Haar measure on a quotient, References for. - MathOverflow

Haar measure on a quotient, References for.

2010-02-05T15:31:34Z

I remember reading Weil's "Basic Number Theory" and giving up after a while. Now I find myself thinking of it(thanks to some comments by Ben Linowitz).

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.

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.

I originally assumed this theorem and went ahead(but not much) with that book.

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).

I had originally seen the construction of Haar measure on H. Royden's "Real Analysis", in which he is not considering any quotients.

Answer by Ben Weiss for Haar measure on a quotient, References for.

2010-02-05T15:39:00Z

"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.

Answer by Steve Huntsman for Haar measure on a quotient, References for.

2010-02-05T15:58:34Z

The ancient (1953) An Introduction to Abstract Harmonic Analysis by Loomis gives a quotient/Fubini type theorem for Haar measure. Here is a link for an online version.

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.

Answer by Harry Gindi for Haar measure on a quotient, References for.

2010-02-05T16:03:27Z

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 integral (a pun for you!) role in proving the Haar measure in full generality.

Answer by Mark Meckes for Haar measure on a quotient, References for.

2010-02-05T16:39:00Z

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".

They say their proof is apparently due to W. Maak, and give a citation to W.F. Donaghue, "Distributions and Fourier Transforms".

Answer by Gerald Edgar for Haar measure on a quotient, References for.

2010-02-05T16:53:34Z

You can find it in: Hewitt & Ross, Abstract Harmonic Analysis

Answer by Mohan Ramachandran for Haar measure on a quotient, References for.

2010-02-05T16:57:23Z

You can find it in Federer Geometric Measure Theory pages 121-129.

Answer by JS Milne for Haar measure on a quotient, References for.

2010-02-05T18:45:14Z

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.

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.