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| Foreword | |
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| Introduction | |
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| Fourier series: completion | |
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| Limits of continuous functions | |
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| Length of curves | |
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| Differentiation and integration | |
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| The problem of measure | |
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| Measure Theory 1 1 Preliminaries | |
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| The exterior measure | |
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| Measurable sets and the Lebesgue measure | |
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| Measurable functions | |
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| Definition and basic properties | |
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| Approximation by simple functions or step functions | |
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| Littlewood's three principles | |
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| The Brunn-Minkowski inequality | |
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| Exercises | |
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| Problems | |
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| Integration Theory | |
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| The Lebesgue integral: basic properties and convergence theorems | |
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| Thespace L 1 of integrable functions | |
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| Fubini's theorem | |
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| Statement and proof of the theorem | |
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| Applications of Fubini's theorem | |
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| A Fourier inversion formula | |
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| Exercises | |
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| Problems | |
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| Differentiation and Integration | |
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| Differentiation of the integral | |
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| The Hardy-Littlewood maximal function | |
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| The Lebesgue differentiation theorem | |
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| Good kernels and approximations to the identity | |
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| Differentiability of functions | |
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| Functions of bounded variation | |
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| Absolutely continuous functions | |
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| Differentiability of jump functions | |
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| Rectifiable curves and the isoperimetric inequality | |
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| Minkowski content of a curve | |
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| Isoperimetric inequality | |
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| Exercises | |
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| Problems | |
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| Hilbert Spaces: An Introduction | |
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| The Hilbert space L 2 | |
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| Hilbert spaces | |
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| Orthogonality | |
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| Unitary mappings | |
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| Pre-Hilbert spaces | |
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| Fourier series and Fatou's theorem | |
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| Fatou's theorem | |
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| Closed subspaces and orthogonal projections | |
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| Linear transformations | |
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| Linear functionals and the Riesz representation theorem | |
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| Adjoints | |
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| Examples | |
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| Compact operators | |
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| Exercises | |
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| Problems | |
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| Hilbert Spaces: Several Examples | |
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| The Fourier transform on L 2 | |
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| The Hardy space of the upper half-plane | |
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| Constant coefficient partial differential equations | |
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| Weaksolutions | |
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| The main theorem and key estimate | |
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| The Dirichlet principle | |
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| Harmonic functions | |
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| The boundary value problem and Dirichlet's principle | |
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| Exercises | |
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| Problems | |
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| Abstract Measure and Integration Theory | |
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| Abstract measure spaces | |
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| Exterior measures and Carathegrave;odory's theorem | |
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| Metric exterior measures | |
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| The extension theorem | |
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| Integration on a measure space | |
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| Examples | |
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| Product measures and a general Fubini theorem | |
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| Integration formula for polar coordinates | |
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| Borel measures on R and the Lebesgue-Stieltjes integral | |
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| Absolute continuity of measures | |
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| Signed measures | |
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| Absolute continuity | |
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| Ergodic theorems | |
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| Mean ergodic theorem | |
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| Maximal ergodic theorem | |
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| Pointwise ergodic theorem | |
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| Ergodic measure-preserving transformations | |
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| Appendix: the spectral theorem | |
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| Statement of the theorem | |
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| Positive operators | |
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| Proof of the theorem | |
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| Spectrum | |
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| Exercises | |
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| Problems | |
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| Hausdorff Measure and Fractals | |
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| Hausdorff measure | |
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| Hausdorff dimension | |
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| Examples | |
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| Self-similarity | |
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| Space-filling curves | |
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| Quartic intervals and dyadic squares | |
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| Dyadic correspondence | |
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| Construction of the Peano mapping | |
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| Besicovitch sets and regularity | |
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| The Radon transform | |
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| Regularity of sets whend3 | |
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| Besicovitch sets have dimension | |
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| Construction of a Besicovitch set | |
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| Exercises | |
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| Problems | |
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| Notes and References | |