F}\{|\mu(A)-\nu(A)|\}$ is a distance on the space
of probability measures. Let $\nu=\nu_a + \nu_s$ (the Lebesgue
decomposition of $\nu$ w/r $\mu$, i.e. $\nu_a\ll \mu$ and
$\nu_s\perp\mu$).
Prove that $d(\mu,\nu)=2\int_{\Omega} \left(1-\frac{d\nu_a}{d\mu}\right)_+
d\mu$
Thanks