The problem is that
Suppose $\nu(E)=\int fd\mu$ where $\mu$ is a positive measure and $f$ is an extended $\mu$-integrable function. Describe the Hahn decompositions of $\nu$ and the positive, negative, and total variations of $\nu$ in terms of $f$ and $\mu$.
I know that $P=\{x\in X: f(x)\geq 0\ \mu-a.e.\} $ and $N=\{x\in X: f(x)\leq 0\ \mu-a.e.\}$ are the Hahn decomposition of $\nu$. But $\nu^{+}(E)=\nu(E\cap P)=\int_{E\cap P}f d\mu$ and $\nu^{-}(E)=-\nu(E\cap N)=-\int_{E\cap N}fd\mu$? and $|\nu|(E)=\int_{E\cap P}f d\mu-\int_{E\cap N}fd\mu$? Can we simplify them?
Thanks for any help!