Chapter Ⅸ Elements of measure theory

1 Measurable spaces

σ-algebras

The Borel σ-algebra

The second countability axiom

Generating the Borel σ-algebra with intervals

Bases of topological spaces

The product topology

Product Borelσ-algebras

Measurability of sections

2 Measures

Set functions

Measure spaces

Properties of measures

Null sets

3 Outer measures

The construction of outer measures

The Lebesgue outer measure

The Lebesgue-Stieltjes outer measure

Hausdorff outer measures

4 Measurable sets

Motivation

Theσ-algebra of μ-measurable sets

Lebesgue measure and Hausdorff measure

Metric measures

5 The Lebesgue 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 ？-valued functions

The lattice of measurable ？-valued functions

Pointwise limits of measurable functions

Radon measures

2 Integrable functions

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 ？-valued functions

The monotone convergence theorem

Fatou's lemma

Integration of ？-valued functions

Lebesgue's dominated convergence theorem

Parametrized integrals

4 Lebesgue spaces

Essentially bounded functions

The H？lder 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 Fubini'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-valued 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

Submanifolds with boundary

Local charts

Tangents and normals

The regular value theorem

One-dimensional manifolds

Partitions of unity

2 Multilinear 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 Poincaré 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 σ-algebra of M

The definition 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