Hey, I'm reading the proof of proposition 6.8 in Big Rudin, pg. 120. I'll just mention where my confusion lies: Suppose K is an arbitrary measure (complex, real, whatever). In the proof of property (f), Rudin says that if K(A) = 0, and F is a subset of A, then K(F) = 0. Rudin has forced that by construction in chapter 1 for positive measures, but why should it work for complex or signed measures? Does there exist two sets A,B such that A is a subset of B, measure of B is 0, but measure of A is non-zero (negative?), for some arbitrary measure?
EDIT: Rudin's prop 6.8, part (f) says: given two arbitrary measures $\lambda_1$, $\lambda_2$, a positive measure $\mu$. If $\lambda_1 \ll \mu$, and $\lambda_2 \perp \mu$, then $\lambda_1 \perp \lambda_2$.
Also, $\ll$ and $\perp$ denote absolute continuity and mutual singularity, respectively.