# Significance and applications of the Riesz Representation Theorem in locally compact Hausdorff spaces

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?

• 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.
– user61527
Mar 7 '14 at 6:50
• 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.
– user99914
Mar 7 '14 at 6:54
• I can even remember what theorem 2.14 is without looking at your description......
– user99914
Mar 7 '14 at 6:59
• The construction/proof of the Borel functional calculus for bounded operators would become seemingly harder without Riesz' representation theorem... Sep 8 '14 at 19:12

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;