# A question on Riesz Representation Theorem

1. Jul 20, 2010

### zzzhhh

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 $$G(f)=G^+(f)-G^-(f)=\int fd\gamma^+ - \int fd\gamma^-$$, where $$\gamma^+$$ and $$\gamma^-$$ are obtained from $$G^+$$ and $$G^-$$ respectively according to Th 9.9. So I guess the charge needed to represent G might be $$\gamma=\gamma^+-\gamma^-$$. If we can prove that $$\gamma$$ is really a charge and that $$\gamma^+$$ and $$\gamma^-$$ is the positive and negative variation of $$\gamma$$ 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 $$G^+$$ and $$G^-$$ by Lemma 8.13, then construct $$g^+$$ (and similarly $$g^-$$) by $$\lim\limits_{n\to\infty}G^+(\varphi_{t,n})$$ as in page 106, and finally construct the Borel-Stieltjes measures $$\gamma^+$$ and $$\gamma^-$$ according to page 105. After these processes, how to prove that $$\gamma^+(E)=\gamma(E\cap P)$$ and $$\gamma^-(E)=-\gamma(E\cap N)$$ where P and N is a Hahn decomposition for $$\gamma$$? 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