دانلود کتاب Aspects of Integration: Novel Approaches to the Riemann and Lebesgue Integrals

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Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Contributors
Preface
To the Reader
Acknowledgement
Part I: A Novel Approach to Riemann Integration
CHAPTER 1. Preliminaries
1.1 Sums of Powers of Positive Integers
1.2 Bernstein Polynomials
CHAPTER 2. The Riemann Integral
2.1 Method of Exhaustion
2.2 Integral of a Continuous Function
2.3 Foundational Theorems of Integral Calculus
2.4 Integration by Substitution
CHAPTER 3. Extension to Higher Dimensions
3.1 Method of Exhaustion
3.2 Bernstein Polynomials in 2 Dimensions
3.3 Integral of a Continuous Function
Integrals Over More General Domains
Iterated Integration
CHAPTER 4. Extension to the Lebesgue Integral
4.1 Convergence and Cauchy Sequences
4.2 Completion of the Rational Numbers
Conundrum
Calculus
4.3 Completion of C in the 1-norm
Part II: Lebesgue Integration
Chapter 5. The Riesz-Nagy Approach to the Lebesgue Integral
5.1 Null Sets and Sets of Measure Zero
5.2 Lemma\'s A and B
5.3 The Class C1(I) of Riesz and Nagy
5.4 The Class C2 of Riesz and Nagy
5.5 Convergence Theorems
5.6 Completeness
5.7 The C2-Integral is the Lebesgue Integral
CHAPTER 6. Comparing Integrals
6.1 Properly Integrable Functions
6.2 Characterization of the Riemann Integral
6.3 Riemann vs. Lebesgue Integrals
6.4 The Novel Approach
Appendix A. Dini\'s Lemma
Appendix B. Semicontinuity
Appendix C. Completion of a Normed Linear Space
References
Index