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Math Help - Question on a definition

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by maya8913 View Post
    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
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  3. #3
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    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}
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