Let $\mu$ be a signed measure. I want to prove the following:
(1) If $A$ is a positive set for $\mu$, then $\mu(A)=|\mu(A)|$.
(2) If $A$ is a negative set for $\mu$, then $\mu(A)=-|\mu(A)|$.
This is what I have done:
From $\mu^+ = \frac{1}{2}(|\mu|+\mu)$, I get that $2\mu^+(A)-\mu(A)=|\mu(A)|$.
Similarly, from $\mu^-=\frac{1}{2}(|\mu|-\mu)$, I get $2\mu^-(A)+\mu(A)=|\mu(A)$.
My questions are the following:
Can I say that $\mu^+(A)=\mu(A)$, and $\mu^-(A)=-\mu(A)$, since $A$ is positive in the first case and negative in the second? If so, how can I prove them?