Riesz Representation Theorem in a compact space

Question

What simplifications would be found going through the proof of the Riesz Representation Theorem assuming the space X to be compact (or even compact metric) rather than just locally compact?

(I'm following the Rudin's Real and Complex Analysis)

I think that the possible situations in which simplifications could be found are:

1. If X is a Hausdorff compact space, a subset is compact if and only if it is closed. Every continuous function would have a compact support, as the support of a function is always closed, and hence (in this case), compact.
2. Every closed set would be in $\mathcal M_F$.
3. All the closed sets would be in $\mathcal M$, as $\mathcal M_F\subset \mathcal M$. As $\mathcal M$ is a $\sigma$-algebra, all the open sets would be also in it.

I guess there are more possible simplifications, but I don't know how to find them.

Thanks in advance!