Foreword Introduction 1 Fourier series: completion 2 Limits of continuous functions 3 Length of curves 4 Differentiation and integration 5 The problem of measure Chapter 1. Measure Theory 1 Preliminaries 2 The exterior measure 3 Measurable sets and the Lebesgue measure 4 Measurable functions 4.1 Definition and basic properties 4.2 Approximation by simple functions or step functions 4.3 Littlewood's three principles 5* The Brunn-Minkowski inequality 6 Exercises 7 Problems Chapter 2. Integration Theory 1 The Lebesgue integral: basic properties and convergence theorems 2 The space L1 of integrable functions 3 Fubini's theorem 3.1 Statement and proof of the theorem 3.2 Applications of Fubini's theorem 4* A Fourier inversion formula 5 Exercises 6 Problems Chapter 3. Differentiation and Integration 1 Differentiation of the integral 1.1 The Hardy-Littlewood maximal function 1.2 The Lebesgue differentiation theorem 2 Good kernels and approximations to the identity 3 Differentiability of functions 3.1 Functions of bounded variation 3.2 Absolutely continuous functions 3.3 Differentiability of jump functions 4 Rectifiable curves and the isoperimetric inequality 4.1 Minkowski content of a curve 4.2* Isoperimetrie inequality 5 Exercises 6 Problems Chapter 4. Hilbert Spaces: An Introduction 1 The Hilbert space L2 2 Hilbert spaces 2.1 Orthogonality 2.2 Unitary mappings 2.3 Pre-Hilbert spaces 3 Fourier series and Fatou's theorem 3.1 Fatou's theorem 4 Closed subspaees and orthogonal projections 5 Linear transformations 5.1 Linear flmetionals and the Riesz representation the-orem 5.2 Adjoints 5.3 Examples 6 Compact operators 7 Exercises 8 Problems Chapter 5. Hilbert Spaces: Several Examples 1 The Fourier transform on L2 2 The Hardy space of the upper half-plane 3 Constant coefficient partial differential equations 3.1 Weak solutions 3.2 The main theorem and key estimate 4* The Dirichlet principle 4.1 Harmonic functions 4.2 The boundary value problem and Diriehlet's principle 5 Exercises 6 Problems Chapter 6.Abstract Measure and Integration Theory Chapter 7.Hausdorff Measure and Fractals Notes and References Bibliography Symbol Glossary Index
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