In Rudin's real complex analysis, the Riesz representation theorem is being proven for locally compact spaces. However, the book also says that the reader can forget about locallcy compact spaces and consider only euclidean spaces without losing any of the principal ideas.
Question: Is the theorem ever used in its most general form (the version of locally compact spaces) ? If not, why are we doing this then ? Thank you
Rudin likes to do things very generally. However, one may become interested in function spaces over manifolds or cell complexes, say, so using Riesz in settings more general than $\mathbb{R}^n$ is not unwarranted.