<< notation on measures

Question

If $\nu$ and $\mu$ are two signed measures of finite variation, what does the notation $\nu << \mu$ means?

Answer 1

If $\mu$ is positive, then we write that $ \nu \ll \mu $ iff $\mu(S)=0\implies\nu(S)=0$ for every measurable set $S$. We say that $\nu$ is absolutely continuous with respect to $\mu.$

If $\mu$ is signed, then we replace $\mu$ by $|\mu|$ in the above.