Rudin's Riesz Representation Theorem assuming X is a compact space

Question

In Rudin's book Real and Complex Analysis there's an exercise in Chapter 2, the number 19, that challenges you to make some simplifications in Riesz Representation Theorem's proof assuming the space to be compact instead of locally compact. I've been trying so, but I think my simplifications are not remarkable at all, and it feels like there should be one or two things in the proof that become much easier. Can anybody give me a hint on where to look? Or maybe how to look at it? Thanks in advance. I'll leave a link to the book. Riesz's theorem is on page 40, and the exercise on page 59. https://59clc.files.wordpress.com/2011/01/real-and-complex-analysis.pdf