Bass Richard. Real Analysis for Graduate Students: Measure and Integration Theory

Bass.math.uconn.edu, Jan 2016. — 422 p.Version 3.1Nearly every Ph.D. student in mathematics needs to take a preliminary or qualifying examination in real analysis. This book provides the necessary tools to pass such an examination. Every effort was made to present the material in as clear a fashion as possible.Preface Preliminaries
Notation and terminology
Some undergraduate mathematics
Proofs of propositionsFamilies of sets
Algebras and σ -algebras
The monotone class theorem
Exercises
Measures
Defenitions and examples
ExercisesConstruction of measures
Outer measures
Lebesgue-Stieltjes measures
Examples and related results
Nonmeasurable sets
The Caratheodory extension theorem
ExercisesMeasurable functions
Measurability
Approximation of functions
Lusin's theorem
ExercisesThe Lebesgue integral
Defenitions
ExercisesLimit theorems
Monotone convergence theorem
Linearity of the integral
Fatou's lemma
Dominated convergence theorem
ExercisesProperties of Lebesgue integrals
Criteria for a function to be zero a.e.
An approximation result
ExercisesRiemann integrals
Comparison with the Lebesgue integral
ExercisesTypes of convergence
Defenitions and examples
ExercisesProduct measures
Product σ -algebras
The Fubini theorem
Examples
ExercisesSigned measures
Positive and negative sets
Hahn decomposition theorem
Jordan decomposition theorem
ExercisesThe Radon-Nikodym theorem
Absolute continuity
The main theorem
Lebesgue decomposition theorem
ExercisesDifferentiation
Maximal functions
Antiderivatives
Bounded variation
Absolutely continuous functions
Approach - differentiability
Approach - antiderivatives
Approach - absolute continuity
ExercisesLp spaces
Norms
Completeness
Convolutions
Bounded linear functionals
ExercisesFourier transforms
Basic properties
The inversion theorem
The Plancherel theorem
ExercisesRiesz representation
Partitions of unity
The representation theorem
Regularity
Bounded linear functionals
ExercisesBanach spaces
Defenitions
The Hahn-Banach theorem
Baire's theorem and consequences
ExercisesHilbert spaces
Inner products
Subspaces
Orthonormal sets
Fourier series
Convergence of Fourier series
ExercisesTopology
Defenitions
Compactness
Tychono's theorem
Compactness and metric spaces
Separation properties
Urysohn's lemma
Tietze extension theorem
Urysohn embedding theorem
Locally compact Hausdor_ spaces
Stone-_Cech compacti_cation
Ascoli-Arzela theorem
Stone-Weierstrass theorems
Connected sets
ExercisesProbability
Defenitions
Independence
Weak law of large numbers
Strong law of large numbers
Conditional expectation
Martingales
Weak convergence
Characteristic functions
Central limit theorem
Kolmogorov extension theorem
Brownian motion
ExercisesHarmonic functions
Defenitions
The averaging property
Maximum principle
Smoothness of harmonic functions
Poisson kernels
Harnack inequality
ExercisesSobolev spaces
Weak derivatives
Sobolev inequalities
ExercisesSingular integrals
Marcinkiewicz interpolation theorem
Maximal functions
Approximations to the identity
The Calderon-Zygmund lemma
Hilbert transform
Lp boundedness
ExercisesSpectral theory
Bounded linear operators
Symmetric operators
Compact symmetric operators
An application
Spectra of symmetric operators
Spectral resolution
ExercisesDistributions
Defenitions and examples
Distributions supported at a point
Distributions with compact support
Tempered distributions
Exercises