Real variable contributions of G. C. Young and W. H. Young

Abstract

Andrew M. Bruckner I and Brian S. Thomson 2 ~Department of Mathematics, University of California, Santa Barbara, California 93106, USA 2Department of Mathematics, Simon Fraser University, Burnaby, B.C., Canada, V5A 1S6 §1. Introduction. The Youngs 1 began to work on real functions in earliest years of 20th century. Then, as now, it was an unfashionable subject. Writing of this time Hardy [40, p. 224] (see also [1D says subjects were not popular, even in France, with conservatively minded mathematicians; in England they were regarded almost as a morbid growth in mathematics .... Fashions are dictated, of course, by vested interests, pride and ignorance. It is hard to imagine, from our perspective at beginning of 21st century, a more profitable time to study this field than at beginning of previous century; Cantor's set theory was very much in air and all of important basic tools were being provided by Baire, Borel and Lebesgue. The whole field of what was then called the theory of functions of a real variable was reworked and rewritten in those first decades. The Youngs played a major role in this effort. It is beyond our ability to present a complete account of their influence on this field. Much of their work was influential in sense that they popularized and made better known seminal contributions of Cantor or important work in integration theory that Lebesgue had produced or category ideas of Baire. Many of their papers are extensions or applications of these themes with new proofs or new techniques. An indica- tion of their impact is evident in Hobson [43] which for many years was main English language reference work on real functions: there are 139 citations of their works in these two volumes. This necessary professional work does not often lead to long lasting fame and renown and by now sources have blurred considerably. A modern graduate course in real functions doubtless owes much to their activity but it is only infrequently explicit. 1 This essay on Youngs' influence on some aspects of real analysis was originally intended to accompany an edition of collected works of Youngs planned as a four volume work with essays covering various aspects of their contributions. Unfortunately this ambitious project had to be cancelled. In its place, a one volume edition of Selected Papers [1] was published by Presses Polytechniques et Universitaires Romandes in 2000, edited by S. D. Chatterji and H. Wefelscheid. Professor Chatterji has done a major service by summarizing in a short introduction many themes that run through vast body of work that Youngs produced in their careers. E-mail addresses: lbmckner@math.ucsb.edu 2thomson@sfu.ca 0732-0869/01/19/4-337 $ 15.00/0