How to find total variation of a signed measure??

Question

Let $(X,F,\nu)$ be a signed measure on the sigma algebra F. now by Jordan-Hahn decomposition theorem $\nu = \nu_1 - \nu_2$, where $\nu_1$ and $\nu_2$ are positive and mutually singular, and such measures are unique.

now to prove

$$\sup_{\text{finite partitions $E_k$ of $X$}} \sum_{1 \leq k \leq n} | \nu(E_k) | = \nu_1 + \nu_2$$

I am stuck to show this plz help me..

Answer 1

Let $|\nu| = \nu_1 + \nu_2$. By the triangle inequality, for any $E \subset X$, $|\nu(E)| \leq |\nu|(E)$. For any finite partition $E_1, \dots, E_n$ of $X$, $$\sum_{k}|\nu(E_k)| \leq \sum_{k}|\nu|(E_k) = |\nu|(X).$$ This shows that the sup is $\leq |\nu|(X)$. To get the other inequality, let $E_1, E_2$ be a Hahn decomposition for $X$.