I'm wondering if these two definitions of mutually singular measure are equivalent? I've been trying to prove the equivalence but none of my attempts have been fruitful. Any suggestions would be much appreciated:
Let $\mu$ be a signed finite measure on $\mathcal{F}$ a sigma algebra on a set $\mathcal{X}$, and $\nu$ a finite positive measure on $\mathcal{F}$. Then $\mu \perp \nu$ if there exists a measurable $B$ such that $\mu(B^c) = 0 = \nu(B)$.
The second definition is with the same setup as above, except we now require for any measurable $E$, $\mu(B^c \cap E) = 0 = \nu(B \cap E)$.
Best Answer
They are equivalent for positive measures. The second one is fixed up so that it can be used for signed measures (or complex measures).