Does anybody know what the current research status is on the topic of Haar measures on uniform spaces? I'm specifically interested in compact uniform spaces and its corresponding Haar probability.

As articles, I currently know of Loomis's article "Haar Measure in Uniform Structures" from 1948, of J.P.R. Christensen's "On Some Measures Analogous to Haar Measure" from 1970 and of D.B. Lewin's "Congruence-invariant Measures in Uniform Spaces" from 1966.

I'm also aware of Norman Howes's book "Modern Analysis and Topology" from 1995, reviewed by John Mack at http://at.yorku.ca/t/o/p/c/23.htm . (I do not have Howes's book, I've only read excerpts of it on Google Books.)

Now, my problems are the following: first, Loomis's article is not freely available (only the first page). To make things worse, it seems to be written in the old mathematical parliance, making it difficult to read. Consequently, I do not know what to make of the other two articles which cite it and are built upon it.

Second, Howes claims in his book that Loomis's proofs contained mistakes - but this is rejected by Mack in his review! As if it weren't enough, I do not know whether I can trust these authors, whose proofs I cannot verify myself (Howes, for instance, never worked as a professional mathematician but rather as a software engineer, which you can check for yourselves at http://www.linkedin.com/pub/norman-howes/13/764/a57 and, in general, Loomis is the only name known to me, affiliated to a known university, namely Harvard.)

Can anybody help me, please? I'm totally lost.

exactly oneuniform structure compatible with the underlying topology (a basis of which is given by the neighbourhoods of the diagonal). This means that, in reality, the uniform structure on a compact topological group does not come from algebra, but from compacity alone! In turn, this means that its Haar measure has nothing to do with the algebraic structure, but only with the uniform structure. It seems that on compact spaces integration goes much deeper than previously thought. – Alex M. Feb 25 '14 at 14:30