This book is first of all designed as a text for the course usually called "theory of functions of a real variable". This course is at present cus­ tomarily otIered as a first or second year graduate course in United States universities, although there are signs that this sort of analysis will soon penetrate upper division undergraduate curricula. We have inc1uded every topic that we think essential for the training of analysts, and we have also gone down a number of interesting bypaths. We hope too that the book will be useful as a reference for mature mathematicians and other scientific workers. Rence we have presented very general and complete versions of a number of important theorems and constructions. Since these sophisticated versions may be difficult for the beginner, we have given elementary avatars of all important theorems, with appro­ priate suggestions for skipping. We have given complete definitions, ex­ planations, and proofs throughout, so that the book should be usable for individual study as weil as for a course text. Prerequisites for reading the book are the foilowing. The reader is assumed to know elementary analysis as the subject is set forth, for example, in TOM M. ApOSTOL'S Mathematical Analysis [Addison-Wesley Publ. Co., Reading, Mass., 1957], OrWALTERRuDIN'S P1'inciplesol Mathe­ 4 matical Analysis [2" Ed., McGraw-Rill Book Co., New York, 1964].

Set Theory and Algebra.- Topology and Continuous Functions.- The Lebesgue Integral.- Function Spaces and Banach Spaces.- Differentiation.- Integration on Product Spaces.