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.

  • $\begingroup$ Yes, I know it. I'm afraid that it is a book that misses the point: it's highly technical, non-conceptual, it introduces concepts that nobody will ever need and it ignores concepts that would be "juicy". For instance, the book is dedicated to the subject of measures (in fact, integrals) on uniform spaces, but I haven't found one single mention of the topic of the Haar measure on uniform spaces. Think about it: when you deal with integration on topological groups, do you use the Haar measure or some general measure? It is a vanity book, that the author wrote just to enhance his cv. $\endgroup$ – Alex M. Feb 18 '14 at 19:08
  • $\begingroup$ If one can get a Haar measure on a general uniform space, then one obtains a Haar measure on every compact space since each compact space can be endowed with a unique uniformity. With this fact in mind, a notion of a Haar measure for a general uniform space could mean absolutely anything. $\endgroup$ – Joseph Van Name Feb 22 '14 at 15:17
  • $\begingroup$ Agreed. But please notice that on any compact space there exist exactly one uniform 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. $\endgroup$ – Alex M. Feb 25 '14 at 14:30
  • $\begingroup$ But the standard Haar measure does depend on the underlying group rather than the uniform structure alone since even the circle can be made into a topological group in many ways that give different Haar measures. I am unsure of what is meant by a Haar measure on a uniform space, but the notion of a "Haar measure" on a compact group does not seem like a true generalization of the notion of a Haar measure on a compact group. $\endgroup$ – Joseph Van Name Feb 25 '14 at 16:25

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