Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question

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 $\nu$.

share|cite|improve this answer
    
so $|v|(E)=v(E \cap P)-v(E \cap N)$, so we can let $E_1=E \cap P$ and $E_2=E \cap N$, clearly they are disjoint and union to $E$. Is this all I have to say? – jack Mar 5 '11 at 15:41

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.