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Full Description
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.
This material in Measure and Integral: Theory and Practice is typically covered in a graduate course and is almost always treated in an abstract way, with little or no motivation. The author employs a plethora of examples and exercises and strives to motivate every concept with its historical background. This textbook is accessible to a wider range of students, including at the undergraduate level.
A major problem facing anyone teaching measure theory is how to combine the elementary approach (measure on the real line or in the plane) and the abstract measure theory. The author develops a theory of measure in the plane, then shows how to generalize these ideas to an abstract setting.
The result is a textbook accessible to a wider range of students.
The material requires a good understanding of topics often referred to as advanced calculus, such as Riemann integration on Euclidean spaces and series of functions. Also, a reader is expected to be proficient in the basics of set theory and point-set topology, preferably including metric spaces.
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
