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The present text resulted from lectures given by the authors at the Rijks Universiteit at Utrecht. These lectures were part of a series on 'History of Contemporary Mathematics'. The need for such an enterprise was generally felt, since the curriculum at many universities is designed to suit an efficient treatment of advanced subjects rather than to reflect the development of notions and techniques. As it is very likely that this trend will continue, we decided to offer lectures of a less technical nature to provide students and interested listeners with a survey of the history of topics in our present-day mathematics. We consider it very useful for a mathematician to have an acquaintance with the history of the development of his subject, especially in the nineteenth century where the germs of many of modern disciplines can be found. Our attention has therefore been mainly directed to relatively young developments. In the lectures we tried to stay clear of both oversimplification and extreme technicality. The result is a text, that should not cause difficulties to a reader with a working knowledge of mathematics. The developments sketched in this book are fundamental for many areas in mathematics and the notions considered are crucial almost everywhere. The book may be most useful, in particular, for those teaching mathematics.

Sets and integration An outline of the development

Set theory from Cantor to Cohen.- Forebodings.- The exploration of the new continent.- The paradoxes.- The axiom of choice.- Zermelo takes over.- Making inconsistent sets respectable.- The consistency of the axiom of choice and the continuum hypothesis.- The independence of the continuum hypothesis.- Large cardinals.- Games and strategies.- The integral from Riemann to Bourbaki.- Introduction; the period before Riemann.- Greek mathematics.- Seventeenth century.- Leibniz and Newton.- Riemann, Lebesgue, real functions.- Riemann.- Measure theory.- Lebesgue.- More about Lebesgue.- Controversies.- Real functions.- Denjoy.- Perron.- Further developments.- Modern theory of the integral.- Integrals as linear functions.- Generalizations.- Carathéodory.- Haar-measure.- List of Mathematicians.