دانلود کتاب The Kurzweil-Henstock integral and its differentials

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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