The lectures on "Operator Theory", of which the present volume constitutes the first part, were given in the academic years 1933-34 and 1934-35, at the Institute for Advanced Study. The notes were prepared in these years by Dr. Robert S. Martin and Dr. Charles C. Torrance, respectively. They were multigraphed and distributed by the Institute for Advanced Study shortly thereafter, but the original edition has been completely exhausted for several years. The interest in these lecture notes appears to have been continuing, and therefore a new edition is now being brought outo The present volume comprises Chapters I-XI, dealing with preliminaries, namely, with the theory of Measures and Integrals. The second volume, on Operator Theory proper, will be published subsequently. The present edition is identical with the original one, except that typographical errors have been corrected and some notations and references have been elaborated. I would like to express my warmest thanks to Dr. H. H. Goldstine, for his advice on this edition, and also for having most obligingly undertaken the exacting task of proof-reading the typescript.