Lineage of the Theory of Invariant Integrals on Groups: Hurwitz, Schur, Weyl, Haar, Neumann, Kakutani, Weil, Kakutani-Kodaira

TL;DR

This paper provides a historical overview of the development of invariant integrals on groups, analyzing original works by Hurwitz, Schur, Weyl, Haar, and others, highlighting their contributions and relationships.

Contribution

It offers a detailed historical analysis of the evolution of invariant integral theory through original papers and personal insights, connecting key mathematicians' works.

Findings

Historical connections among key mathematicians' contributions

Clarification of the development of invariant integrals

Personal interpretation of original papers

Abstract

This is mainly a translation of Proc. of the 29th Symp. on History of Mathematics, Tsuda University, held Oct. 2018, of my talk. The first occasion when I studied the history of the theory of invariant integrals (or measures) was an unintended opportunity where I was asked to write "Kaisetsu" (explanatory and commentary article) to the new book of Prof. M. Saito, a first translation into Japanese of the famous Weil's book "L'int\'egrations dans les groupes topologiques et...". Personally, for my professional work, it was needed only to read this Weil's original book and several text books on measure theory. For writing the Kaisetsu mentioned above, other than several mathematical papers and also Weil's non-mathematical works, it was sufficient for me to read Haar's original paper roughly and similarly for other historical classics. Thus I felt the necessity of further study and now I make intendedly a historical study. Reading original papers due to Hurwitz, Schur, Weyl and so on in detail, I explain their contents from my point of view, and give the relationships among them. I make intendedly a historical study. Reading original papers due to Hurwitz, Schur, Weyl and so on in detail, I explain their contents from my point of view, and give the relationships among them.