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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 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 partially useful theorem in Bourbaki's "Elements of Mathematics - Integration II", chapter IX, paragraph 5, page IX.59. I have also found a hint in "Bounded Continuous Functions On A Completely Regular Space" by Dennis F. Sentilles, appeared in Transactions of the AMS, vol.168, June 1972, page 311 (you can find it on Jstor).

My question is: does anybody know of a "definitive" statement concerning the dual of the algebra of bounded continuous functions on a completely regular space? What is lost when one gives up local-compacity? (Please notice that I am not interested in the algebra of functions with compact support or vanishing at infinity.)

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You certainly do need continuity in the locally compact case. A discontinuous linear functional is not a measure, and Zorn's lemma shows that discontinuous linear functionals exist. You may be thinking of the fact that a positive linear functional is automatically continuous. –  Nate Eldredge Mar 9 at 14:53
    
By the way, I don't know the answer offhand, but the first place I would look is in Dunford and Schwartz. They have lots of results classifying the duals of various spaces. –  Nate Eldredge Mar 9 at 14:54
    
Thank you @NateEldredge, I have corrected my question following your comment. Indeed, I was ignoring the fact that positivity implies continuity. –  Alex M. Mar 9 at 16:29
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@NateEldredge: I've just looked in Dunford & Schwartz: if X is normal then $C(X)^*$ is isometrically and order-preserving isomorphic to $rba(X)$ (the space of finitely-additive regular complex measures); if X is compact and Hausdorff then $C(X)^*$ is isometrically and order-preserving isomorphic to $rca(X)$ (as above, but countably-additive). These results are weaker than what I'm looking for, but this is natural given that the edition that I have dates back to 1957, when the results that I'm looking for were still in their infancy. –  Alex M. Mar 9 at 16:54
    
I'm curious: do you have a non-locally compact space in mind for which you want to use the Riesz representation theorem? –  Paul Siegel Mar 9 at 22:19

2 Answers 2

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)

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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.

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