I am trying to solve this exercise: Let $\mu=\mu^+-\mu^-$ be the Hahn-Jordan decomposition of a finite signed measure on a measurable space $(X, A)$. Show that for any bounded measurable $f:X\rightarrow\mathbb{R}$ we have
$$\left\lvert\int fd\mu\right\rvert\leq\int\lvert f\rvert d\lvert\mu\rvert$$
where $\lvert\mu\rvert=\mu^++\mu^-$ is the total variation measure.
I have some questions:
- Is it true that $\int fd\mu=\int fd\mu^+-\int fd\mu^-$?
- In case 1. is true, I had written
$$\int fd\mu=\int fd\mu^+-\int fd\mu^-\leq\int fd\mu^++\int fd\mu^-=\int f d\lvert\mu\rvert$$
and hence
$$\left\lvert\int fd\mu\right\rvert\leq\int\lvert f\rvert d\mu\leq\int\lvert f\rvert d\lvert\mu\rvert$$
I think my approach is correct, buy maybe 1 requires a proof. How do I prove it? Thank you!
Best Answer
Your inequality $\displaystyle \left| \int f \, d\mu \right| \le \int |f| d\mu$ on the last line isn't correct. What if $\mu$ is a negative measure?
The Hahn-Jordan decomposition is (essentially) unique, so your equation 1. may as well be the definition of the integral with respect to the signed measure $\mu$. (If it isn't what is the definition you are using?)
In this case you can write $f = f^+ - f^-$ and $\mu = \mu^+ - \mu^-$ to get $$ \int f \, d\mu = \int f^+ \, d\mu^+ - \int f^+ d\mu^- - \int f^- d\mu^+ + \int f^- d\mu^-.$$ Since each of these integrals is nonnegative you have $$ \int f \, d\mu \le \int f^+ \, d\mu^+ + \int f^+ d\mu^- + \int f^- d\mu^+ + \int f^- d\mu^-$$ and this last expression can be rearranged to $\displaystyle \int |f| \, d|\mu|$.