Preface Acknowledgments Preliminaries 1 Countable sets 2 The Cantor set 3 Cardinality 3.1 Some examples 4 Cardinality of some infinite Cartesian products 5 Orderings, the maximal principle, and the axiom of choice 6 Well-ordering 6.1 The first uncountable Problems and Complements Ⅰ Topologies and Metric Spaces 1 Topological spaces 1.1 Hausdorff and normal spaces 2 Urysohns lemma 3 The Tietze extension theorem 4 Bases, axioms of countability, and product topologies 4.1 Product topologies 5 Compact topological spaces 5.1 Sequentially compact topological spaces 6 Compact subsets of RN 7 Continuous functions on countably compact spaces 8 Products of compact spaces 9 Vector spaces 9.1 Convex sets 9.2 Linear maps and isomorphisms 10 Topological vector spaces 10.1 Boundedness and continuity 11 Linear functionals 12 Finite-dimensional topological vector spaces 12.1 Locally compact spaces 13 Metric spaces 13.1 Separation and axioms of countability 13.2 Equivalent metrics 13.3 Pseudometrics 14 Metric vector spaces 14.1 Maps between metric spaces 15 Spaces of continuous functions 15.1 Spaces of continuously differentiable functions 16 On the structure of a complete metric space 17 Compact and totally bounded metric spaces 17.1 Precompact subsets of X Problems and Complements Ⅱ Measuring Sets 1 Partitioning open subsets of RN 2 Limits of sets, characteristic functions, and or-algebras 3 Measures 3.1 Finite,a-finite, and complete measures 3.2 Some examples 4 Outer measures and sequential coverings 4.1 The Lebesgue outer measure in RN 4.2 The Lebesgue-Stieltjes outer measure 5 The Hausdorff outer measure in RN 6 Constructing measures from outer measures 7 The Lebesgue——Stieltjes measure on R 7.1 Borel measures 8 The Hausdorff measure on RN 9 Extending measures from semialgebras to a-algebras 9.1 On the Lebesgue-Stieltjes and Hausdorff measures 10 Necessary and sufficient conditions for measurability 11 More on extensions from semialgebras to a-algebras 12 The Lebesgue measure of sets in RN 12.1 A necessary and sufficient condition of naeasurability 13 A nonmeasurable set 14 Borel sets, measurable sets, and incomplete measures 14.1 A continuous increasing function f : [0, 1] → [0, 1] 14.2 On the preimage of a measurable set 14.3 Proof of Propositions 14.1 and 14.2 15 More on Borel measures 15.1 Some extensions to general Borel measures 15.2 Regular Borel measures and Radon measures 16 Regular outer measures and Radon measures 16.1 More on Radon measures 17 Vitali coverings 18 The Besicovitch covering theorem 19 Proof of Proposition 18.2 20 The Besicovitch measure-theoretical covering theorem Problems and Complements Ⅲ The Lebesgue Integral 1 Measurable functions 2 The Egorov theorem 2.1 The Egorov theorem in RN 2.2 More on Egorovs theorem 3 Approximating measurable functions by simple functions 4 Convergence in measure 5 Quasi-continuous functions and Lusins theorem 6 Integral of simple functions 7 The Lebesgue integral of nonnegative functions 8 Fatous lemma and the monotone convergence theorem 9 Basic properties of the Lebesgue integral 10 Convergence theorems 11 Absolute continuity of the integral 12 Product of measures 13 On the structure of （A*p ） 14 The Fubini-Tonelli theorem 14.1 The Tonelli version of the Fubini theorem 15 Some applications of the Fubini-Tonelli theorem 15.1 Integrals in terms of distribution functions 15.2 Convolution integrals 15.3 The Marcinkiewicz integral 16 Signed measures and the Hahn decomposition 17 The Radon-Nikodym theorem 18 Decomposing measures 18.1 The Jordan decomposition 18.2 The Lebesgue decomposition 18.3 A general version of the Radon-Nikodym theorem Problems and Complements IV Topics on Measurable Functions of Real Variables 1 Functions of bounded variations 2 Dini derivatives 3 Differentiating functions of bounded variation 4 Differentiating series of monotone functions 5 Absolutely continuous functions 6 Density of a measurable set 7 Derivatives of integrals 8 Differentiating Radon measures 9 Existence and measurability of Dvv 9.1 Proof of Proposition 9.2 10 Representing Dvv 10.1 Representing Duv for v << # 10.2 Representing Duv for v u 11 The Lebesgue differentiation theorem 11.1 Points of density 11.2 Lebesgue points of an integrable function 12 Regular families 13 Convex functions 14 Jensens inequality 15 Extending continuous functions 16 The Weierstrass approximation theorem 17 The Stone-Weierstrass theorem 18 Proof of the Stone-Weierstrass theorem 18.1 Proof of Stones theorem 19 The Ascoli-Arzela theorem 19.1 Precompact subsets of C（E） Problems and Complements V The LP（E） Spaces 1 Functions in Lp（E） and their norms 1.1 The spaces LP for 0 < p < 1 1.2 The spaces Lq for q < 0 2 The HOlder and Minkowski inequalities 3 The reverse Holder and Minkowski inequalities 4 More on the spaces Lp and their norms 4.1 Characterizing the norm fp for 1 < p < oo 4.2 The norm II I1 for E of finite measure 4.3 The continuous version Of the Minkowski inequality 5 LP（E） for 1 < p < oo as normed spaces of equivalence classes 5.1 Lp（E） for 1 < p < as ametric topological vector space 6 A metric topology for LP（E） when 0 < p < 1 6.1 Open convex subsets of LP （E） when0 < p < 1 7 Convergence in LP（E） and completeness 8 Separating LP（E） by simple functions Ⅵ Banach Spaces Ⅶ Spaces of Continuous Functions,Distributions,and Weak Ⅷ Topics on Integrable Functions of Real Variables Ⅸ Embeddings of W1,p（E)into Lq（E) References Index
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