There are two concurrent theories of measure/integration on a locally compact topological spaces: either as positive linear forms on the space of continuous functions with compact support, or as Borel measures. The theorem of Riesz establishes the relation between these two point of views, which are essentially equivalent: if one imposes the Borel measure to be inner-regular (the measure of any Borel set is the supremum of the measures of its compact subsets), then there is indeed a bijection.

However, this result is hard to find in the litterature (I found it in Fremlin's book, as well as in a 1975 paper of Pollard and Topsoe). Rudin's *Real and complex analysis* proves a variant where he requires outer-regularity. Of course, the main source of examples for a book like Rudin's consists in open subsets of $\mathbf R^n$, for which all Borel measures are automatically inner-regular. On the other hand, having a bijection is particularly nice in some contexts, such as locally compact groups. (Actually, in my own research, I have to construct measures on some non-metrizable locally compact spaces, and we do it by working on each compact set at a time, hence the inner-regularity point of view is both natural and necessary.)

After this presentation, here are my questions :

What are the good reasons to favor at first the outer-regularity instead of having a nice Riesz theorem, and proving the outer-regularity when needed?

What is a standard reference for the form of the theorem of Riesz that gives a bijection between positive linear forms and inner-regular Borel measures on a locally compact space?