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Thread: Question on a definition

  1. #1
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    Question on a definition

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

    I tried:

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

    I don't know the definitions for positive and negative variations of $\displaystyle \lambda^+$ and $\displaystyle \lambda^-$. Thanks.
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  2. #2
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    Quote Originally Posted by maya8913 View Post
    If $\displaystyle \lambda$ is a signed measure on $\displaystyle (S, \Sigma)$, define the positive and negative variations of $\displaystyle \lambda$, $\displaystyle \lambda^+$, and $\displaystyle \lambda^-$.

    I tried:

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

    I don't know the definitions for positive and negative variations of $\displaystyle \lambda^+$ and $\displaystyle \lambda^-$. Thanks.
    $\displaystyle \lambda^+$ and $\displaystyle \lambda^-$ are the positive and negative variations of $\displaystyle \lambda$
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  3. #3
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    Hello,

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

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

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

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