Riesz's representation theorem for non-locally compact spaces

The basic idea is that of the strict topology on $C^b(X)$. This locally convex topology was introduced in the case of a locally compact space by R.C. Buck in the fifties using weighted seminorms and generalised to the completely regular case by many authors in the sixties and seventies. It can be succinctly described as the finest locally convex topology which agrees with that of compact convergence on the unit ball for the supremum norm and the dual is the space of bounded Radon measures. One of many references: "Bounded measures in topological spaces" by Fremlin, Garling and Haydon (Proc. Lond. Math. Soc. 25 (1972) 115-135). The role of complete regularity is to ensure that the space of continuous functions is large enough for the purposes of this result.

Historical notes of chapter 9 of Bourbaki's integration give the following as original reference for the case of completely regular spaces:

A. D. Alexandroff, Additive set functions in abstract spaces, Mat. Sbornik, I (chap. 1), t. VIII (1940), p. 307-348; II (chap. 2 et 3), t. IX (1941), p. 563-628; III (chap. 4 6), t. XIII (1943), p. 169-238.

Google scholar finds them online

http://www.mathnet.ru/rus/sm6032

http://www.mathnet.ru/eng/sm6105

http://www.mathnet.ru/rus/sm6177

and obviously also papers citing them and so on.

Specifically, you want theorem 1 in the second paper taking into account the definition of space in the first paper (the zero sets of continuous real functions on a completely regular topological space form the closed sets of a normal space in Alexandroff terminology)

Heinz König has proved a very general version of the the Riesz Representation Theorem on an arbitrary Hausdorff topological space.

See Chapter 5 of the book: H. König, Measure and Integration: An Advanced Course in Basic Procedures and Applications, Springer, 1997, corr. reprint 2009, pp. XXI+260.

The monograph H. König, Measure and Integration: Publications 1997–2011, Birkhäuser, Springer Basel, 2012, pp. XI+512 is a collection of some related papers of Konig that appeared after the book, including some on Konig's representation theorems.

The following article provides a nice overview of the machinery König develops.

H. König, Measure and integral: new foundations after one hundred years. (English summary) Functional analysis and evolution equations, 405–422, Birkhäuser, Basel, 2008.