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.)