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? Thanks in advance.

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.