Question on a definition

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March 11th 2009, 09:54 AM
maya8913

Question on a definition

If $\lambda$ is a signed measure on $(S, \Sigma)$ , define the positive and negative variations of $\lambda$ , $\lambda^+$ , and $\lambda^-$ .

I tried:

Let $\lambda$ is a signed measure on $(S, \Sigma)$ , let $(P, N)$ be a Hahn decomposition of $S$ w.r.t. $\lambda$ ; $\lambda^+$ , $\lambda^-$ are pointwise finite measures such that $\lambda^+:=\lambda(E \cap P)$ , $\lambda^-:=-\lambda(E \cap N)$ $\forall E \in \Sigma$ .

I don't know the definitions for positive and negative variations of $\lambda^+$ and $\lambda^-$ . Thanks.
March 12th 2009, 07:49 AM
GaloisTheory1

Quote:

Originally Posted by maya8913

If $\lambda$ is a signed measure on $(S, \Sigma)$ , define the positive and negative variations of $\lambda$ , $\lambda^+$ , and $\lambda^-$ .

I tried:

Let $\lambda$ is a signed measure on $(S, \Sigma)$ , let $(P, N)$ be a Hahn decomposition of $S$ w.r.t. $\lambda$ ; $\lambda^+$ , $\lambda^-$ are pointwise finite measures such that $\lambda^+:=\lambda(E \cap P)$ , $\lambda^-:=-\lambda(E \cap N)$ $\forall E \in \Sigma$ .

I don't know the definitions for positive and negative variations of $\lambda^+$ and $\lambda^-$ . Thanks.

$\lambda^+$ and $\lambda^-$ are the positive and negative variations of $\lambda$
March 12th 2009, 09:20 AM
Moo

Hello,

$\lambda^+=\left\{\begin{array}{ll} \lambda \text{ if } \lambda>0 \\ 0 \text{ otherwise}\end{array} \right.$

$\lambda^-=\left\{\begin{array}{ll} -\lambda \text{ if } \lambda<0 \\ 0 \text{ otherwise}\end{array} \right.$
these are the definitions for $\lambda^+$ and $\lambda^-$

it follows that :
$\lambda=\lambda^+-\lambda^-$
$|\lambda|=\lambda^++\lambda^-$

so if you want other formulae, you can add or substract these last 2 equalities :
$\lambda^+=\frac{\lambda+|\lambda|}{2}$
$\lambda^-=\frac{|\lambda|-\lambda}{2}$