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Kurtz D.S., Swartz C.W. Theories of Integration: The Integrals of Riemann, Lebesgue, Henstock-Kurzweil, and McShane

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Kurtz D.S., Swartz C.W. Theories of Integration: The Integrals of Riemann, Lebesgue, Henstock-Kurzweil, and McShane
Singapore: World Scientific Publishing Company, 2011. - 294p.
Contents
Introduction
Areas
Exercises
Riemann integral
Riemann's definition
Basic properties
Cauchy criterion
Darboux's definition
Necessary and sufficient conditions for Darboux integrability
Equivalence of the Riemann and Darboux definitions
Lattice properties
Integrable functions
Additivity of the integral over intervals
Fundamental Theorem of Calculus
Integration by parts and substitution
Characterizations of integrability
Lebesgue measure zero
Improper integrals
Exercises
Convergence theorems and the Lebesgue integral
Lebesgue's descriptive definition of the integral
Measure
Outer measure
Lebesgue measure
The Cantor set
Lebesgue measure in R^n
Measurable functions
Lebesgue integral
Integrals depending on a parameter
Riemann and Lebesgue integrals
Mikusinski's characterization of the Lebesgue integral
Fubini's Theorem
Convolution
The space of Lebesgue integrable functions
Exercises
Fundamental Theorem of Calculus and the Henstock-Kurzweil integral
Denjoy and Perron integrals
A General Fundamental Theorem of Calculus
Basic properties
Cauchy criterion
The integral as a set function
Unbounded intervals
Henstock's Lemma
Absolute integrability
Bounded variation
Absolute integrability and indefinite integrals
Lattice properties
Convergence theorems
Henstock-Kurzweil and Lebesgue integrals
Differentiating indefinite integrals
Characterizations of indefinite integrals
Derivatives of monotone functions
Indefinite Lebesgue integrals
Indefinite Riemann integrals
The space of Henstock-Kurzweil integrable functions
Henstock-Kurzweil integrals on R^n
Exercises
Absolute integrability and the McShane integral
Definitions
Basic properties
Absolute integrability
Fundamental Theorem of Calculus
Convergence theorems
The McShane integral as a set function
The space of McShane integrable functions
McShane, Henstock-Kurzweil and Lebesgue integrals
McShane integrals on R^n
Fubini and Tonelli Theorems
McShane, Henstock-Kurzweil and Lebesgue integrals in R^n
Exercises
Bibliography
Index
Back Cover
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