实分析

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

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