实分析-(影印版)

forewordintroduction 1 fourier series： 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： 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最 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： 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： 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最 ergodic theorems 5.1 mean ergodic theorem 5.2 maximal ergodic theorem 5.3 pointwise ergodic theorem 5.4 ergodic measure-preserving traformatio 6最 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最 besicovitch sets and regularity 4.1 the radon traform 4.2 regularity of sets when d ≥ 3 4.3 besicovitch sets have dimeion 2 4.4 cotruction of a besicovitch set 5 exercises 6 problemsnotes and referencesbibliographysymbol glossaryindex