This is an exercise in Royden and Fitzpatrick Real Analysis.
Let $\mu$ be a measure and $\mu_1$ and $\mu_2$ be mutually singular measures on a measurable space $(X, \mu)$ for which $\mu = \mu_1 – \mu_2$. Show that $\mu_2 = 0$.
I'm quite beginner and have difficulty to understand the definitions well.
First it seems that $\mu_1$ and $\mu_2$ are not necessarily signed measures. Is this the meaning of the question's requirement: "for any measurable subset $E \subset X$ we have $\mu_2(E) = 0$"
by definition of singularity of measures there exists measurable subsets of $X$ , such that $X= A\cup A^c$ with $\mu_1(A)=0=\mu_2(A^c)$. How do I use this to show the result?
Best Answer
As defined, all of them are measures, not signed measures.
Yes. However, since $\mu_2$ is a measure, not a signed measure, it's enough to show $\mu_2(X) = 0.$ [If you haven't proven it before, show that this property suffices to show $\mu_2=0.$]
As for your final question, you should evaluate $\mu(A)$ for the set $A$ you defined.