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A question on Riesz Representation Theorem

  1. Jul 20, 2010 #1
    This question comes from the following paragraph of Bartle's "The Elements of Integration and Lebesgue Measure" in page 108.
    I stuck first at the phrase "integration with respect to a charge"(some author call it signed measure). I didn't find its definition in this book and later find it in page 88 of Folland's book "real analysis". Then, for arbitrary bounded linear functional G, we have [tex]G(f)=G^+(f)-G^-(f)=\int fd\gamma^+ - \int fd\gamma^-[/tex], where [tex]\gamma^+[/tex] and [tex]\gamma^-[/tex] are obtained from [tex]G^+[/tex] and [tex]G^-[/tex] respectively according to Th 9.9. So I guess the charge needed to represent G might be [tex]\gamma=\gamma^+-\gamma^-[/tex]. If we can prove that [tex]\gamma[/tex] is really a charge and that [tex]\gamma^+[/tex] and [tex]\gamma^-[/tex] is the positive and negative variation of [tex]\gamma[/tex] respectively, we can apply Folland's definition to get the desired extended representation of G. The former is easy, but the latter seems too complicated to me -- we first decompose G to [tex]G^+[/tex] and [tex]G^-[/tex] by Lemma 8.13, then construct [tex]g^+[/tex] (and similarly [tex]g^-[/tex]) by [tex]\lim\limits_{n\to\infty}G^+(\varphi_{t,n})[/tex] as in page 106, and finally construct the Borel-Stieltjes measures [tex]\gamma^+[/tex] and [tex]\gamma^-[/tex] according to page 105. After these processes, how to prove that [tex]\gamma^+(E)=\gamma(E\cap P)[/tex] and [tex]\gamma^-(E)=-\gamma(E\cap N)[/tex] where P and N is a Hahn decomposition for [tex]\gamma[/tex]? I have no idea on this problem, could you please help me? Thanks!

    PS: Bartle's book is available online << links deleted by berkeman >>
    Last edited by a moderator: Jul 21, 2010
  2. jcsd
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