The Kurzweil Henstock Integral Its Differentials Pure and Applied Mathematics 1st Edition by Solomon Leader ISBN 0367397153 978-0367397159

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Product details: 

ISBN 10: 0367397153

ISBN 13: 978-0367397159

Author: Solomon Leader 

A comprehensive review of the Kurzweil-Henstock integration process on the real line and in higher dimensions. It seeks to provide a unified theory of integration that highlights Riemann-Stieljes and Lebesgue integrals as well as integrals of elementary calculus. The author presents practical applications of the definitions and theorems in each section as well as appended sets of exercises.

Table of contents: 

1. Integration of Summants:

   • Summation of measurable functions.

   • Interchange of limits and integrals (Monotone Convergence Theorem, Dominated Convergence Theorem).

2. Differentials and Their Integrals:

   • Differentials as linear approximations.

   • Integration of differentials in the context of the Fundamental Theorem of Calculus for Lebesgue integrals.

3. Differentials with Special Properties:

   • Functions of bounded variation.

   • Absolute continuity and its connection with differentiability.

4. Measurable Sets and Functions:

   • σ-algebras and measure spaces.

   • Properties of Lebesgue measurable sets.

   • Characterization of measurable functions.

5. Vitali Covering Theorem Applied to Differentials:

   • Vitali’s covering lemma.

   • Differentiation of measures using Vitali’s theorem.

6. Derivatives and Differentials:

   • Radon-Nikodym derivative.

   • Lebesgue differentiation theorem.

7. Essential Properties of Functions:

   • Lp spaces and norms.

   • Essential supremum and infimum.

   • Almost everywhere (a.e.) properties.

8. Absolute Continuity:

   • Relationship with bounded variation.

   • Connection with Lebesgue integration.

9. Conversion of Lebesgue-Stieltjes Integrals into Lebesgue Integrals:

   • Representation in terms of measures induced by monotone functions.

   • Applications to probability theory and distribution functions.

10. Some Results on Higher Dimensions:

• Fubini’s theorem.

• Product measures.

• Multivariable integration and change of variables formula.

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Tags: Solomon Leader, Kurzweil Henstock, Integral Its, Differentials Pure