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.
Best Answer
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.