A Course on Integration Theory including more than 150 exercises with detailed answers 1st Edition by Nicolas Lerner ISBN 3034806949 9783034806947

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ISBN 10: 3034806949 
ISBN 13: 9783034806947
Author: Nicolas Lerner

This textbook provides a detailed treatment of abstract integration theory, construction of the Lebesgue measure via the Riesz-Markov Theorem and also via the Carathéodory Theorem. It also includes some elementary properties of Hausdorff measures as well as the basic properties of spaces of integrable functions and standard theorems on integrals depending on a parameter. Integration on a product space, change of variables formulas as well as the construction and study of classical Cantor sets are treated in detail. Classical convolution inequalities, such as Young’s inequality and Hardy-Littlewood-Sobolev inequality are proven. The Radon-Nikodym theorem, notions of harmonic analysis, classical inequalities and interpolation theorems, including Marcinkiewicz’s theorem, the definition of Lebesgue points and Lebesgue differentiation theorem are further topics included.   A detailed appendix provides the reader with various elements of elementary mathematics, such as a discussion around the calculation of antiderivatives or the Gamma function. The appendix also provides more advanced material such as some basic properties of cardinals and ordinals which are useful in the study of measurability.​  

A Course on Integration Theory including more than 150 exercises with detailed answers 1st Table of contents:

Chapter 1 General Theory of Integration
1.1 Measurable spaces, σ-algebras
1.2 Measurable spaces and topological spaces
1.3 Structure of measurable functions
1.4 Positive measures
1.5 Integrating non-negative functions
1.6 Three basic convergence theorems
1.7 Space L1(µ) and negligible sets
1.8 Notes
1.9 Exercises
Elementary set theory
Topology
Measure theory
Chapter 2 Actual Construction of Measure Spaces
2.1 Partitions of unity
2.2 The Riesz–Markov representation theorem
2.3 Producing positive Radon measures
2.4 The Lebesgue measure on Rm, properties and characterization
2.5 Carathéodory theorem on outer measures
2.6 Hausdorff measures, Hausdorff dimension
Definition, first properties
Hausdorff dimension
Hausdorff measures on Rm
2.7 Notes
2.8 Exercises
Topology
Measure theory
Calculations
Chapter 3 Spaces of Integrable Functions
3.1 Convexity inequalities (Jensen, Hölder, Minkowski)
3.2 Lp spaces
3.3 Integrals depending on a parameter
Continuity
Differentiability
Holomorphy
3.4 Continuous functions in Lp spaces
3.5 On various notions of convergence
3.6 Notes
3.7 Exercises
Chapter 4 Integration on a Product Space
4.1 Product of measurable spaces
4.2 Tensor product of sigma-finite measures
4.3 The Lebesgue measure on Rm and tensor products
4.4 Notes
4.5 Exercises
Chapter 5 Diffeomorphisms of Open Subsets of Rn and Integration
5.1 Differentiability
5.2 Linear transformations
5.3 Change-of-variables formula
5.4 Examples, polar coordinates in Rn
Polar coordinates in R²
Spherical coordinates in R³
Polar coordinates in Rn
5.5 Integration on a C¹ hypersurface of the Euclidean Rn
5.6 More on Hausdorff measures on Rm
5.7 Cantor sets
Perfect sets, Nowhere dense sets
Cantor ternary set
The Cantor function
Lebesgue and Borel measurability
A Cantor set with positive measure
5.8 Category and measure
5.9 Notes
5.10 Exercises
Chapter 6 Convolution
6.1 The Banach algebra L1(Rn)
6.2 Lp Estimates for convolution, Young’s inequality
6.3 Weak Lp spaces
6.4 The Hardy–Littlewood–Sobolev inequality
6.5 Notes
6.6 Exercises
Chapter 7 Complex Measures
7.1 Complex measures
7.2 Total variation of a complex measure
7.3 Absolute continuity, mutually singular measures
7.4 Radon–Nikodym theorem
7.5 The dual of Lp(X,M,µ), 1 ≤ p < +∞
Main result
The Banach spaces c0, lp
Duality results
Examples of weak convergence
7.6 Notes
7.7 Exercises
Chapter 8 Basic Harmonic Analysis on Rn
8.1 Fourier transform of tempered distributions
The Fourier transformation on J(Rn)
The Fourier transformation on J'(Rn)
The Fourier transformation on L1(Rn)
The Fourier transformation on L2(Rn)
Some standard examples of Fourier transform
Fourier transform of Gaussian functions
Multipliers of J'(Rn)
8.2 The Poisson summation formula
Wave packets
Poisson’s formula
8.3 Periodic distributions
The Dirichlet kernel
Pointwise convergence of Fourier series
Periodic distributions
8.4 Notes
8.5 Exercises
Chapter 9 Classical Inequalities
9.1 Riesz–Thorin interpolation theorem
9.2 Marcinkiewicz Interpolation Theorem
9.3 Maximal function
9.4 Lebesgue differentiation theorem, Lebesgue points
9.5 Gagliardo–Nirenberg inequality
9.6 Sobolev spaces, Sobolev injection theorems
9.7 Notes
9.8 Exercises
Chapter 10 Appendix
10.1 Set theory, cardinals, ordinals
Set theory
Cardinals
Ordinals
Introduction
Ordering of ordinals
Addition of ordinals
Uncountable well-ordered sets
10.2 Topological matters
Filters
General properties of filters
Filters in a topological space
Compactness and Tychonoff’s Theorem
Connectedness of topological spaces
Partitions of unity in a topological space
Hahn–Banach Theorem
Baire category theorem and its consequences
Banach–Steinhaus
General properties of filters
Filters in a topological space
Banach–Steinhaus
Open mapping Theorem
10.3 Duality in Banach spaces
Definitions
Weak convergence
Weak-* convergence on E*
Reflexivity
10.4 Calculating antiderivatives
Table of classical antiderivatives
Integrating rational fractions
Antiderivatives of rational fractions of cos x, sin x
Abelian integrals
The function φ in 10.4.4 enjoys a parametric unicursal representation
Some Fourier integrals
Gaussian integrals
10.5 Some special functions
The complex logarithm
Logarithm on CR_
Logarithm of a nonsingular symmetric matrix
The Γ function
Wallis integrals
Laplace equation in spherical coordinates
More calculations on the Laplace operator
Laplace–Beltrami operator
10.6 Classical volumes and areas
Platonic polyhedra
Two-dimensional polygons
Three-dimensional regular polyhedrons

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