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This accessible introduction to the topic covers the theory of measure and integral, as introduced by Lebesgue and developed in the first half of the 20th century. It leads naturally to Banach spaces of functions and linear operators acting on them.
List of contents
Prologue
I Preliminaries
1 Set Theory
1.1 Sets
1.2 Functions
1.3 Cardinal and Ordinal Numbers
1.4 The Axiom of Choice
2 Metric Spaces
2.1 Elementary Theory of Metric Spaces
2.2 Completeness
2.3 Compactness
2.4 Limits of Functions
2.5 Baire’s Theorem
3 Geometry of the Line and the Plane
II Measure Theory
4 Lebesgue Measure on R2
4.1 Jordan Measure
4.2 Lebesgue Measure
4.3 The σ-Algebra of Lebesgue Measurable Sets
5 Abstract Measure
5.1 Measures and Measurable Sets
5.2 Carath´eodory Extension of Measure
5.3 Lebesgue Measure on Euclidean Spaces
5.4 Beyond Lebesgue σ-Algebra
5.5 Signed Measures
6 Measurable Functions
6.1 Definition and Basic Facts
6.2 Fundamental Properties of Measurable Functions
6.3 Sequences of Measurable Functions
III Integration Theory
7 The Integral
7.1 About Riemann Integral
7.2 Integration of Nonnegative Measurable Functions
7.3 The Integral of a Real-Valued Function
7.4 Computing Lebesgue Integral
8 Integration on Product Spaces
8.1 Measurability on Cartesian Products
8.2 Product Measures
8.3 The Fubini Theorem
9 Differentiation and Integration
9.1 Dini Derivatives
9.2 Monotone Functions
9.3 Functions of Bounded Variation
9.4 Absolutely Continuous Functions
9.5 The Radon–Nikodym Theorem
IV An Introduction to Functional Analysis
10 Banach Spaces
10.1 Normed Linear Spaces
10.2 The Space Lp(X, µ)
10.3 Completeness of Lp(X, µ)
10.4 Dense Sets in Lp(X, µ)
10.5 Hilbert Space
10.6 Bessel’s Inequality and Orthonormal Bases
10.7 The Space C(X)
11 Continuous Linear Operators Between Banach Spaces
11.1 Linear Operators
11.2 Banach Space Isomorphisms
11.3 The Uniform Boundedness Principle
11.4 The Open Mapping and Closed Graph Theorems
12 Duality
12.1 Linear Functionals
12.2 The Hahn–Banach Theorem
12.3 The Dual of Lp(X, µ)
12.4 The Dual Space of L∞(X, µ)
12.5 The Dual Space of C(X)
12.6 Weak Convergence
Epilogue
Solutions and Answers to Selected Exercises
Bibliography
Subject Index
Author Index
About the author
John Srdjan Petrovic was born in Belgrade, Yugoslavia. He earned his PhD from the University of Michigan under the direction of Dr.Carl Pearcy. His research area is the theory of operators on Hilbert space, and he has published more than 30 articles in prestigious journals. He is a professor of mathematics at Western Michigan University and his visiting positions include Texas A&M University, Indiana University, and University of North Carolina Charlotte. His text, Advanced Caluclus: Theory and Practice, is in its second edition (CRC Press).
Summary
This accessible introduction to the topic covers the theory of measure and integral, as introduced by Lebesgue and developed in the first half of the 20th century. It leads naturally to Banach spaces of functions and linear operators acting on them.
