Krantz S.G. The Integral: A Crux for Analysis

Morgan & Claypool Publishers, 2011. — 106 p.This book treats all of the most commonly used theories of the integral. After motivating the idea of integral, we devote a full chapter to the Riemann integral and the next to the Lebesgue integral. Another chapter compares and contrasts the two theories. The concluding chapter offers brief introductions to the Henstock integral, the Daniell integral, the Stieltjes integral, and other commonly used integrals. The purpose of this book is to provide a quick but accurate (and detailed) introduction to all aspects of modern integration theory. It should be accessible to any student who has had calculus and some exposure to upper division mathematics.Introduction
The Riemann Integral
The Lebesgue Integral
Comparison of the Riemann and Lebesgue Integrals
Other Theories of the Integral