Timeline for The Haar integral on uniform spaces
Current License: CC BY-SA 3.0
5 events
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| Feb 25, 2014 at 16:25 | comment | added | Joseph Van Name | 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. | |
| Feb 25, 2014 at 14:30 | comment | added | Alex M. | 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. | |
| Feb 22, 2014 at 15:17 | comment | added | Joseph Van Name | 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. | |
| Feb 18, 2014 at 19:08 | comment | added | Alex M. | 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. | |
| Feb 18, 2014 at 17:27 | history | asked | Alex M. | CC BY-SA 3.0 |