This question comes from the following paragraph of Bartle's "The Elements of Integration and Lebesgue Measure" in page 108.(adsbygoogle = window.adsbygoogle || []).push({});

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 representGmight 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 ofG. The former is easy, but the latter seems too complicated to me -- we first decomposeGto [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] wherePandNis 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 >>

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A question on Riesz Representation Theorem

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads - question Riesz Representation | Date |
---|---|

B Function rules question | Saturday at 9:49 AM |

B Question about a limit definition | Feb 27, 2018 |

A Angular Moment Operator Vector Identity Question | Feb 10, 2018 |

I A question regarding Logistic population model | Feb 1, 2018 |

A question on proof of Riesz Representation Theorem when p=1 | Jun 23, 2010 |

**Physics Forums - The Fusion of Science and Community**