I am studying measure theory and had a few questions. In the text I am using, it defines total variation of a measure $\nu$ as
$$|\nu| = \nu^+ + \nu^-$$ My first question is why are we interested in this quantity to begin with? Secondly, the text claims that the above implies $\nu$ is of the form $\nu(E) = \int_E f d|\nu|$, for any measurable set $E$. I cannot see how this integral definition follows the total variation definition. Lastly, the text claims the inequality $|\nu(E)| \leq |\nu|(E)$ is obvious, but I cannot quite see how. One last question I have is one of notation. I see notation such as $g(\frac{d\nu}{d\mu})$, what does this mean exactly?
Thank you.
(1) Why are we interested in the total variation?
Firstly, because of (2) and (3), below. We can often prove things about signed measures using this.
(2) Why is $\nu(E) = \int_E f d|\nu|$?
Hahn decomposition.
(3) Why is $|\nu(E)| \leq |\nu|(E)$?
Use your solution to (2).
(4) I doubt you often see $g(\frac{d\nu}{d\mu})$, that is a function $g$ evaluated on a Radon-Nikodym derivative. But you do often see $g\;\frac{d\nu}{d\mu}$, that is a function $g$ multiplied by a Radon-Nikodym derivative.