The goal of this book is to provide an extensive collection of results which generalize classical real analysis. Besides discussing density, approximate continuity, and approximate derivatives in detail, culminating with the Denjoy-Saks-Young Theorem, the authors also present an interesting example due to Ruziewicz onan infinite number of functions with the same derivative (not everywhere finite) but the difference of any two is not a constant and Sierpinski's theorem on the extension of approximate continuity to nonmeasurable functions.
The goal of this book is to provide an extensive collection of results which generalize classical real analysis. Besides discussing density, approximate continuity, and approximate derivatives in detail, culminating with the Denjoy-Saks-Young Theorem, the authors also present an interesting example due to Ruziewicz onan infinite number of functions with the same derivative (not everywhere finite) but the difference of any two is not a constant and Sierpinski's theorem on the extension of approximate continuity to nonmeasurable functions.
0 Preliminaries.- 0.1 Lebesgue Measure.- 0.2 The Lebesgue Integral.- 0.3 Vitali Covering Theorem.- 0.4 Baire Category Theorem and Baire Class Functions.- 1 Monotone Functions.- 1.1 Continuity Properties.- 1.2 Differentiability Properties.- 1.3 Reconstruction of f from f?.- 1.4 Series of Monotone Functions.- Exercises.- 2 Density and Approximate Continuity.- 2.1 Preliminaries and Definitions.- 2.2 The Lebesgue Density Theorem.- 2.3 Approximate Continuity.- 2.4 Approximate Continuity and Integrability.- 2.5 Further Results on Approximate Continuity.- 2.6 Sierpinski's Theorem.- 2.7 The Darboux Property and the Density Topology.- Exercises.- 3 Dini Derivatives.- 3.1 Preliminaries and Definitions.- 3.2 Simple Properties of Derivatives.- 3.3 Ruziewicz's Example.- 3.4 Further Properties of Derivatives.- 3.5 The Denjoy-Saks-Young Theorem.- 3.6 Measurability of Dini Derivatives.- 3.7 Dini Derivatives and Convex Functions.- Exercises.- 4 Approximate Derivatives.- 4.1 Definitions.- 4.2 Measurability of Approximate Derivatives.- 4.3 Analogue of the Denjoy-Saks-Young Theorem.- 4.4 Category Results for Approximate Derivatives.- 4.5 Other Properties of Approximate Derivatives.- Exercises.- 5 Additional Results on Derivatives.- 5.1 Derivatives.- 5.2 Derivates.- 5.3 Approximate Derivatives.- 5.4 The Denjoy Property.- 5.5 Metrically Dense.- 6 Bounded Variation.- 6.1 Bounded Variation of Finite Intervals.- 6.2 Stieltjes Integral.- 6.3 The Space BV[a,b].- BVloc and L1loc.- 6.5 Additional Remarks on Fubini's Theorem.- Exercises.- 7 Absolute Continuity.- 7.1 Absolute Continuity.- 7.2 Rectifiable Curves.- Exercises.- 8 Cantor Sets and Singular Functions.- 8.1 The Cantor Ternary Set and Function.- 8.2 Hausdorff Measure.- 8.3 Generalized Cantor Sets-Part I.- 8.4 Generalized CantorSets-Part II.- 8.5 Cantor-like Sets.- 8.6 Strictly Increasing Singular Functions.- Exercises.- 9 Spaces of BV and AC Functions.- 9.1 Convergence in Variation.- 9.2 Convergence in Length.- 9.3 Norms on AC.- 9.4 Norms on BV.- 10 Metric Separability.- Exercises.
0 Preliminaries.- 0.1 Lebesgue Measure.- 0.2 The Lebesgue Integral.- 0.3 Vitali Covering Theorem.- 0.4 Baire Category Theorem and Baire Class Functions.- 1 Monotone Functions.- 1.1 Continuity Properties.- 1.2 Differentiability Properties.- 1.3 Reconstruction of f from f?.- 1.4 Series of Monotone Functions.- Exercises.- 2 Density and Approximate Continuity.- 2.1 Preliminaries and Definitions.- 2.2 The Lebesgue Density Theorem.- 2.3 Approximate Continuity.- 2.4 Approximate Continuity and Integrability.- 2.5 Further Results on Approximate Continuity.- 2.6 Sierpinski's Theorem.- 2.7 The Darboux Property and the Density Topology.- Exercises.- 3 Dini Derivatives.- 3.1 Preliminaries and Definitions.- 3.2 Simple Properties of Derivatives.- 3.3 Ruziewicz's Example.- 3.4 Further Properties of Derivatives.- 3.5 The Denjoy-Saks-Young Theorem.- 3.6 Measurability of Dini Derivatives.- 3.7 Dini Derivatives and Convex Functions.- Exercises.- 4 Approximate Derivatives.- 4.1 Definitions.- 4.2 Measurability of Approximate Derivatives.- 4.3 Analogue of the Denjoy-Saks-Young Theorem.- 4.4 Category Results for Approximate Derivatives.- 4.5 Other Properties of Approximate Derivatives.- Exercises.- 5 Additional Results on Derivatives.- 5.1 Derivatives.- 5.2 Derivates.- 5.3 Approximate Derivatives.- 5.4 The Denjoy Property.- 5.5 Metrically Dense.- 6 Bounded Variation.- 6.1 Bounded Variation of Finite Intervals.- 6.2 Stieltjes Integral.- 6.3 The Space BV[a,b].- BVloc and L1loc.- 6.5 Additional Remarks on Fubini's Theorem.- Exercises.- 7 Absolute Continuity.- 7.1 Absolute Continuity.- 7.2 Rectifiable Curves.- Exercises.- 8 Cantor Sets and Singular Functions.- 8.1 The Cantor Ternary Set and Function.- 8.2 Hausdorff Measure.- 8.3 Generalized Cantor Sets-Part I.- 8.4 Generalized CantorSets-Part II.- 8.5 Cantor-like Sets.- 8.6 Strictly Increasing Singular Functions.- Exercises.- 9 Spaces of BV and AC Functions.- 9.1 Convergence in Variation.- 9.2 Convergence in Length.- 9.3 Norms on AC.- 9.4 Norms on BV.- 10 Metric Separability.- Exercises.
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