Measure, Integration and Real Analysis.

• Superior document: Graduate Texts in Mathematics Series ; v.282
• Author: Axler, Sheldon.
• Place / Publishing House: Cham : : Springer International Publishing AG,, 2019.; ©2020.
• Year of Publication: 2019
• Edition: 1st ed.
• Language: English
• Series: Graduate Texts in Mathematics Series
• Physical Description: 1 online resource (430 pages)

Table of Contents

• Intro
• About the Author
• Contents
• Preface for Students
• Preface for Instructors
• Acknowledgments
• Riemann Integration
• Review: Riemann Integral
• Exercises 1A
• Riemann Integral Is Not Good Enough
• Exercises 1B
• Measures
• Outer Measure on R
• Motivation and Definition of Outer Measure
• Good Properties of Outer Measure
• Outer Measure of Closed Bounded Interval
• Outer Measure is Not Additive
• Exercises 2A
• Measurable Spaces and Functions
• -Algebras
• Borel Subsets of R
• Inverse Images
• Measurable Functions
• Exercises 2B
• Measures and Their Properties
• Definition and Examples of Measures
• Properties of Measures
• Exercises 2C
• Lebesgue Measure
• Additivity of Outer Measure on Borel Sets
• Lebesgue Measurable Sets
• Cantor Set and Cantor Function
• Exercises 2D
• Convergence of Measurable Functions
• Pointwise and Uniform Convergence
• Egorov's Theorem
• Approximation by Simple Functions
• Luzin's Theorem
• Lebesgue Measurable Functions
• Exercises 2E
• Integration
• Integration with Respect to a Measure
• Integration of Nonnegative Functions
• Monotone Convergence Theorem
• Integration of Real-Valued Functions
• Exercises 3A
• Limits of Integrals &amp
• Integrals of Limits
• Bounded Convergence Theorem
• Sets of Measure 0 in Integration Theorems
• Dominated Convergence Theorem
• Riemann Integrals and Lebesgue Integrals
• Approximation by Nice Functions
• Exercises 3B
• Differentiation
• Hardy-Littlewood Maximal Function
• Markov's Inequality
• Vitali Covering Lemma
• Hardy-Littlewood Maximal Inequality
• Exercises 4A
• Derivatives of Integrals
• Lebesgue Differentiation Theorem
• Derivatives
• Density
• Exercises 4B
• Product Measures
• Products of Measure Spaces
• Products of -Algebras
• Monotone Class Theorem
• Products of Measures.
• Exercises 5A
• Iterated Integrals
• Tonelli's Theorem
• Fubini's Theorem
• Area Under Graph
• Exercises 5B
• Lebesgue Integration on Rn
• Borel Subsets of Rn
• Lebesgue Measure on Rn
• Volume of Unit Ball in Rn
• Equality of Mixed Partial Derivatives Via Fubini's Theorem
• Exercises 5C
• Banach Spaces
• Metric Spaces
• Open Sets, Closed Sets, and Continuity
• Cauchy Sequences and Completeness
• Exercises 6A
• Vector Spaces
• Integration of Complex-Valued Functions
• Vector Spaces and Subspaces
• Exercises 6B
• Normed Vector Spaces
• Norms and Complete Norms
• Bounded Linear Maps
• Exercises 6C
• Linear Functionals
• Bounded Linear Functionals
• Discontinuous Linear Functionals
• Hahn-Banach Theorem
• Exercises 6D
• Consequences of Baire's Theorem
• Baire's Theorem
• Open Mapping Theorem and Inverse Mapping Theorem
• Closed Graph Theorem
• Principle of Uniform Boundedness
• Exercises 6E
• Lp Spaces
• Lp()
• Hölder's Inequality
• Minkowski's Inequality
• Exercises 7A
• Lp()
• Definition of Lp()
• Lp() Is a Banach Space
• Duality
• Exercises 7B
• Hilbert Spaces
• Inner Product Spaces
• Inner Products
• Cauchy-Schwarz Inequality and Triangle Inequality
• Exercises 8A
• Orthogonality
• Orthogonal Projections
• Orthogonal Complements
• Riesz Representation Theorem
• Exercises 8B
• Orthonormal Bases
• Bessel's Inequality
• Parseval's Identity
• Gram-Schmidt Process and Existence of Orthonormal Bases
• Riesz Representation Theorem, Revisited
• Exercises 8C
• Real and Complex Measures
• Total Variation
• Properties of Real and Complex Measures
• Total Variation Measure
• The Banach Space of Measures
• Exercises 9A
• Decomposition Theorems
• Hahn Decomposition Theorem
• Jordan Decomposition Theorem
• Lebesgue Decomposition Theorem
• Radon-Nikodym Theorem
• Dual Space of Lp().
• Exercises 9B
• Linear Maps on Hilbert Spaces
• Adjoints and Invertibility
• Adjoints of Linear Maps on Hilbert Spaces
• Null Spaces and Ranges in Terms of Adjoints
• Invertibility of Operators
• Exercises 10A
• Spectrum
• Spectrum of an Operator
• Self-adjoint Operators
• Normal Operators
• Isometries and Unitary Operators
• Exercises 10B
• Compact Operators
• The Ideal of Compact Operators
• Spectrum of Compact Operator and Fredholm Alternative
• Exercises 10C
• Spectral Theorem for Compact Operators
• Orthonormal Bases Consisting of Eigenvectors
• Singular Value Decomposition
• Exercises 10D
• Fourier Analysis
• Fourier Series and Poisson Integral
• Fourier Coefficients and Riemann-Lebesgue Lemma
• Poisson Kernel
• Solution to Dirichlet Problem on Disk
• Fourier Series of Smooth Functions
• Exercises 11A
• Fourier Series and Lp of Unit Circle
• Orthonormal Basis for L2 of Unit Circle
• Convolution on Unit Circle
• Exercises 11B
• Fourier Transform
• Fourier Transform on L1(R)
• Convolution on R
• Poisson Kernel on Upper Half-Plane
• Fourier Inversion Formula
• Extending Fourier Transform to L2(R)
• Exercises 11C
• Probability Measures
• Probability Spaces
• Independent Events and Independent Random Variables
• Variance and Standard Deviation
• Conditional Probability and Bayes' Theorem
• Distribution and Density Functions of Random Variables
• Weak Law of Large Numbers
• Exercises 12
• Photo Credits
• Bibliography
• Notation Index
• Index
• Colophon: Notes on Typesetting.