Haar measure on a quotient, References for.

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.

• This is a total shot in the dark because I don't own the book myself, but have you tried Halmos' Measure Theory? I have some vague recollections that a former officemate of mine went there to learn about exactly what you are asking about. Feb 5 '10 at 16:15
• When $H$ is compact, as in your example (in fact for any homogenous space), can't you just define the measure on $G/H$ as the pushforward? If $q\colon G \to G/H$ is the quotient, for $A \subset G/H$ set $\mu(A) = m(q^{-1}(A))$. Feb 5 '10 at 19:31
• @TomChurch: Why would the pushforward operation require compactness? Mar 16 '18 at 18:55
• @AlexM. when $H$ is noncompact, the resulting measure could be infinite everywhere. (Think about defining, rather stupidly, $\mathbb{R}^1$ as the quotient of $\mathbb{R}^2$ by a translation action. If you try to define a quotient measure on $\mathbb{R}^1$ by pushing forward the Lebesgue measure on $\mathbb{R}^2$, it is infinite for every set.) Oct 8 '19 at 18:29

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.

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.

• +1 For your enthusiasm for Bourbaki. :) But I am maimed by reading many books of Weil and do not want to repeat the experience again. Feb 5 '10 at 16:06
• Oh, but sir! Dieudonne wrote the final drafts. Nicolas Bourbaki is one of the great writers of the 20th century. Feb 5 '10 at 16:07

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

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

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.

• To amuse Harry (though I hope not to dissuade you, Anweshi), Loomis says the following: "The method of procedure is taken quite directly from Weil". Feb 5 '10 at 16:11
• I have great admiration for Weil, starting with his first achievement, the Mordell-weil theorem. I have no problem about reading about the work of Weil, in fact that is what I was doing much of time. I only said that his writing so hard to figure out. So I will be much happier if somebody else is writing an explanation for his work. Note that people say, "After Grothendieck algebraic geometry is so much easier", though Groth. is so much more abstract and voluminous. This shows how hopeless Weil's writing style is. Feb 5 '10 at 16:20

"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.

• @BW: Do you know where in R&V this Haar measure on homogeneous spaces is discussed? My copy was 18 inches away when I read the question, so I just flipped through it and didn't find any discussion of this point. Feb 5 '10 at 16:13
• @Pete: You're right! I didn't have my copy, and just misremembered. I tried skimming through google just now, and didn't see it. It may be in chapter 3 (which Google didn't let me see), but I doubt it. Feb 5 '10 at 16:20

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".

A very detailed account is given in the book "Lectures on Spectrum of $L^{2}(\Gamma \backslash G)$" by Floyd Williams. The first chapter does exactly what is required.