Swartz C. Introduction to Gauge Integrals

Singapore: World Scientific, 2001. - 169p.This book presents the Henstock/Kurzweil integral and the McShane integral. These two integrals are obtained by changing slightly the definition of the Riemann integral. These variations lead to integrals which are much more powerful than the Riemann integral. The Henstock/Kurzweil integral is an unconditional integral for which the fundamental theorem of calculus holds in full generality, while the McShane integral is equivalent to the Lebesgue integral in Euclidean spaces.
A basic knowledge of introductory real analysis is required of the reader, who should be familiar with the fundamental properties of the real numbers, convergence, series, differentiation, continuity, etc.Introduction to the Gauge or Henstock-Kurzweil Integral
Basic Properties of the Gauge Integral
Henstock's Lemma and Improper Integrals
The Gauge Integral over Unbounded Intervals
Convergence Theorems
Integration over More General Sets: Lebesgue Measure
The Space of Gauge Integrable Functions
Multiple Integrals and Fubini's Theorem
Definition and Basic Properties
Convergence Theorems
Integrability of Products and Integration by Parts
More General Convergence Theorems
The Space of McShane Integrable Functions
Multiple Integrals and Fubini's Theorem
McShane Integrability is Equivalent to Absolute Henstock-Kurzweil IntegrabilityThe Riemann Integral
Functions of Bounded Variations
Differentiating Indefinite Integrals
Equivalence of Lebesgue and McShane Integrals
Change of Variable in Multiple Integrals