In Folland's text he states,
The total variation of a complex measure $\nu$ is the positive measure $|\nu|$ determined by the property that if $d\nu = f d\mu$ where $\mu$ is a positive measure, then $d|\nu| = |f|d\mu$. To see that this is well defind, we observe first that every $\nu$ is of the form $fd\mu$ for some finite measure $\mu$ and some $L^1(\mu)$; indeed we can take $\mu = |\nu_r| + |\nu_i|$ and use Theorem 3.12 to obtain $f$.
Here $\nu_r$ and $\nu_i$ represent the real and imaginary portions of the complex measure $\nu$. The theorem he references is the Lebesgue-Radon-Nikodym Theorem for complex measures:
Theorem 3.12 If $\nu$ is a complex measure and $\mu$ is a $\sigma$-finite positive measure on $(X, \mathcal{M})$, there exists a complex measure $\lambda$ and an $f \in L^1(\mu)$ such that $\lambda \perp \mu$ and $d\nu = d\lambda + f d\mu$.
What I am unclear on is his claim that $\nu = \int f d\mu$. To my understanding, Theorem 3.12 states we should instead have $\nu = \lambda + \int fd\,\mu$, for some complex measure $\lambda$. Why is the $\lambda$ omitted here?
Best Answer
He is using $\mu = |\nu_i| + |\nu_r|$. For this $\mu$, we have $\nu << \mu$, so $\lambda = 0$.