# Weil's book L'intégration dans les groupes topologiques et ses applications

The book L'intégration dans les groupes topologiques et ses applications published by André Weil in 1940 is regarded as one of the classical references for harmonic analysis on topological groups.

Unfortunately I am not fluent in French, so reading the book in all details is simply impossible. However, the reason why I am interested in reading this book is because it seems that Weil's treatment is different from than the now "standard treatment" of so-called abstract harmonic analysis, which is in general characterised by its extensive use of Gelfand theory.

This book by Weil was never translated into English, although there are Russian and Japanese editions, as noted in the comments. Therefore, I was curious if someone could point out references to treatments of harmonic analysis which are similar to Weil's, but are available in English.

I am aware of the book Classical Harmonic Analysis and Locally Compact Groups by Hans Reiter, which was a student of Weil and which seems to have a similar approach to harmonic analysis as Weil had. However, the proofs of classical results such as the existence and uniqueness of the Haar measure and Pontryagin's Duality Theorem are all omitted in this text.

Other books where I expect a similar approach as in Weil's book are, of course, the books by Bourbaki. However, I do not think that, for example, Pontryagin's Duality Theorem is proved in any of these books. I am aware that the uniqueness and existence of the Haar measure is proved in the book on Integration though.

Any reference is highly appreciated.

• the statement "the above mentioned book by Weil is never translated into any other language" is not quite correct: there is a Russian translation from 1950. – Carlo Beenakker Oct 2 '16 at 10:25
• Thanks for pointing this out. However, the sentence starts with "As far as I know". I was not aware of this translation. The Russian translation does not help me either though as I am only able to read English, Dutch and German. – user92170 Oct 2 '16 at 10:26
• just for the record, there is also a Japanese edition from 2015 – Carlo Beenakker Oct 2 '16 at 10:30
• This doesn't address your question, but if you are able to read mathematical English and German then I think you may have a better chance with mathematical French than you imply, especially if your goal is to understand the proofs rather than every phrase – Yemon Choi Oct 2 '16 at 14:48
• How about volume 1 of Hewitt--Ross? – Yemon Choi Oct 2 '16 at 14:49

Leopoldo Nachbin's book "The Haar Integral" has Weil's proofs of existence and uniqueness of Haar measure, as well as Cartan's.

Weil establishes the Pontryagin duality theorem by an argument very similar to the original one by Pontryagin (in the compact/discrete case), which was extended to more general groups by van Kampen. Duality is established for $\mathbb{R}$, compact groups and discrete groups, and then generalized using structure theory of locally compact abelian groups. Hewitt & Ross has a proof along these lines, as does the second edition of Pontryagin's own Topological Groups (which was translated into English). Van Kampen's original paper is in English and available online, but it's not the easiest to follow.

• Rudin's book "Fourier Analysis on Groups" gives a uniform proof of the Pontryagin Duality Theorem for all locally compact Hausdorff abelian groups (i.e., without using any structure theory of such groups). It appears quite early in the book. – nfdc23 Oct 3 '16 at 13:12
• Rudin's proof is based on Gelfand theory, which the asker was trying to avoid. Also, Rudin's proof is incorrect as written, which is noted (with the fix) in the errata at the back of the second edition. – Cameron Zwarich Oct 3 '16 at 23:28
• Ah, thanks for clarifying. It's been a long time since I had read that part of Rudin's book... – nfdc23 Oct 4 '16 at 1:34
• By the way, Nachbin's book (originally in Portuguese) is rather classical. Apparently the most up-to-date information on its publication history is here: ams.org/mrlookup – Jim Humphreys Jul 31 '17 at 20:25

It seems that Bourbaki has integrated at least some of Weil's ideas, and all of Bourbaki has been translated, as far as I know.

• Thanks. I was not aware of Hewitt's review, but it explains in detail what is covered in Bourbaki and what is not. However, as I also mentioned in my question much of the theory of locally compact Abelian groups is not in Bourbaki's volume on Integration, but is in Weil's book. – user92170 Oct 2 '16 at 11:51
• Pontryaguin duality and the structure of locally compact abelian groups are in the book "Théorie spectrales" of Bourbaki (specifically in chapter 2). – Denis Chaperon de Lauzières Oct 2 '16 at 12:35
• Besides Bourbaki, there is also the series of books Elements d'Analyse'' by J. Dieudonné, in particular Vol.2, a simplified version of Bourbaki's Intégration: Bourbaki gets to deep water only after constructing a fleet of nuclear submarines; the author (J. Dieudonné) has decided to get there by rowing, in a glass-bottomed boat (from MR0235946 by S.K. Berberian). – user111 Oct 2 '16 at 12:48
• @DenisChaperondeLauzières. Thanks for pointing this out. It seems that this particular book is one of the few of Bourbaki's that is not translated into English. – user92170 Oct 2 '16 at 13:27

I really like Cartan and Godement's Théorie de la dualité et analyse harmonique dans les groupes abéliens localement compacts (1947). It isn't quite what you hope for, since it is again untranslated. But it is very much in Weil's spirit — improved to avoid the structure theory, but unlike Bourbaki's Théories spectrales, still eschewing Gelfand theory (replaced by use of the Kreĭn-Milman theorem).

The French is more accessible than Weil's, and it's much shorter: just 20 pages to progress through the theorems Riemann-Lebesgue, Bochner, Fourier inversion, Plancherel, and Pontryagin duality. Add two pages for Godement's generalized Stone's theorem (1944), and all the bases are covered.