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
