9
$\begingroup$

Can anyone tell me the signification of Theorem $2.14$ (The Riesz Representation Theorem in locally compact Hausdorff spaces), page $40, 41$ in Rudin - Real and Complex Analysis? And some applications of that theorem?

Thanks in advance.

$\endgroup$
4
  • 2
    $\begingroup$ A few pages later, he uses this theorem to construct Lebesgue measure by considering the positive functional that is Riemann integration; a lot of the nicest properties drop out pretty quickly. Many other authors proceed through some variant of outer measure, instead. $\endgroup$
    – user61527
    Mar 7 '14 at 6:50
  • 4
    $\begingroup$ Extremely significant! It tells you that (together with some results in chatper 6) the dual space of $C_0(X)$ is the space of Borel regular measure. Also in chapter 6 you have a neat representation of measure. $\endgroup$
    – user99914
    Mar 7 '14 at 6:54
  • $\begingroup$ I can even remember what theorem 2.14 is without looking at your description...... $\endgroup$
    – user99914
    Mar 7 '14 at 6:59
  • 1
    $\begingroup$ The construction/proof of the Borel functional calculus for bounded operators would become seemingly harder without Riesz' representation theorem... $\endgroup$ Sep 8 '14 at 19:12
1
$\begingroup$

Just a summary of what have been said in the comments (by user61527, John Ma and Freeze_S):

  1. Rudin uses the theorem a few pages later to construct Lebesgue measure by considering the positive functional that is Riemann integration; a lot of the nicest properties drop out pretty quickly. Many other authors proceed through some variant of outer measure, instead;
  2. The theorem is very significant, because together with results from Chapter 6 tells you that the dual space of $C_0(X)$ is the space of Borel regular measure;
  3. The construction and the proof of the Borel functional calculus for bounded operators would become harder without the theorem.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.