Haar measure on a quotient, References for. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T00:06:43Z http://mathoverflow.net/feeds/question/14278 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14278/haar-measure-on-a-quotient-references-for Haar measure on a quotient, References for. Anweshi 2010-02-05T15:31:34Z 2010-02-05T18:45:14Z <p>I remember reading Weil's "Basic Number Theory" and giving up after a while. Now I find myself thinking of it(thanks to <a href="http://mathoverflow.net/questions/13106/map-of-number-theory/13134#13134" rel="nofollow">some comments</a> by Ben Linowitz).</p> <p>Right from the very beginning, Weil uses the fact that when you have a locally compact topolgocal group \$G\$ and a locally compact subgroup \$H\$, in addition to the Haar measures on \$G\$ and \$H\$, there exists a "Haar measure" on the coset space \$G/H\$, with some properties.</p> <p>For instance, the upper half plane \$\mathbb H\$ is the quotient \$SL_2(\mathbb R)/SO_2(\mathbb R)\$ and the usual measure there which gives rise to the usual hyperbolic metric, is arising in this way.</p> <p>I originally assumed this theorem and went ahead(but not much) with that book.</p> <p>I want to have a reference for the above theorem. A reference which is not written by Weil. I find him very hard to penetrate. This should exclude Bourbaki's "Integration", as I supppose it would be heavily influenced by him, and thus is a horrible book(note to Harry: this is personal opinion; spare me the brickbats).</p> <p>I had originally seen the construction of Haar measure on H. Royden's "Real Analysis", in which he is not considering any quotients.</p> http://mathoverflow.net/questions/14278/haar-measure-on-a-quotient-references-for/14279#14279 Answer by Ben Weiss for Haar measure on a quotient, References for. Ben Weiss 2010-02-05T15:39:00Z 2010-02-05T15:39:00Z <p>"Fourier Analysis on Number Fields" by Ramakrishnan and Valenza deals with many of the same topics, but starts in chapter 1 with exactly this material and works up to Tate's thesis in chapter 7. I hope this helps.</p> http://mathoverflow.net/questions/14278/haar-measure-on-a-quotient-references-for/14286#14286 Answer by Steve Huntsman for Haar measure on a quotient, References for. Steve Huntsman 2010-02-05T15:58:34Z 2010-02-05T17:13:37Z <p>The ancient (1953) <em>An Introduction to Abstract Harmonic Analysis</em> by Loomis gives a quotient/Fubini type theorem for Haar measure. <a href="http://www.archive.org/stream/introductiontoab031610mbp#page/n143/" rel="nofollow">Here is a link for an online version</a>.</p> <p>BTW, the one time I ever cited Bourbaki as a reference was because its description of Haar measures on the affine group (Integration II, §7-9) is more concrete than any other I could find.</p> http://mathoverflow.net/questions/14278/haar-measure-on-a-quotient-references-for/14288#14288 Answer by Harry Gindi for Haar measure on a quotient, References for. Harry Gindi 2010-02-05T16:03:27Z 2010-02-05T16:03:27Z <p>Bourbaki's section on the Haar measure is one of the best sections on the Haar measure in any book, plus it's one of the best pieces of Bourbaki writing. This is of course because Weil played an <em>integral</em> (a pun for you!) role in proving the Haar measure in full generality.</p> http://mathoverflow.net/questions/14278/haar-measure-on-a-quotient-references-for/14291#14291 Answer by Mark Meckes for Haar measure on a quotient, References for. Mark Meckes 2010-02-05T16:39:00Z 2010-02-05T16:39:00Z <p>In the case that \$G/H\$ is compact and can be given a \$G\$-invariant metric (I mean metric-space metric, not necessarily a Riemannian metric), a nice proof and discussion is given in the very first section of Milman and Schechtman's book "Asymptotic Theory of Finite Dimensional Normed Spaces". </p> <p>They say their proof is apparently due to W. Maak, and give a citation to W.F. Donaghue, "Distributions and Fourier Transforms".</p> http://mathoverflow.net/questions/14278/haar-measure-on-a-quotient-references-for/14292#14292 Answer by Gerald Edgar for Haar measure on a quotient, References for. Gerald Edgar 2010-02-05T16:53:34Z 2010-02-05T16:53:34Z <p>You can find it in: Hewitt &amp; Ross, <em>Abstract Harmonic Analysis</em></p> http://mathoverflow.net/questions/14278/haar-measure-on-a-quotient-references-for/14294#14294 Answer by Mohan Ramachandran for Haar measure on a quotient, References for. Mohan Ramachandran 2010-02-05T16:57:23Z 2010-02-05T16:57:23Z <p>You can find it in Federer Geometric Measure Theory pages 121-129.</p> http://mathoverflow.net/questions/14278/haar-measure-on-a-quotient-references-for/14305#14305 Answer by JS Milne for Haar measure on a quotient, References for. JS Milne 2010-02-05T18:45:14Z 2010-02-05T18:45:14Z <p>The book I always look at for such things is Nachbin, The Haar Integral, which is short, and has a whole chapter on Integration on Locally Compact Homogeneous Spaces. </p> <p>And a plus: he gives you a choice of reading the proof of the existence and uniqueness of the Haar integral according to Weil or according to Henri Cartan.</p>