The Kurzweil-Henstock Integral and Its Differential: A Unified Theory of Integration on R and Rn (Pure and Applied Mathematics)

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. Cover Title page Preface Introduction 0.1 The Gauge-Directed Integral 0.2 Differentials 0.3 Guidance for the Reader Chapter 1. Integration of Summants 1.1 Cells, Figures and Partitions 1.2 Tagged CeIls, Divisions, and Gauges 1.3 The Upper and Lower Integrais of a Summant over a Figure 1.4 Summants with Special Properties 1.5 Upper and Lower Integrals as Functions on the Boolean Algebra of Figures 1.6 Uniform Integrability and Its Consequences 1.7 Term-by-Term Integration of Series 1.8 Applications of Term-by- Term Upper Integration 1.9 Integration over Arbitrary Intervals Chapter 2. Differentials and Their Integrals 2.1 Differential Equivalence and Differentials 2.2 The Riesz Space D = D(K) of All Differentials on K 2.3 Differential Norm and Summable Differentials 2.4 Conditionally and Absolutely Integrable Differentials 2.5 The Differential dg of a Function g 2.6 The Total Variation of a Function on a Cell K 2.7 Functions as Differential Coefficients 2.8 The Lebesgue Space £1 and Convergence Theorems Chapter 3. Differentials with Special Properties 3.1 Products Involving Tag-Finite Summants and Differentials 3.2 Continuous Differentials 3.3 Archimedean Properties for Differentials 3.4 Differentials on Open-Ended Intervals 3.5 σ-Nullity of the Union of All σ-Null Cells 3.6 Mappings of Differentials Induced by Lipschitz Functions 3.7 n-Differentials on a Cell K Chapter 4. Measurable Sets and Functions 4.1 Measurable Sets 4.2 The Hahn Decomposition for Differentials 4.3 Measurable Functions 4.4 Step Functions and Regulated Functions 4.5 The Radon-Nikodym Theorem for Differentials 4.6 Minimal Measurable Dominators Chapter 5. The Vitali Covering Theorem Applied to Differentials 5.1 The Vitali Covering Theorem with some Applications to Upper Integrals 5.2 ν(l_Edf) and Lebesgue Outer Measure of f(E) 5.3 Continuity σ-Everywhere of ρ Given ρσ = 0 Chapter 6. Derivatives and Differentials 6.1 Differential Coefficients from the Gradient 6.2 Integration by Parts and Taylor's Formula 6.3 A Generalized Fundamental Theorem of Calculus 6.4 L'Hôpital's Rule and the Limit Comparison Test Using Essential Limits 6.5 Differentiation Under the Integral Sign Chapter 7. Essential Properties of Functions 7.1 Essentially Bounded Functions 7.2 Essentially Regulated Functions 7.3 Essential Variation Chapter 8. Absolute Continuity 8.1 Various Concepts of Absolute Continuity for Differentials 8.2 Absolute Continuity for Restricted Classes of Differentials 8.3 Absolutely Continuous Functions 8.4 The Vitali Convergence Theorem Chapter 9. Conversion of Lebesgue-Stieltjes Integrals into Lebesgue Integrals 9.1 Banach's Indicatrix Theorem 9.2 A Generalization of the Indicatrix Theorem with Applications Chapter 10. Some Results on Higher Dimensions 10.1 Integral and Differential on n-Cells 10.2 Direct Products of Summants 10.3 A Fubini Theorem 10.4 Integration on Paths in R^n 10.5 Green's Theorem Chapter 11. Mathematical Background 11.1 Filterbases, Lower and Upper Limits 11.2 Metric Spaces 11.3 Norms and Inner Products 11.4 Topological Spaces 11.5 Regular Closed Sets 11.6 Riesz Spaces 11.7 The Inclusion-Exclusion Formula References Index Reviews the Kurzweil-Henstock integration process on the real line and in higher dimensions. This book seeks to provide a unified theory of integration that highlights Riemann-Stieljes and Lebesgue integrals as well as integrals of elementary calculus.