This is an invaluable book that presents the original work published in French, in 1904, by Henry Leon Lebesgue, the creator of the theory of integration. Translated into English for the first time, the book offers readers a unique opportunity to explore Lebesgue?s groundbreaking ideas and delves into the mind of one of the greatest mathematicians in history. The book provides historical context and explanations that enhance readers? comprehension and appreciation of the material. Covering a wide range of topics, from the integration before Riemann to the search for primitive functions, it offers a comprehensive understanding of Lebesgue?s theories and their significance in the field of mathematics. It inspires readers to explore further in the field, stimulates new ideas, and opens avenues for future research. The book bridges the gap between theory and practice by providing examples and applications that contributed to the development of Lebesgue integration theory.

The book serves as a valuable resource for courses in analysis, measure theory, and Lebesgue integration theory, providing students with the opportunity to study the original work of Lebesgue and deepen their understanding of integration theory. It is meant for a broad audience, including advanced undergraduate and graduate students, mathematics scholars, researchers, educators, and enthusiasts, seeking a comprehensive understanding of Lebesgue?s theories and the historical development of integration theory. Mathematicians and researchers will find this book essential for its historical significance and the preservation of important mathematical literature.