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.