suppose $v$ is a signed measure on $(X,M)$ and $E\in M$
how do i go about showing that $|v|(E)=sup\{\sum_1^n |v(E_j)|: E_j\cap E_i=0 \forall i\neq j, \cup_1^n E_j=E\}$
sorry it took me awhile to fix the latex
suppose $v$ is a signed measure on $(X,M)$ and $E\in M$
how do i go about showing that $|v|(E)=sup\{\sum_1^n |v(E_j)|: E_j\cap E_i=0 \forall i\neq j, \cup_1^n E_j=E\}$
sorry it took me awhile to fix the latex
Hint: Consider the Hahn Decomposition Theorem, which allows us to split $X$ into a positive and negative set for $\nu$. That is, we may write $X=P\cup N$, $P\cap N=\emptyset$ where $P$ is a positive set for $\nu$ and $N$ is a negative set for $\nu$.