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forewordintroduction 1 fourier series£º completion 2 limits of continuous functio 3 length of curves 4 differentiation and integration 5 the problem of measurechapter 1. measure theory 1 prelhninaries 2 the exterior measure 3 measurable sets and the lebesgue measure 4 measurable functio 4.1 definition and basic properties 4.2 approximation by simple functio or step functio 4.3 littlewood's three principles 5 the brunn-minkowski inequality 6 exercises 7 problemschapter 2. integration theory 1 the lebesgue integral£º basic properties and convergencetheorems 2 the space l1 ofintegrable functio 3 fubini's theorem 3.1 statement and proof of the theorem 3.2 applicatio of fubini's theorem 4* a fourier inveion formula 5 exercises 6 problemschapter 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 approximatio to the identity 3 differentiability of functio 3.1 functio of bounded variation 3.2 absolutely continuous functio 3.3 differentiability ofjump functio 4 rectifiable curves and the isoperimetric inequality 4.1 minkowski content of a curve 4.2 isoperimetric inequality 5 exercises 6 problemschapter 4. hilbert spaces£º 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 subspaces and orthogonal projectio 5 linear traformatio 5.1 linear functionals and the riesz representation theorem 5.2 adjoints 5.3 examples 6 compact operato 7 exercises 8 problemschapter 5. hilbert spaces£º several examples 1 the fourier traform on l2 2 the hardy space of the upper half-plane 3 cotant coefficient partial differential equatio 3.1 weaak solutio 3.2 the main theorem and key estimate 4 the dirichlet principle 4.1 harmonic functio 4.2 the boundary value problem and dirichlet's principle 5 exercises 6 problemschapter 6. abstract measure and integration theory 1 abstract measure spaces 1.1 exterior measures and carathodory's theorem 1.2 metric exterior measures 1.3 the exteion theorem 2 integration o a measure space 3 examples 3.1 product measures and a general fubini theorem 3.2 integration formula for polar coordinates 3.3 borel measures on and the lebesgue-stieltjes integral 4 absolute continuity of measures 4.1 signed measures 4.2 absolute continuity 5* ergodic theorems 5.1 mean ergodic theorem 5.2 maximal ergodic theorem 5.3 pointwise ergodic theorem 5.4 ergodic measure-preserving traformatio 6* appendix: the spectral theorem 6.1 statement of the theorem 6.2 positive operato 6.3 proof of the theorem 6.4 spectrum 7 exercises 8 problemschapter 7. hausdorff measure and fractals 1 hausdorff measure 2 hausdorff dimeion 2.1 examples 2.2 self-similarity 3 space-filling curves 3.1 quartic intervals and dyadic squares 3.2 dyadic correspondence 3.3 cotruction of the peano mapping 4* besicovitch sets and regularity 4.1 the radon traform 4.2 regularity of sets when d ¡Ý 3 4.3 besicovitch sets have dimeion 2 4.4 cotruction of a besicovitch set 5 exercises 6 problemsnotes and referencesbibliographysymbol glossaryindex