نظریه‌های یکپارچگی کلاسیک و مدرن

Classical and Modern Integration Theories discusses classical integration theory, particularly that part of the theory directly associated with the problems of area. The book reviews the history and the determination of primitive functions, beginning from Cauchy to Daniell. The text describes Cauchy's definition of an integral, Riemann's definition of the R-integral, the upper and lower Darboux integrals. The book also reviews the origin of the Lebesgue-Young integration theory, and Borel's postulates that define measures of sets. W.H. Young's work provides a construction of the integral equivalent to Lebesque's construction with a different generalization of integrals leading to different approaches in solutions. Young's investigations aim at generalizing the notion of length for arbitrary sets by means of a process which is more general than Borel's postulates. The text notes that the Lebesgue measure is the unique solution of the measure problem for the class of L-measurable sets. The book also describes further modifications made into the Lebesgue definition of the integral by Riesz, Pierpont, Denjoy, Borel, and Young. These modifications bring the Lebesgue definition of the integral closer to the Riemann or Darboux definitions, as well as to have it associated with the concepts of classical analysis. The book can benefit mathematicians, students, and professors in calculus or readers interested in the history of classical mathematics. Content: Front Matter, Page iii Copyright, Page iv Dedication, Page v FOREWORD, Pages xi-xii PREFACE, Pages xiii-xiv NOTATION AND TERMINOLOGY, Pages xv-xviii 1 - FROM CAUCHY TO RIEMANN, Pages 3-9 2 - DEVELOPMENT OF INTEGRATION IDEAS IN THE SECOND HALF OF THE 19TH CENTURY, Pages 10-41 INTRODUCTION TO THE ORIGIN OF LEBESGUE–YOUNG INTEGRAHON THEORY, Page 43 3 - THE BOREL MEASURE, Pages 45-47 4 - LEBESGUE'S MEASURE AND INTEGRATION, Pages 48-76 5 - YOUNG'S INTEGRAL, Pages 77-85 6 - OTHER DEFINITIONS RELATED TO THE DEFINITION OF LEBESGUE'S INTEGRAL, Pages 86-105 7 - STIELTJES' INTEGRAL, Pages 106-129 8 - THE PROBLEM OF THE PRIMITIVE—THE DENJOY-KHINCHIN INTEGRAL, Pages 133-159 9 - PERRON'S INTEGRAL, Pages 160-171 10 - DANIELL'S INTEGRAL, Pages 172-177 CONCLUSION, Pages 178-189 AUTHOR INDEX, Pages 191-192 SUBJECT INDEX, Pages 193-195 Probability and Mathematical Statistics: A Series of Monographs and Textbooks, Pages ibc1-ibc2