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Real Analysis Measure and Integration 1st Edition by Marat V Markin ISBN 3110600978 9783110600971

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Author(s): Marat V. Markin
ISBN(s): 9783110600971, 3110600978
Edition: 1
File Details: PDF, 4.49 MB
Year: 2019
Language: english
SKU: EB-10666132 Category: Tag:

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ISBN 10: 3110600978 
ISBN 13: 9783110600971
Author: Marat V Markin

The philosophy of the book, which makes it quite distinct from many existing texts on the subject, is based on treating the concepts of measure and integration starting with the most general abstract setting and then introducing and studying the Lebesgue measure and integration on the real line as an important particular case.　 The book consists of nine chapters and appendix, with the material flowing from the basic set classes, through　measures, outer measures and the general procedure of measure extension, through measurable functions　and　various types of convergence of sequences of such based on the idea of measure, to the fundamentals of the abstract Lebesgue integration, the basic limit theorems, and the comparison of the Lebesgue and Riemann integrals. Also, studied are Lp spaces, the basics of normed vector spaces, and signed measures. The novel approach based on the Lebesgue measure and integration theory is applied to develop a better understanding of differentiation and extend the classical total change formula linking differentiation with integration to a substantially wider class of functions. Being designed as a text to be used in a classroom, the book constantly calls for the student’s actively mastering the knowledge of the subject matter. There are problems at the end of each chapter, starting with Chapter 2 and totaling at 125. Many important statements are given as problems and frequently referred to in the main body. There are also 358 Exercises throughout the text, including Chapter 1 and the Appendix, which require of the student to prove or verify a statement or an example, fill in certain details in a proof, or provide an intermediate step or a counterexample. They are also an inherent part of the material. More difficult problems are marked with an asterisk, many problems and exercises are supplied with “existential” hints.　 The book is generous on Examples and contains numerous Remarks accompanying definitions, examples, and statements to discuss certain subtleties, raise questions on whether the converse assertions are true, whenever appropriate, or whether the conditions are essential.　 　　 With　plenty of examples, problems, and exercises, this well-designed text is ideal for a one-semester Master’s level graduate course on real analysis with emphasis on the measure and integration theory for students majoring in mathematics, physics, computer science, and engineering.　 A concise but profound and detailed presentation of the basics of real analysis with emphasis on the measure and integration theory. Designed　for a one-semester graduate course, with plethora of examples, problems, and exercises. Is of interest to students and instructors in mathematics, physics, computer science, and engineering. Prepares the students for more advanced courses in functional analysis and operator theory.　　 Contents Preliminaries Basic Set Classes Measures Extension of Measures Measurable Functions Abstract Lebesgue Integral Lp Spaces Differentiation and Integration Signed Measures The Axiom of Choice and Equivalents

Real Analysis Measure and Integration 1st Table of contents:

1 Preliminaries
1.1 Set Theoretic Basics
1.1.1 Some Terminology and Notations
1.1.2 Cardinality and Countability
1.2 Terminology Related to Functions
1.3 Upper and Lower Limits
1.4 Fundamentals of Metric Spaces
1.4.1 Definition and Examples
1.4.2 Convergence
1.4.3 Completeness
1.4.4 Balls and Boundedness
1.4.5 Interior Points, Open Sets
1.4.6 Limit Points, Closed Sets
1.4.7 Dense Sets and Separable Spaces
1.4.8 Compactness
1.4.9 Continuity
2 Basic Set Classes
2.1 Semirings, Semi-algebras
2.2 Rings, Algebras
2.3 σ-Rings, σ-Algebras
2.4 Monotone Classes
2.5 Generated Set Classes
2.5.1 Intersection Lemma
2.5.2 Generated Set Classes
2.5.3 Borel Sets
2.6 Problems
3 Measures
3.1 Set Functions
3.2 Measure
3.2.1 Definition and Examples
3.2.2 Properties of Measure
3.2.3 Continuity of Measure
3.2.4 More Examples of Measures
3.3 Problems
4 Extension of Measures
4.1 Extension of a Set Function
4.2 Extension From a Semiring
4.3 Outer Measure
4.3.1 Definition and Examples
4.3.2 Construction of Outer Measures
4.4 μ∗-Measurable Sets, Carathéodory’s Theorem
4.4.1 μ∗-Measurable Sets
4.4.2 Carathéodory’s Theorem
4.5 Completeness
4.5.1 Null Sets, Completeness
4.5.2 Addendum to Carathéodory’s Theorem
4.5.3 Completion
4.6 Measure Extension From a Ring
4.6.1 Carathéodory’s Extension Theorem
4.6.2 Approximation
4.7 Lebesgue–Stieltjes Measures
4.7.1 The Construct
4.7.2 Relationships Between Various Extensions of Length
4.7.3 Existence of a Non-Lebesgue Measurable Set in ℝ
4.7.4 Multidimensional Lebesgue Measure
4.8 Problems
5 Measurable Functions
5.1 Measurable Space and Measure Space
5.2 Definition and Examples
5.3 A Characterization of Σ-Σ′-Measurability
5.4 Borel and Lebesgue Measurable Functions
5.5 Properties of Measurable Functions
5.5.1 Compositions of Measurable Functions
5.5.2 Combinations of Measurable Functions
5.5.3 Sequences of Measurable Functions
5.6 Simple Functions
5.7 Luzin’s Theorem
5.8 Notion of Almost Everywhere
5.8.1 Definition and Examples
5.8.2 Equivalence of Functions
5.8.3 A. E. Characterization of Measurability
5.9 Convergence Almost Everywhere
5.9.1 Definition, Examples, and Properties
5.9.2 Uniqueness A. E. of Limit A. E.
5.9.3 Measurability of Limit A. E.
5.9.4 Egorov’s Theorem
5.10 Convergence in Measure
5.10.1 Definition, Examples, and Properties
5.10.2 Uniqueness A. E. of Limit in Measure
5.10.3 Lebesgue and Riesz Theorems
5.11 Probabilistic Terminology
5.12 Problems
6 Abstract Lebesgue Integral
6.1 Definitions and Examples
6.2 Properties of Lebesgue Integral
6.3 Countable Additivity
6.4 Further Properties
6.5 Monotone Convergence Theorem
6.6 Linearity of Lebesgue Integral
6.7 Basic Limit Theorems
6.7.1 Levi’s Theorem
6.7.2 Fatou’s Lemma
6.7.3 Lebesgue’s Dominated Convergence Theorem
6.8 Change of Variable Theorem
6.9 Approximation by Continuous Functions
6.10 Comparison of Riemann and Lebesgue Integrals
6.10.1 Riemann Integral Basics
6.10.2 Characterization of Riemann Integrability
6.10.3 Improper Integrals and Lebesgue Integral
6.11 Problems
7 Lp Spaces
7.1 Hölder’s and Minkowski’s Inequalities
7.1.1 Conjugate Indices
7.1.2 Two Important Inequalities
7.1.3 Essential Supremum
7.1.4 p-Norms
7.1.5 Hölder’s Inequality
7.1.6 Minkowski’s Inequality
7.2 Convergence in p-Norm
7.2.1 Definitions and Examples
7.2.2 Convergence in ∞-Norm
7.2.3 Uniqueness A. E. of Limit in p-Norm
7.2.4 Relationships Between Different Types of Convergence
7.3 Fundamentals of Normed Vector Spaces
7.3.1 Definitions and Examples
7.3.2 Incompleteness of R[a, b]7.4 Lp Spaces
7.4.1 Definition
7.4.2 Important Particular Cases
7.4.3 Hölder’s and Minkowski’s Inequalities in Lp Spaces
7.4.4 Completeness of Lp Spaces
7.4.5 Approximation in Lp Spaces
7.5 Problems
8 Differentiation and Integration
8.1 Derivative Numbers
8.2 Vitali Covers and Vitali Covering Lemma
8.3 Monotone Functions
8.3.1 Definition and Certain Properties
8.3.2 Total Jump and Jump Function
8.3.3 Derivative Numbers of Increasing Functions
8.3.4 Differentiability of Monotone Functions
8.3.5 Total Change Estimate
8.3.6 The Cantor Function
8.4 Functions of Bounded Variation
8.4.1 Definition, Examples, Properties
8.4.2 Additivity of Total Variation, Total Variation Function
8.4.3 Jordan Decomposition Theorem
8.4.4 Derivative of a Function of Bounded Variation
8.5 Absolutely Continuous Functions
8.5.1 Definition, Examples, Properties
8.5.2 Characterization of Absolute Continuity
8.5.3 Derivative of an Absolutely Continuous Function
8.5.4 Singular Functions
8.5.5 Antiderivative and Total Change Formula
8.5.6 Lebesgue Decomposition Theorem
8.6 Problems
9 Signed Measures
9.1 Definition and Examples
9.2 Elementary Properties
9.3 Hahn Decomposition
9.3.1 Positive, Negative, and Null Set
9.3.2 Negative Subset Lemma
9.3.3 Hahn Decomposition Theorem
9.4 Jordan Decomposition Theorem
9.4.1 Mutual Singularity of Measures
9.4.2 Jordan Decomposition Theorem
9.4.3 Total Variation of a Signed Measure
9.5 Radon–Nikodym Theorem
9.5.1 Absolute Continuity of Signed Measure
9.5.2 Radon–Nikodym Theorem
9.5.3 Radon–Nikodym Derivative
9.6 Lebesgue Decomposition Theorem
9.7 Problems
A The Axiom of Choice and Equivalents
A.1 The Axiom of Choice
A.1.1 The Axiom of Choice
A.1.2 Controversy
A.1.3 Timeline
A.2 Ordered Sets
A.3 Equivalents

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