分析. 第3卷

Foreword

Chapter Ⅸ Elements of measure theory

1 Measurable spaces

σ-algebras

The Borel σ-algebra

The second countability axiom

Generating the Borel a-algebra with intervals

Bases of topological spaces

The product topology

Product Borel a-algebras

Measurability of sections

2 Measures

Set functions

Measure spaces

Properties of measures

Null sets

Outer measures

The construction of outer measures

The Lebesgue outer measure

The Lebesgue-Stieltjes outer measure

Hausdorff outer measures

4 Measurable sets

Motivation

The a-algebra of/μ*-measurable sets

Lebesgue measure and Hausdorff measure

Metric measures

5 The Lebasgue measure

The Lebesgue measure space

The Lebesgue measure is regular

A characterization of Lebesgue measurability

Images of Lebesgue measurable sets

The Lebesgue measure is translation invariant

A characterization of Lebesgue measure

The Lebesgue measure is invariant under rigid motions

The substitution rule for linear maps

Sets without Lebesgue measure

Chapter Ⅹ Integration theory

1 Measurable functions

Simple functions and measurable functions

A measurability criterion

Measurable R-valued functions

The lattice of measurable T-valued functions

Pointwise limits of mensurable functions

Radon measures

2 Integrable fuuctions

The integral of a simple function

The L1-seminorm

The Bochner-Lebesgue integral

The completeness of L1

Elementary properties of integrals

Convergence in L1

3 Convergence theorems

Integration of nonnegative T-valued functions

The monotone convergence theorem

Fatou's lemma

Integration of R-valued functions

Lebesgue's dominated convergence theorem

Parametrized integrals

4 Lebesgue spaces

Essentially bounded functions

The Holder and Minkowski inequalities

Lebesgue spaces are complete

Lp-spaces

Continuous functions with compact support

Embeddings

Continuous linear functionals on Lp

5 The n-dimensional Bochner-Lebesgue integral

Lebesgue measure spaces

The Lebesgue integral of absolutely integrable functions

A characterization of Riemann integrable functions

6 Fubiul's theorem

Maps defined almost everywhere

Cavalieri's principle

Applications of Cavalieri's principle

Tonelli's theorem

Fubini's theorem for scalar functions

Fubini's theorem for vector-vained functions

Minkowski's inequality for integrals

A characterization of Lp （Rm+n, E）

A trace theorem

7 The convolution

Defining the convolution

The translation group

Elementary properties of the convolution

Approximations to the identity

Test functions

Smooth partitions of unity

Convolutions of E-valued functions

Distributions

Linear differential operators

Weak derivatives

8 The substitution rule

Pulling back the Lebesgue measure

The substitution rule: general case

Plane polar coordinates

Polar coordinates in higher dimensions

Integration of rotationally symmetric functions

The substitution rule for vector-valued functions

9 The Fourier transform

Definition and elementary properties

The space of rapidly decreasing functions

The convolution algebra S

Calculations with the Fourier transform

The Fourier integral theorem

Convolutions and the Fourier transform

Fourier multiplication operators

Plancherel's theorem

Symmetric operators

The Heisenberg uncertainty relation

Chapter Ⅺ Manifolds and differential forms

1 Submanifolds

Definitions and elementary properties

Submersions

Submanifo]ds with boundary

Local charts

Tangents and normals

The regular value theorem

One-dimensional manifolds

Partitions of unity

2 MultUinear algebra

Exterior products

Pull backs

The volume element

The Riesz isomorphism

The Hodge star operator

Indefinite inner products

Tensors

3 The local theory of differential forms

Definitions and basis representations

Pull backs

The exterior derivative

The Poincare lemma

Tensors

4 Vector fields and differential forms

Vector fields

Local basis representation

Differential forms

Local representations

Coordinate transformations

The exterior derivative

Closed and exact forms

Contractions

Orientability

Tensor fields

5 Riemannian metrics

The volume element

Riemannian manifolds

The Hodge star

The codifferential

6 Vector analysis

The Riesz isomorphism

The gradient

The divergence

The Laplace-Beltrami operator

The curl

The Lie derivative

The Hodge-Laplace operator

The vector product and the curl

Chapter Ⅻ Integration on manifolds

1 Volume measure

The Lebesgue a-algebra of M

The defiaition of the volume measure

Properties

Integrability

Calculation of several volumes

2 Integration of differential forms

Integrals of m-forms

Restrictions to submanifolds

The transformation theorem

Fubini's theorem

Calculations of several integrals

Flows of vector fields

The transport theorem

3 Stokes's theorem

Stokes's theorem for smooth manifolds

Manifolds with singularities

Stokes's theorem with singularities

Planar domains

Higher-dimensional problems

Homotopy invariance and applications

Gauss's law

Green's formula

The classical Stokes's theorem

The star operator and the coderivative

References

This third volume concludes our introduction to analysis, where in we finish laying the groundwork needed for further study of the subject. As with the first two, this volume contains more material than can treated in a single course. It is therefore important in preparing lectures to choose a suitable subset of its content; the remainder can be treated in seminars or left to independent study. For a quick overview of this content, consult the table of contents and the chapter introductions.