Riesz's representation theorem for non-locally compact spaces Every version of Riesz's representation theorem (the one expressing linear functionals as integrals) that I have found so far assumes that the underlying topological space is locally-compact. (For instance, Rudin in "Real and Complex Analysis" uses this assumption in the proof of Urysohn's lemma, upon which he bases the proof of Riesz's theorem.) Nevertheless, in an answer on MO, @jbc (inactive on MO since 2013 and with no real name or contact information available) claims that the theorem can be stated in much more generality, for Hausdorff completely regular spaces (which suggests that Urysohn's lemma is indeed the essential ingredient here). Unfortunately, no detail or bibliography is given in that answer. Currently, I have only found a practically useless theorem in Bourbaki's "Elements of Mathematics - Integration II", chapter IX, paragraph 5, page IX.59, and a very difficult to use version in "Bounded Continuous Functions On A Completely Regular Space" by Dennis F. Sentilles ("Transactions of the AMS", vol.168, June 1972, page 311, theorems 9.1.c and 9.2).
My question is: does anybody know of a "definitive", recent statement concerning the dual of the algebra of bounded continuous functions on non-locally-compact spaces? What is lost when one gives up local-compactness? (Please notice that I am not interested in the algebra of functions with compact support or vanishing at infinity.)
 A: 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)
A: 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.
A: 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.
