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Real Analysis: Measure and Integration
Real Analysis: Measure and Integration
Marat V. Markin
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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 welldesigned text is ideal for a onesemester 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.
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 welldesigned text is ideal for a onesemester 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.
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2019
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De Gruyter
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english
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358
ISBN 10:
3110600994
ISBN 13:
9783110600995
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PDF, 4.49 MB
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theorem^{892}
function^{684}
measurable^{578}
lebesgue^{501}
functions^{408}
exercise^{374}
download date^{340}
authenticated download^{340}
definition^{255}
continuous^{254}
university of warwick^{252}
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lim^{247}
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Marat V. Markin Real Analysis Also of Interest Elementary Functional Analysis Marat V. Markin, 2018 ISBN 9783110613919, eISBN (PDF) 9783110614039, eISBN (EPUB) 9783110614091 Elementary Operator Theory Marat V. Markin, 2019 ISBN 9783110600964, eISBN (PDF) 9783110600988, eISBN (EPUB) 9783110598889 Functional Analysis. A Terse Introduction Gerardo Chacón, Humberto Rafeiro, Juan Camilo Vallejo, 2016 ISBN 9783110441918, eISBN (PDF) 9783110441925, eISBN (EPUB) 9783110433647 Complex Analysis. A Functional Analytic Approach Friedrich Haslinger, 2017 ISBN 9783110417234, eISBN (PDF) 9783110417241, eISBN (EPUB) 9783110426151 Single Variable Calculus. A First Step Yunzhi Zou, 2018 ISBN 9783110524628, eISBN (PDF) 9783110527780, eISBN (EPUB) 9783110527858 Marat V. Markin Real Analysis  Measure and Integration Mathematics Subject Classification 2010 2801, 28A10, 28A12, 28A15, 28A20, 28A25 Author Prof. Dr. Marat V. Markin California State University, Fresno Department of Mathematics 5245 North Backer Avenue Fresno, CA 93740 USA mmarkin@csufresno.edu ISBN 9783110600971 eISBN (PDF) 9783110600995 eISBN (EPUB) 9783110598827 Library of Congress Control Number: 2019931612 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2019 Walter de Gruyter GmbH, Berlin/Boston Cover image: Merrymoonmary / Getty Images Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com  With utmost appreciation to all my teachers. Preface The author discusses valueless measures in pointless spaces. Paul Halmos The Purpose of the Book and Targeted Audience The book is intended as a text for a onesemester Master’s level graduate course in real analysis with emphasis on the measure and integration theory to be taught wit; hin the existing constraints of the standard for the United States graduate curriculum (fifteen weeks with two seventyfiveminute lectures per week). Real analysis, being, as a rule, a core course in every graduate program in mathematics, is also of significant interest to a wider audience of STEM (science, technology, engineering, and mathematics) graduate students or advanced undergraduates with a solid background in proofbased intermediate analysis. Book’s Philosophy, Scope, and Specifics 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 an appendix taking the reader from the basic set classes, through measures, outer measures and the general procedure of measure extension described in the celebrated Carathéodory’s Extension Theorem and applied to the construct of the Lebesgue–Stieltjes measures as a central particular case. It further treats measurable functions and various types of convergence of sequences of such based on the idea of measure, the treatment including the classical Luzin’s and Egorov’s Theorem as well as the Lebesgue and Riesz Theorems. Then the fundamentals of the abstract Lebesgue integration and the basic limit theorems, such as Fatou’s Lemma and Lebesgue’s Dominated Convergence Theorem, are furnished, the discourse culminating into the comparison of the Lebesgue and Riemann integrals and characterization of the Riemann integrable functions. Chapter 1 outlines certain necessary preliminaries, including the fundamentals of metric spaces. The course is designed to be taught starting with Chapter 2, Chapter 1 being referred to whenever the need arises, for instance when dealing with the concepts of upper and lower limits of numeric or set sequences, using the properties of the inverse image operation, or defining the Borel sets. Chapter 7 is dedicated to studying convergence in pnorm, Lp spaces, and the basics of normed vector spaces. It further demonstrates the deficiencies of the Riemann https://doi.org/10.1515/9783110600995201 VIII  Preface integral relative to its Lebesgue counterpart and prepares the students for more advanced courses in functional analysis and operator theory. Chapter 8 is dedicated to using the novel approach based on the Lebesgue measure and integration theory machinery 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. Chapter 9 on signed measures can be considered as a “bonus” chapter to be taught should the time constraints of a onesemester course permit. The Appendix gives a concise treatise of the Axiom of Choice, its equivalents (the Hausdorff Maximal Principle, Zorn’s Lemma, and Zermello’s WellOrdering Principle), and ordered sets, which is fundamental for proving the famed Vitali Theorem on the existence of a nonLebesgue measurable set in ℝ. 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. These problems are indispensable for understanding the material and moving forward. Many important statements, such as the Approximation of Borel Sets Proposition (Proposition 4.6) (Section 4.8, Problem 13), 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, such as Section 8.6, Problem 11, 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. As amply demonstrated by experience, students tend to better remember statements by their names rather than by numbers. Thus, a distinctive feature of the book is that every theorem, proposition, corollary, and lemma, unless already possessing a name, is endowed with a descriptive one, making it easier to remember, which, in this author’s humble opinion, is quite a bargain when the price for better understanding and retention of the material is a little clumsiness while making a longer reference. Each statement is referred to by its name and not just the number, e. g., the Characterization of Riemann Integrability (Theorem 6.10), as opposed to merely Theorem 6.10. With no pretense on furnishing the history of the subject, the text provides certain dates and lists every related name as a footnote. Acknowledgments  IX Acknowledgments First and foremost, I wish to express my heartfelt gratitude to my mother, Svetlana A. Markina, for believing in me and all the moral support in the course of this undertaking. My utmost appreciation goes to Mr. Edward Sichel, my pupil and graduate advisee, for his invaluable assistance while having tirelessly worked with me on proofreading the manuscript and making a number of insightful suggestions, which have contributed to its improvement. I am very thankful to Dr. Przemyslaw Kajetanowicz (Department of Mathematics, CSU, Fresno) for his kind assistance with graphics. My sincere acknowledgments are also due to the following associates of the Walter de Gruyter GmbH: Dr. Apostolos Damialis, Acquisitions Editor in Mathematics, for seeing value in my manuscript and making authors his highest priority, Ms. Nadja Schedensack, Project Editor in Mathematics and Physics, for her superb efficiency in managing all project related matters, as well as Ms. Ina Talandienė and Ms. Ieva Spudulytė, VTeX Book Production, for their expert editorial and LATEX typesetting contributions. Clovis, California, USA December 2018–January 2019 Marat V. Markin Contents Preface  VII 1 1.1 1.1.1 1.1.2 1.2 1.3 1.4 1.4.1 1.4.2 1.4.3 1.4.4 1.4.5 1.4.6 1.4.7 1.4.8 1.4.9 Preliminaries  1 Set Theoretic Basics  1 Some Terminology and Notations  1 Cardinality and Countability  4 Terminology Related to Functions  6 Upper and Lower Limits  8 Fundamentals of Metric Spaces  9 Definition and Examples  9 Convergence  10 Completeness  12 Balls and Boundedness  13 Interior Points, Open Sets  15 Limit Points, Closed Sets  16 Dense Sets and Separable Spaces  18 Compactness  19 Continuity  22 2 2.1 2.2 2.3 2.4 2.5 2.5.1 2.5.2 2.5.3 2.6 Basic Set Classes  25 Semirings, Semialgebras  25 Rings, Algebras  26 σRings, σAlgebras  29 Monotone Classes  31 Generated Set Classes  32 Intersection Lemma  33 Generated Set Classes  34 Borel Sets  39 Problems  40 3 3.1 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.3 Measures  43 Set Functions  43 Measure  45 Definition and Examples  45 Properties of Measure  46 Continuity of Measure  49 More Examples of Measures  51 Problems  62 XII  Contents 4 4.1 4.2 4.3 4.3.1 4.3.2 4.4 4.4.1 4.4.2 4.5 4.5.1 4.5.2 4.5.3 4.6 4.6.1 4.6.2 4.7 4.7.1 4.7.2 4.7.3 4.7.4 4.8 Extension of Measures  65 Extension of a Set Function  65 Extension From a Semiring  65 Outer Measure  68 Definition and Examples  68 Construction of Outer Measures  69 μ∗ Measurable Sets, Carathéodory’s Theorem  72 μ∗ Measurable Sets  72 Carathéodory’s Theorem  74 Completeness  77 Null Sets, Completeness  77 Addendum to Carathéodory’s Theorem  78 Completion  79 Measure Extension From a Ring  81 Carathéodory’s Extension Theorem  82 Approximation  90 Lebesgue–Stieltjes Measures  92 The Construct  92 Relationships Between Various Extensions of Length  95 Existence of a NonLebesgue Measurable Set in ℝ  96 Multidimensional Lebesgue Measure  98 Problems  100 5 5.1 5.2 5.3 5.4 5.5 5.5.1 5.5.2 5.5.3 5.6 5.7 5.8 5.8.1 5.8.2 5.8.3 5.9 5.9.1 5.9.2 5.9.3 Measurable Functions  105 Measurable Space and Measure Space  105 Definition and Examples  105 A Characterization of ΣΣ Measurability  107 Borel and Lebesgue Measurable Functions  108 Properties of Measurable Functions  110 Compositions of Measurable Functions  110 Combinations of Measurable Functions  111 Sequences of Measurable Functions  112 Simple Functions  114 Luzin’s Theorem  120 Notion of Almost Everywhere  124 Definition and Examples  124 Equivalence of Functions  125 A. E. Characterization of Measurability  126 Convergence Almost Everywhere  128 Definition, Examples, and Properties  128 Uniqueness A. E. of Limit A. E.  132 Measurability of Limit A. E.  133 Contents  XIII 5.9.4 5.10 5.10.1 5.10.2 5.10.3 5.11 5.12 Egorov’s Theorem  134 Convergence in Measure  137 Definition, Examples, and Properties  137 Uniqueness A. E. of Limit in Measure  140 Lebesgue and Riesz Theorems  141 Probabilistic Terminology  145 Problems  145 6 Abstract Lebesgue Integral  151 6.1 Definitions and Examples  151 6.2 Properties of Lebesgue Integral  160 6.3 Countable Additivity  165 6.4 Further Properties  167 6.5 Monotone Convergence Theorem  173 6.6 Linearity of Lebesgue Integral  177 6.7 Basic Limit Theorems  180 6.7.1 Levi’s Theorem  180 6.7.2 Fatou’s Lemma  182 6.7.3 Lebesgue’s Dominated Convergence Theorem  184 6.8 Change of Variable Theorem  186 6.9 Approximation by Continuous Functions  188 6.10 Comparison of Riemann and Lebesgue Integrals  192 6.10.1 Riemann Integral Basics  192 6.10.2 Characterization of Riemann Integrability  194 6.10.3 Improper Integrals and Lebesgue Integral  200 6.11 Problems  204 7 7.1 7.1.1 7.1.2 7.1.3 7.1.4 7.1.5 7.1.6 7.2 7.2.1 7.2.2 7.2.3 7.2.4 7.3 7.3.1 Lp Spaces  211 Hölder’s and Minkowski’s Inequalities  211 Conjugate Indices  211 Two Important Inequalities  212 Essential Supremum  213 pNorms  215 Hölder’s Inequality  217 Minkowski’s Inequality  219 Convergence in pNorm  221 Definitions and Examples  221 Convergence in ∞Norm  224 Uniqueness A. E. of Limit in pNorm  225 Relationships Between Different Types of Convergence  225 Fundamentals of Normed Vector Spaces  226 Definitions and Examples  226 XIV  Contents 7.3.2 7.4 7.4.1 7.4.2 7.4.3 7.4.4 7.4.5 7.5 Incompleteness of R[a, b]  230 Lp Spaces  233 Definition  233 Important Particular Cases  235 Hölder’s and Minkowski’s Inequalities in Lp Spaces  236 Completeness of Lp Spaces  236 Approximation in Lp Spaces  241 Problems  245 8 8.1 8.2 8.3 8.3.1 8.3.2 8.3.3 8.3.4 8.3.5 8.3.6 8.4 8.4.1 8.4.2 8.4.3 8.4.4 8.5 8.5.1 8.5.2 8.5.3 8.5.4 8.5.5 8.5.6 8.6 Differentiation and Integration  249 Derivative Numbers  249 Vitali Covers and Vitali Covering Lemma  251 Monotone Functions  257 Definition and Certain Properties  257 Total Jump and Jump Function  258 Derivative Numbers of Increasing Functions  261 Differentiability of Monotone Functions  269 Total Change Estimate  270 The Cantor Function  272 Functions of Bounded Variation  274 Definition, Examples, Properties  274 Additivity of Total Variation, Total Variation Function  278 Jordan Decomposition Theorem  279 Derivative of a Function of Bounded Variation  280 Absolutely Continuous Functions  281 Definition, Examples, Properties  281 Characterization of Absolute Continuity  283 Derivative of an Absolutely Continuous Function  286 Singular Functions  287 Antiderivative and Total Change Formula  289 Lebesgue Decomposition Theorem  292 Problems  294 9 9.1 9.2 9.3 9.3.1 9.3.2 9.3.3 9.4 9.4.1 Signed Measures  297 Definition and Examples  297 Elementary Properties  299 Hahn Decomposition  300 Positive, Negative, and Null Set  300 Negative Subset Lemma  301 Hahn Decomposition Theorem  303 Jordan Decomposition Theorem  306 Mutual Singularity of Measures  306 Contents  XV 9.4.2 9.4.3 9.5 9.5.1 9.5.2 9.5.3 9.6 9.7 A A.1 A.1.1 A.1.2 A.1.3 A.2 A.3 Jordan Decomposition Theorem  307 Total Variation of a Signed Measure  309 Radon–Nikodym Theorem  309 Absolute Continuity of Signed Measure  309 Radon–Nikodym Theorem  312 Radon–Nikodym Derivative  317 Lebesgue Decomposition Theorem  317 Problems  318 The Axiom of Choice and Equivalents  321 The Axiom of Choice  321 The Axiom of Choice  321 Controversy  321 Timeline  322 Ordered Sets  322 Equivalents  326 Bibliography  331 Index  333 1 Preliminaries In this chapter, we outline certain terminology, notations, and preliminary facts essential for our subsequent discourse. 1.1 Set Theoretic Basics 1.1.1 Some Terminology and Notations – – – – – – – – – The logic quantifiers ∀, ∃, and ∃! stand for “for all,”“there exist(s),” and “there exists a unique,” respectively. ℕ := {1, 2, 3, . . . } is the set of natural numbers. ℤ := {0, ±1, ±2, . . . } is the set of integers. ℚ is the set of rational numbers. ℝ is the set of real numbers. ℂ is the set of complex numbers. ℤ+ , ℚ+ , and ℝ+ are the sets of nonnegative integers, rationals, and reals, respectively. ℝ := [−∞, ∞] is the set of extended real numbers (extended real line). For n ∈ ℕ, ℝn and ℂn are the nspaces of all ordered ntuples of real and complex numbers, respectively. Let X be a set. Henceforth, all sets are supposed to be subsets of X. – P (X) is the power set of X, i. e., the collection of all subsets of X. – 2X is the set of all binary functions f : X → {0, 1} provided X ≠ 0. – Sets A, B ⊆ X with A ∩ B = 0 are called disjoint. – Let I be a nonempty indexing set. The sets of a collection {Ai }i∈I of subsets of X are said to be pairwise disjoint if Ai ∩ Aj = 0, i, j ∈ I, i ≠ j. – – – For A, B ⊆ X, A \ B := {x ∈ X  x ∈ A, but x ∉ B} is the difference of A and B, in particular, Ac := X \ A = {x ∈ X  x ∉ A} is the complement of A and A \ B = A ∩ Bc . For A, B ⊆ X, A △ B := (A \ B) ∪ (B \ A) is the symmetric difference of A and B. Let I be a nonempty indexing set and {Ai }i∈I be a collection of subsets of X. De Morgan’s laws state c c (⋃ Ai ) = ⋂ Aci and (⋂ Ai ) = ⋃ Aci . i∈I i∈I i∈I i∈I More generally, B \ ⋃ Ai = ⋂ B \ Ai and B \ ⋂ Ai = ⋃ B \ Ai . i∈I i∈I i∈I i∈I https://doi.org/10.1515/9783110600995001 Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 2  1 Preliminaries – The Cartesian product of sets Ai ⊆ X, i = 1, . . . , n (n ∈ ℕ), A1 × ⋅ ⋅ ⋅ × An := {(x1 , . . . , xn )  xi ∈ Ai , i = 1, . . . , n} . Definition 1.1 (Monotone Set Sequences). A sequence (An )n∈ℕ (another notation: {An }∞ n=1 ) of subsets of X is said to increase (to be increasing) if An ⊆ An+1 , n ∈ ℕ. We also say that (An )n∈ℕ increases to ⋃∞ n=1 An and write ∞ An ↑ ⋃ Ai . i=1 A sequence (An )n∈ℕ of subsets of X is said to decrease (to be decreasing) if An ⊇ An+1 , n ∈ ℕ. We also say that (An )n∈ℕ decreases to ⋂∞ n=1 An and write ∞ An ↓ ⋂ Ai . i=1 An increasing or decreasing sequence (An )n∈ℕ of subsets of X is called monotone. Exercise 1.1. Give an example of: (a) an increasing set sequence, (b) a decreasing set sequence, (c) a set sequence that is not monotone. Definition 1.2 (Upper and Lower Limits of a Set Sequence). For an arbitrary sequence (An )n∈ℕ of subsets of X, – upper limit or limit superior of (An )n∈ℕ is ∞ ∞ lim An := ⋂ ⋃ Ak , n→∞ – n=1 k=n another notation is lim supn→∞ An ; lower limit or limit inferior of (An )n∈ℕ is ∞ ∞ lim An := ⋃ ⋂ Ak , n→∞ n=1 k=n another notation is lim infn→∞ An . Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 1.1 Set Theoretic Basics  3 Example 1.1. For X = ℝ and [0, n], n ∈ ℕ is odd, An := { [−n, n] n ∈ ℕ is even, lim An = ℝ and lim An = [0, ∞) n→∞ n→∞ Exercise 1.2. Verify. Remarks 1.1. – Upper and lower limits are welldefined and exist for any set sequence and, by the definition, ∞ ∞ Un := ⋃ Ak ↓ lim An , Vn := ⋂ Ak ↑ lim An . n→∞ k=n – k=n n→∞ As follows from the definition, lim An = {x ∈ X  x ∈ An frequently, i. e., for infinitely many n ∈ ℕ} n→∞ and lim An = {x ∈ X  x ∈ An eventually, i. e., for n ≥ N(x) with some N(x) ∈ ℕ} . n→∞ Exercise 1.3. Explain. This immediately implies the inclusion lim An ⊆ lim An ; n→∞ n→∞ Definition 1.3 (Limit of a Set Sequence). If, for a sequence (An )n∈ℕ of subsets of X, lim An = lim An , n→∞ n→∞ we use the notation limn→∞ An for the common value of the lower and upper limits and call this set the limit of (An )n∈ℕ . Remarks 1.2. – As Example 1.1 shows, the limit of a set sequence need not exist. – For an increasing set sequence (An )n∈ℕ , ∞ lim An = ⋃ An n→∞ n=1 and, for a decreasing set sequence (An )n∈ℕ , ∞ lim An = ⋂ An . n→∞ n=1 Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 4  1 Preliminaries Exercise 1.4. Verify. Example 1.2. ∞ ∞ lim [0, n) = ⋃ [0, n) = [0, ∞) and lim [n, ∞) = ⋂ [n, ∞) = 0. n→∞ n→∞ n=1 n=1 Exercise 1.5. Give an example of a nonmonotone set sequence (An )n∈ℕ for which limn→∞ An exists. 1.1.2 Cardinality and Countability Definition 1.4 (Similarity of Sets). Sets A and B are said to be similar if there exists a onetoone correspondence (bijection) between them. Notation. A ∼ B. Remark 1.3. Similarity is an equivalence relation (reflexive, symmetric, and transitive) on the power set P (X) of a nonempty set X. Exercise 1.6. Verify. Thus, in the context, we can use the term “equivalence” synonymously to “similarity.” Definition 1.5 (Cardinality). Equivalent sets are said to have the same number of elements or cardinality. Cardinality is a characteristic of an equivalence class of similar sets. Notation. P (X) ∋ A → A. Remark 1.4. Thus, A ∼ B iff A = B, i. e., two sets are equivalent iff they share the same cardinality. Examples 1.3. 1. For a nonempty set X, P (X) ∼ 2X . 2. ℕ = ℤ = ℚ := ℵ0 . 3. [0, 1] = ℝ = ℂ := c. See, e. g., [6, 8]. Definition 1.6 (Domination). If sets A and B are such that A is equivalent to a subset of B, we write A⪯B and say that B dominates A. If, in addition, A ≁ B, we write A≺B and say that B strictly dominates A. Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 1.1 Set Theoretic Basics  5 Remark 1.5. The relation ⪯ is a partial order (reflexive, antisymmetric, and transitive) on the power set P (X) of a nonempty set X (see Appendix A). Exercise 1.7. Verify reflexivity and transitivity. The antisymmetry of ⪯ is the subject of the following celebrated theorem. Theorem 1.1 (Schröder–Bernstein Theorem). If, for sets A and B, A ⪯ B and B ⪯ A, then A ∼ B.1 For a proof, see, e. g., [6]. Remark 1.6. The set partial order ⪯ defines a partial order ≤ on the set of cardinals: A ≤ B ⇔ A ⪯ B. Thus, the Schröder–Bernstein Theorem can be equivalently reformulated in terms of cardinalities as follows: If, for sets A and B, A ≤ B and B ≤ A, then A = B. Theorem 1.2 (Cantor’s Theorem). Every set X is strictly dominated by its power set P (X):2 X ≺ P (X). Equivalently, X < P (X). For a proof, see, e. g., [6]. In view of Examples 1.3, we obtain the following. Corollary 1.1. For a nonempty set X, X ≺ 2X , i. e., X < 2X . Definition 1.7 (Countable/Uncountable Set). A countable set is a set with the same cardinality as a subset of the set ℕ of natural numbers, i. e., equivalent to a subset of ℕ. A set, which is not countable, is called uncountable. Remarks 1.7. – A countable set A is either finite, i. e., equivalent to a set of the form {1, . . . , n} ⊂ ℕ with some n ∈ ℕ, in which case, we say that A has n elements, or countably infinite, i. e., equivalent to the entire ℕ. 1 Ernst Schröder (1841–1902), Felix Bernstein (1878–1956). 2 Georg Cantor (1845–1918). Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 6  1 Preliminaries – For a finite set A of n elements (n ∈ ℕ), A = {1, . . . , n} = n. For a countably infinite set A, A = ℕ = ℵ0 – (see Examples 1.3). In some sources, the term “countable” is used in the sense of “countably infinite.” To avoid ambiguity, the term “at most countable” can be used when finite sets are included in consideration. The subsequent statement immediately follows from Cantor’s Theorem (Theorem 1.2). Proposition 1.1 (Uncountable Sets). The sets P (ℕ) and 2ℕ (the set of all binary sequences) are uncountable. Theorem 1.3 (Properties of Countable Sets). (1) Every infinite set contains a countably infinite subset (based on the Axiom of Choice (see Appendix A)). (2) Any subset of a countable set is countable. (3) The union of countably many countable sets is countable. (4) The Cartesian product of finitely many countable sets is countable. Exercise 1.8. Prove that (a) the set ℤ of all integers and the set of all rational numbers are countable; (b) for any n ∈ ℕ, ℤn and ℚn are countable; (c) the set of all algebraic numbers (the roots of polynomials with integer coefficients) is countable. Subsequently, we also need the following useful result. Proposition 1.2 (Cardinality of the Collection of Finite Subsets). The cardinality of the collection of all finite subsets of an infinite set coincides with the cardinality of the set. For a proof, see, e. g., [8, 11, 18]. 1.2 Terminology Related to Functions Let X and Y be nonempty sets and 0 ≠ D ⊆ X, f : D → Y. Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 1.2 Terminology Related to Functions  7 – – – The set D is called the domain (of definition) of f . The value of f corresponding to an x ∈ D is designated by f (x). The set {f (x)  x ∈ D} – of all values of f is called the range of f (also codomain or target set). For a set A ⊆ D, the set of values of f corresponding to all elements of A f (A) := {f (x)  x ∈ A} – is called the image of A under the function f . Thus, the range of f is the image f (D) of the whole domain D. For a set B ⊆ Y, the set of all elements of the domain that map to the elements of B, f −1 (B) := {x ∈ D  f (x) ∈ B} is called the inverse image (or preimage) of B. Example 1.4. For X = Y := ℝ and f (x) := x2 with D := [−1, 2], – f ([−1, 2]) = [0, 4] and f ([1, 2]) = [1, 4]. – f −1 ([−2, −1]) = 0, f −1 ([0, 1]) = [−1, 1], and f −1 ([1, 4]) = {−1} ∪ [1, 2]. Theorem 1.4 (Properties of Inverse Image). Let X and Y be nonempty sets and 0 ≠ D ⊆ X f : D → Y. Then, for an arbitrary nonempty collection {Bi }i∈I of subsets of Y, (1) f −1 (⋃i∈I Bi ) = ⋃i∈I f −1 (Bi ), (2) f −1 (⋂i∈I Bi ) = ⋂i∈I f −1 (Bi ), and (3) for any B1 , B2 ⊆ Y, f −1 (B1 \ B2 ) = f −1 (B1 ) \ f −1 (B2 ), i. e., preimage preserves all set operations. Exercise 1.9. (a) Prove. (b) Show that image preserves unions, i. e., for an arbitrary nonempty collection {Ai }i∈I of subsets of D, f (⋃ Ai ) = ⋃ f (Ai ), i∈I i∈I and unions only. Give corresponding counterexamples for intersections and differences. Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 8  1 Preliminaries 1.3 Upper and Lower Limits Definition 1.8 (Upper and Lower Limits). Let (xn )n∈ℕ (another notation is {xn }∞ n=1 ) be a sequence of real numbers. The upper limit or limit superior of (xn )n∈ℕ is defined as follows: lim x n→∞ n := lim sup xk = inf sup xk ∈ ℝ. n→∞ k≥n n∈ℕ k≥n The lower limit or limit inferior of (xn )n∈ℕ is defined as follows: lim xn := lim inf xk = sup inf xk ∈ ℝ. n→∞ n→∞ k≥n n∈ℕ k≥n Alternative notations are lim supn→∞ xn and lim infn→∞ xn , respectively. Example 1.5. For n, n ∈ ℕ is odd, xn := { −1/n, n ∈ ℕ is even, lim x n→∞ n = ∞ and lim xn = 0. n→∞ Exercise 1.10. (a) Verify. (b) Explain why the upper and lower limits, unlike the regular limit, are guaranteed to exist for an arbitrary sequence of real numbers. (c) Show that lim xn ≤ lim xn . n→∞ n→∞ Proposition 1.3 (Characterization of Limit Existence). For a sequence of real numbers (xn )n∈ℕ , lim x n→∞ n ∈ℝ exists iff lim xn = lim xn , n→∞ n→∞ in which case lim x n→∞ n = lim xn = lim xn . n→∞ n→∞ Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 1.4 Fundamentals of Metric Spaces  9 1.4 Fundamentals of Metric Spaces In this section, we furnish a brief discourse of metric spaces, i. e., sets endowed with a notion of distance, whose properties mimic those of the regular distance in threedimensional space. Distance brings to life various topological notions such as limit, continuity, openness, closedness, compactness, and denseness, the geometric notion of boundedness, as well as the notions of fundamentality of sequences and completeness. We are to touch upon these concepts here. 1.4.1 Definition and Examples Definition 1.9 (Metric Space). A metric space is a nonempty set X with a metric (or distance function), i. e., a mapping ρ(⋅, ⋅) : X × X → ℝ subject to the following metric axioms: 1. ρ(x, y) ≥ 0, x, y ∈ X. 2. ρ(x, y) = 0 iff x = y. 3. ρ(x, y) = ρ(y, x), x, y ∈ X. 4. ρ(x, z) ≤ ρ(x, y) + ρ(y, z), x, y, z ∈ X. Nonnegativity Separation Symmetry Triangle inequality For any fixed x, y ∈ X, the number ρ(x, y) is called the distance of x from y, or from y to x, or between x and y. Notation. (X, ρ). Examples 1.6. 1. Any nonempty set X is a metric space relative to the discrete metric 0 if x = y, X ∋ x, y → ρd (x, y) := { 1 if x ≠ y. 2. 3. The real line ℝ or the complex plane ℂ is a metric space relative to the regular distance function ρ(x, y) := x − y. Let n ∈ ℕ and 1 ≤ p ≤ ∞. The real/complex nspace, ℝn or ℂn , is a metric space relative to pmetric, 1/p [∑n xk − yk p ] ρp (x, y) = { k=1 max1≤k≤n xk − yk  if 1 ≤ p < ∞, if p = ∞, where x := (x1 , . . . , xn ) and y := (y1 , . . . , yn ), designated by lp(n) (real or complex, respectively). Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 10  1 Preliminaries Remarks 1.8. – For n = 1, all these metrics coincide with ρ(x, y) = x − y. – For n = 2, 3 and p = 2, we have the usual Euclidean distance. – (ℂ, ρ) = (ℝ2 , ρ2 ). 4. Let 1 ≤ p ≤ ∞. The set lp of all real or complex sequences (xk )k∈ℕ satisfying ∞ ∑ xk p < ∞ (1 ≤ p < ∞), sup xk  < ∞ (p = ∞) k=1 k∈ℕ (psummable/bounded sequences, respectively) is a metric space relative to pmetric 1/p [∑∞ xk − yk p ] ρp (x, y) = { k=1 supk∈ℕ xk − yk  5. if 1 ≤ p < ∞, if p = ∞, where x := (xk )k∈ℕ , y := (yk )k∈ℕ ∈ lp . The set C[a, b] of all real/complexvalued functions continuous on [a, b] (−∞ < a < b < ∞) is a metric space relative to pmetric 1/p {[∫b f (x) − g(x)p dx] C[a, b] ∋ f , g → ρp (f , g) = { a {maxa≤x≤b f (x) − g(x) if 1 ≤ p < ∞, if p = ∞. Exercise 1.11. Verify examples 3–5 for p = 1 and p = ∞ only. 1.4.2 Convergence Definition 1.10 (Convergence and Limit of a Sequence). A sequence of points (xn )n∈ℕ (another notation is {xn }∞ n=1 ) in a metric space (X, ρ) is said to converge (to be convergent) to a point x ∈ X if ∀ ε > 0 ∃ N ∈ ℕ ∀ n ≥ N : ρ(xn , x) < ε, i. e., lim ρ(xn , x) = 0 (ρ(xn , x) → 0, n → ∞). n→∞ We write in this case lim x n→∞ n =x or xn → x, n → ∞ and say that x is the limit of (xn )n∈ℕ . A sequence (xn )n∈ℕ in a metric space (X, ρ) is called convergent if it converges to some x ∈ X and divergent otherwise. Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 1.4 Fundamentals of Metric Spaces  11 Theorem 1.5 (Uniqueness of Limit). The limit of a convergent sequence (xn )n∈ℕ in a metric space (X, ρ) is unique. Exercise 1.12. Prove. Hint. Use the triangle inequality. Theorem 1.6 (Characterization of Convergence). A sequence (xn )n∈ℕ in a metric space (X, ρ) converges to a point x ∈ X iff every subsequence (xn(k) )k∈ℕ of (xn )n∈ℕ contains a subsequence (xn(k(j)) )j∈ℕ such that xn(k(j)) → x, j → ∞. Exercise 1.13. Prove. Hint. Prove the “if” part by contrapositive. Examples 1.7. 1. A sequence (xn )n∈ℕ is convergent in a discrete space (X, ρd ) iff it is eventually constant, i. e., ∃ N ∈ ℕ ∀ n ≥ N : xn = xN . 2. Convergence of a sequence in the space lp(n) (n ∈ ℕ and 1 ≤ p ≤ ∞) is equivalent to componentwise convergence, i. e., (x1(k) , . . . , xn(k) ) → (x1 , . . . , xn ), k → ∞ ⇔ ∀ i = 1, . . . , n : xi(k) → xi , k → ∞. 3. Convergence of a sequence in the space lp (1 ≤ p ≤ ∞) (xn(k) )n∈ℕ =: x(k) → x := (xn )n∈ℕ , k → ∞, implies termwise convergence, i. e., ∀ n ∈ ℕ : xn(k) → xn , k → ∞, the converse not being true. Indeed, in lp (1 ≤ p ≤ ∞), the sequence (ek := (δnk )n∈ℕ )k∈ℕ , where δnk is the Kronecker3 delta, converges to the zero sequence 0 := (0, 0, 0, . . . ) componentwise, but does not converge. 4. Convergence in (C[a, b], ρ∞ ) (see Examples 1.6) is uniform convergence on [a, b], i. e., fn → f , n → ∞, in (C[a, b], ρ∞ ) iff ∀ ε > 0 ∃ N ∈ ℕ ∀n ≥ N ∀ x ∈ [a, b] : fn (x) − f (x) < ε. 3 Leopold Kronecker (1823–1891). Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 12  1 Preliminaries Uniform convergence on [a, b] implies pointwise convergence on [a, b], i. e., ∀ ε > 0 ∀ x ∈ [a, b] ∃ N ∈ ℕ ∀n ≥ N : fn (x) − f (x) < ε. Exercise 1.14. (a) Verify. (b) Give an example showing that a function sequence (fn )n∈ℕ in C[a, b] pointwise convergent on [a, b] need not converge on [a, b] uniformly. 1.4.3 Completeness The concept of completeness is a basic property of metric spaces underlying many important facts (see, e. g., [14]). 1.4.3.1 Cauchy/Fundamental Sequences Definition 1.11 (Cauchy/Fundamental Sequence). A sequence (xn )n∈ℕ in a metric space (X, ρ) is called a Cauchy sequence, or a fundamental sequence, if4 ∀ ε > 0 ∃ N ∈ ℕ ∀ m, n ≥ N : ρ(xm , xn ) < ε. Remark 1.9. The latter is equivalent to ρ(xm , xn ) → 0, m, n → ∞, or to sup ρ(xn+k , xn ) → 0, n → ∞. k∈ℕ Examples 1.8. 1. A sequence is fundamental in a discrete space (X, ρd ) iff it is eventually constant. 2. The sequence (1/n)n∈ℕ is fundamental in ℝ and the sequence (n)n∈ℕ is not. 3. The sequence (xn := (1, 1/2, . . . , 1/n, 0, 0, . . . ))n∈ℕ is fundamental in lp (1 < p ≤ ∞) but not in l1 . 4. The sequence (en := (δnk )k∈ℕ )n∈ℕ , where δnk is the Kronecker delta, is not fundamental in lp (1 ≤ p ≤ ∞). Exercise 1.15. Verify. Remark 1.10. Every convergent sequence (xn )n∈ℕ is fundamental but not vice versa, i. e., a fundamental sequence need not converge. For instance, the sequence (1/n)n∈ℕ is fundamental, but divergent in ℝ \ {0} with the regular distance. Exercise 1.16. Verify. 4 AugustinLouis Cauchy (1789–1857). Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 1.4 Fundamentals of Metric Spaces  13 1.4.3.2 Complete Metric Spaces Definition 1.12 (Complete Metric Space). A metric space (X, ρ), in which every Cauchy/fundamental sequence converges, is called complete and incomplete otherwise. Examples 1.9. 1. The spaces ℝ and ℂ are complete relative to the regular distance as is known from intermediate analysis courses. 2. The spaces ℝ \ {0}, (0, 1), and ℚ are incomplete relative to the regular distance. 3. A discrete metric space (X, ρd ) is complete. Exercise 1.17. Verify 2 and 3. 4. Theorem 1.7 (Completeness of (C[a, b], ρ∞ )). The (real or complex) space (C[a, b], ρ∞ ) (−∞ < a < b < ∞) is complete. See, e. g., [14]. 1.4.4 Balls and Boundedness Definition 1.13 (Balls and Spheres). Let (X, ρ) be a metric space and r ≥ 0. – The open ball of radius r centered at a point x0 ∈ X is the set B(x0 , r) := {x ∈ X  ρ(x, x0 ) < r} . – The closed ball of radius r centered at a point x0 ∈ X is the set B(x0 , r) := {x ∈ X  ρ(x, x0 ) ≤ r} . – The sphere of radius r centered at a point x0 ∈ X is the set S(x0 , r) := {x ∈ X  ρ(x, x0 ) = r} = B(x0 , r) \ B(x0 , r). Remarks 1.11. – When contextually important to indicate which space the balls/spheres are considered in, the letter designating the space in question is added as a subscript. For example, for (X, ρ), we use the notation BX (x0 , r), BX (x0 , r), and SX (x0 , r), x0 ∈ X, r ≥ 0, – respectively. As is easily seen, for an arbitrary x0 ∈ X, B(x0 , 0) = 0 and B(x0 , 0) = S(x0 , 0) = {x0 } (trivial cases). Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 14  1 Preliminaries Exercise 1.18. (a) Explain the latter. (b) Describe balls and spheres in ℝ and ℂ with the regular distance, give some examples. (c) Sketch the unit sphere S(0, 1) in (ℝ2 , ρ1 ), (ℝ2 , ρ2 ), and (ℝ2 , ρ∞ ). (d) Describe balls and spheres in (C[a, b], ρ∞ ). (e) Let (X, ρd ) be a discrete metric space and x0 ∈ X be arbitrary. Describe B(x0 , r), B(x0 , r), and S(x0 , r) for different values of r ≥ 0. Convergence can be equivalently redefined using ball terminology. Definition 1.14 (Equivalent Definition of Convergence). A sequence of points (xn )n∈ℕ in a metric space (X, ρ) is said to converge (to be convergent) to a point x ∈ X if ∀ ε > 0 ∃ N ∈ ℕ ∀ n ≥ N : xn ∈ B(x0 , ε), in which case we say that the sequence (xn )n∈ℕ is eventually in the εball B(x, ε). Definition 1.15 (Bounded Set). Let (X, ρ) be a metric space. A nonempty set A ⊆ X is called bounded if diam(A) := sup ρ(x, y) < ∞. x,y∈A The number diam(A) is called the diameter of A. The empty set 0 is regarded to be bounded with diam(0) := 0. Examples 1.10. 1. In a metric space (X, ρ), an open/closed ball of radius r > 0 is a bounded set of diameter at most 2r. 2. In (ℝ, ρ), the sets (0, 1], {1/n}n∈ℕ are bounded and the sets (−∞, 1), {n2 }n∈ℕ are not. 3. In l∞ , the set {(xn )n∈ℕ  xn  ≤ 1, n ∈ ℕ} is bounded and, in lp (1 ≤ p < ∞), it is not. 4. In (C[0, 1], ρ∞ ), the power set {xn }n∈ℤ is bounded and, in (C[0, 2], ρ∞ ), it is not. + Exercise 1.19. (a) Verify. (b) Show that a set A is bounded iff it is contained in some (closed) ball, i. e., ∃ x ∈ X ∃ r ≥ 0 : A ⊆ B(x, r). (c) Describe all bounded sets in a discrete metric space (X, ρd ). (d) Give an example of a metric space (X, ρ), in which, for a ball B(x, r) with some x ∈ X and r > 0, diam(B(x, r)) < 2r. Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 1.4 Fundamentals of Metric Spaces  15 Definition 1.16 (Bounded Function). Let T be a nonempty set and (X, ρ) be a metric space. A function f : T → X is called bounded if the set of its values f (T) is bounded in (X, ρ). Remark 1.12. As a particular case, for T = ℕ, we obtain the definition of a bounded sequence. Theorem 1.8 (Boundedness of Fundamental Sequences). Each fundamental sequence (xn )n∈ℕ in a metric space (X, ρ) is bounded. Exercise 1.20. (a) Prove. (b) Give an example showing that a bounded sequence (xn )n∈ℕ in a metric space (X, ρ) need not be fundamental. Corollary 1.2 (Boundedness of Convergent Sequences). Each convergent sequence (xn )n∈ℕ in a metric space (X, ρ) is bounded. 1.4.5 Interior Points, Open Sets Definition 1.17 (Interior Point). Let (X, ρ) be a metric space. A point x ∈ X is called an interior point of a nonempty set A ⊆ X if A contains a nontrivial open ball centered at x, i. e., ∃ r > 0 : B(x, r) ⊆ A. Examples 1.11. 1. In an arbitrary metric space (X, ρ), any point x ∈ X is, obviously, an interior point of an open ball B(x, r) or a closed ball B(x, r) with an arbitrary r > 0. 2. For the set [0, 1) in ℝ with the regular distance, the points 0 < x < 1 are interior and the point x = 0 is not. 3. A singleton {x} in ℝ with the regular distance, has no interior points. Exercise 1.21. Verify. Definition 1.18 (Interior of a Set). The interior of a nonempty set A in a metric space (X, ρ) is the set of all interior points of A. Notation. int(A). Remark 1.13. Thus, int(A) ⊆ A. As the prior examples demonstrate, this inclusion can be proper and int(A) can be empty. Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 16  1 Preliminaries Definition 1.19 (Open Set). A nonempty set A in a metric space (X, ρ) is called open if each point of A is its interior point, i. e., A = int(A). Remark 1.14. The empty set 0 is regarded to be open and the whole space X is trivially open as well. Exercise 1.22. (a) Verify that, in ℝ with the regular distance, the intervals of the form (a, ∞), (−∞, b), and (a, b) (−∞ < a < b < ∞) are open sets. (b) Prove that, in a metric space (X, ρ), an open ball B(x0 , r) (x0 ∈ X, r ≥ 0) is an open set. (c) Describe all open sets in a discrete metric space (X, ρd ). Theorem 1.9 (Properties of Open Sets). The open sets in a metric space (X, ρ) have the following properties: (1) 0 and X are open sets; (2) an arbitrary union of open sets is an open set; (3) an arbitrary finite intersection of open sets is an open set. Exercise 1.23. (a) Prove. (b) Give an example showing that an infinite intersection of open sets need not be open. Definition 1.20 (Metric Topology). The collection G of all open sets in a metric space (X, ρ) is called the metric topology on X generated by metric ρ. See, e. g., [18, 21, 25]. 1.4.6 Limit Points, Closed Sets Definition 1.21 (Limit Point, Derived Set). Let (X, ρ) be a metric space. A point x ∈ X is called a limit point (also an accumulation point or a cluster point) of a set A in X if every open ball centered at x contains a point of A distinct from x, i. e., ∀ r > 0 : B(x, r) ∩ (A \ {x}) ≠ 0. The set A of all limit points of A is called the derived set of A. Example 1.12. In ℝ, [0, 1) = [0, 1], ℤ = 0, ℚ = ℝ. Exercise 1.24. Verify. Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 1.4 Fundamentals of Metric Spaces  17 Remarks 1.15. – A limit point x of a set A need not belong to A. It may even happen that none of them does, i. e., A ⊆ Ac . – – Each open ball centered at a limit point x of a set A in a metric space (X, ρ) contains infinitely many points of A distinct from x. To have a limit point, a set A in a metric space (X, ρ) must necessarily be nonempty and even infinite. However, an infinite set need not have limit points. Exercise 1.25. (a) Verify and give corresponding examples. (b) Describe the situation in a discrete metric space (X, ρd ). (c) Give examples showing that an interior point of a set need not be its limit point and vise versa. Definition 1.22 (Closure of a Set). The closure A of a set A in a metric space (X, ρ) is the set consisting of all points, which are either points of A or limit points of A, i. e., A := A ∪ A . Example 1.13. In ℝ, [0, 1) = [0, 1], ℤ = ℤ, ℚ = ℝ. Exercise 1.26. Verify (see Example 1.12). Remarks 1.16. – Obviously 0 = 0, and hence, 0 = 0. – We always have the inclusion A ⊆ A, – which may be proper. A point x ∈ A iff every nontrivial open ball centered at x contains a point of A (not necessarily distinct from x). Exercise 1.27. Verify and give a corresponding example. Definition 1.23 (Closed Set). Let (X, ρ) be a metric space. A set A in X is called closed if it contains all its limit points, i. e., A ⊆ A, and hence, A = A. Remarks 1.17. – The whole space X is trivially closed. – Also closed are the sets with no limit points, in particular, finite sets, including the empty set 0. Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 18  1 Preliminaries – A set in a metric space (X, ρ), which is simultaneously closed and open is called clopen. There are always at least two (trivial) clopen sets: 0 and X. However, there can exist nontrivial ones. Exercise 1.28. (a) Verify that in ℝ with the regular distance the intervals of the form [a, ∞), (−∞, b], and [a, b] (−∞ < a < b < ∞) are closed sets. (b) Verify that the sets (0, 1) and {2} are clopen in the metric space (0, 1) ∪ {2} with the regular distance. (c) Describe all closed sets in a discrete metric space (X, ρd ). Theorem 1.10 (Characterizations of Closed Sets). For an arbitrary nonempty set A ⊆ X in a metric space (X, ρ), the following statements are equivalent: 1. The set A is closed in (X, ρ). 2. The complement Ac of the set A is an open set in (X, ρ). 3. (Sequential Characterization) For any sequence (xn )n∈ℕ in A convergent in (X, ρ), limn→∞ xn ∈ A, i. e., the set A contains the limits of all its convergent sequences. Theorem 1.11 (Properties of Closed Sets). The closed sets in a metric space (X, ρ) have the following properties: (1) 0 and X are closed sets; (2) an arbitrary intersection of closed sets is a closed set; (3) a finite union of closed sets is a closed set. Exercise 1.29. (a) Prove. (b) Give an example showing that an infinite union of closed sets need not be closed. 1.4.7 Dense Sets and Separable Spaces Definition 1.24 (Dense Set). A set A in a metric space (X, ρ) is called dense if A = X. Example 1.14. The set ℚ of the rational numbers is dense in ℝ (see Example 1.13). Theorem 1.12 (Sequential Characterization of Dense Sets). A set A is dense in a metric space (X, ρ) iff ∀x ∈ X ∃ (xn )n∈ℕ ⊆ A : xn → x, n → ∞. Definition 1.25 (Separable Metric Space). A metric space (X, ρ) containing a countable dense subset is a called separable. Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 1.4 Fundamentals of Metric Spaces  19 Remark 1.18. Any countable metric space is, obviously, separable. However, as the following examples show, a metric space need not be countable to be separable. Examples 1.15. 1. The spaces lp(n) are separable for (n ∈ ℕ, 1 ≤ p ≤ ∞), which includes the cases of ℝ and ℂ with the regular distances. Indeed, as a countable dense set here, one can consider that of all ordered ntuples with (real or complex) rational components. 2. The spaces lp are separable for 1 ≤ p < ∞. Indeed, as a countable dense set here, one can consider that of all eventually zero sequences with real or complex rational terms. 3. The space l∞ is not separable (see, e. g., [14]). 4. The space (C[a, b], ρ∞ ) (−∞ < a < b < ∞) is separable, which follows from Weierstrass Approximation Theorem (see, e. g., [14]) when we consider as a countable dense set that of all polynomials with rational coefficients. Exercise 1.30. When is a discrete metric space (X, ρ) separable? Open sets in separable metric spaces allow the following description. Proposition 1.4 (Open Sets in Separable Spaces). Every open set in a separable metric space is a countable a union of open balls. See, e. g., [5, 10, 18]. For in ℝ with the regular distance, we have the following useful one: Proposition 1.5 (Open Sets of the Real Line). Every open set in ℝ is a countable union of pairwise disjoint open intervals. 1.4.8 Compactness Definition 1.26 (Cover, Subcover, Open Cover). A collection C = {Ci }i∈I of subsets of a nonempty set X is said to be a cover of a set A ⊆ X, or to cover A, if A ⊆ ⋃ Ci . (1.1) i∈I A subcollection C of a cover C of A, which is also a cover of A, is called a subcover of C . If (X, ρ) is a metric space, a cover of a set A ⊆ X consisting of open sets is called an open cover of A. Remark 1.19. In particular, when A = X, (1.1) acquires the form X = ⋃ Ci . i∈I Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 20  1 Preliminaries Examples 1.16. 1. The collection {[n, n + 1)}n∈ℤ is a cover for ℝ. 2. The collection {(n, n + 1)}n∈ℤ is not a cover for ℤ. 3. The collection of all concentric open balls in a metric space (X, ρ) centered at a fixed point x ∈ X {B(x, r)  r > 0} is an open cover of X, the subcollection {B(x, n)  n ∈ ℕ} being its countable subcover. 4. Let A be a dense set in a metric space (X, ρ). For any ε > 0, the collection of εballs {B(x, ε)  x ∈ A} , 5. is an open cover of X. Let {rn }n∈ℕ be a countably infinite subset of ℝ. The collection of intervals {[rn − 1/2n+1 , rn + 1/2n+1 ] n ∈ ℕ} does not cover ℝ. This is true even when the set is dense in ℝ, as is the case, e. g., for ℚ. Exercise 1.31. Verify. Definition 1.27 (Compactness). A set A is said to be compact in a metric space (X, ρ) if each open cover O of A contains a finite subcover O . A metric space (X, ρ) is called compact if the set X is compact in (X, ρ). Definition 1.28 (Sequential Compactness). A nonempty set A in a metric space (X, ρ) is said to be sequentially compact in (X, ρ) if every sequence (xn )n∈ℕ in A has a subsequence (xn(k) )k∈ℕ convergent to a limit in A. A metric space (X, ρ) is called sequentially compact if the set X is sequentially compact in (X, ρ). Remarks 1.20. – Compactness in the sense of the prior definition is also called compactness in the Heine5 –Borel6 sense. 5 Heinrich Heine (1821–1881). 6 Émile Borel (1871–1956). Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 1.4 Fundamentals of Metric Spaces  21 – – Sequential compactness is also called compactness in the Bolzano7 –Weierstrass8 sense. The notions of compactness and sequential compactness in a metric space are equivalent. Examples 1.17. 1. Finite sets, including the empty set 0, are compact in an arbitrary metric space (X, ρ) and only finite sets are compact in a discrete metric space (X, ρd ). 2. The sets [0, ∞), (0, 1], and {1/n}n∈ℕ are not compact in ℝ and the set {0} ∪ {1/n}n∈ℕ is. 3. The set E := {en := (δnk )k∈ℕ }n∈ℕ , where δnk is the Kronecker delta, is closed and bounded, but not compact in l∞ since its open cover by 1/2balls {B(en , 1/2)}n∈ℕ , has no finite subcover. The same example works in lp (1 ≤ p < ∞). Exercise 1.32. Verify. Theorem 1.13 (Heine–Borel Theorem). A set A is compact in ℝn (n ∈ ℕ) with the Euclidean metric iff it is closed and bounded. See, e. g., [14]. Example 1.18. Thus, any closed and bounded interval [a, b] (−∞ < a < b < ∞) is a compact set in ℝ. Theorem 1.14 (Properties of Compact Sets). The compact sets in a metric space (X, ρ) have the following properties: (1) a compact set is necessarily bounded, but not vice versa; (2) a compact set is necessarily closed, but not vice versa; (3) a closed subset of a compact set is compact, in particular, a closed set in a compact metric space (X, ρ) is compact; (4) an arbitrary intersection of compact sets is compact; (5) a finite union of compact sets is compact. See, e. g., [14]. The following classical result is a direct implication of the equivalence of different forms of compactness. 7 Bernard Bolzano (1781–1848). 8 Karl Weierstrass (1815–1897). Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 22  1 Preliminaries Theorem 1.15 (Bolzano–Weierstrass Theorem). Each bounded sequence of real or complex numbers contains a convergent subsequence. 1.4.9 Continuity Definition 1.29 (Continuity of a Function). Let (X, ρ) and (Y, σ) be metric spaces. A function f : X → Y is called continuous at a point x0 ∈ X if ∀ ε > 0 ∃ δ > 0 ∀ x ∈ X with ρ(x, x0 ) < δ : σ(f (x), f (x0 )) < ε. A function f : X → Y is called continuous on X if it is continuous at every point of X. The set of all such functions is designated as C(X, Y) and we write f ∈ C(X, Y). Remarks 1.21. – When X and Y are subsets of ℝ with the regular distance, we obtain the familiar calculus (ε, δ)definitions. – When Y = ℝ or Y = ℂ, the shorter notation C(X) is used. Definition 1.30 (Equivalent Definition of Continuity). Let (X, ρ) and (Y, σ) be metric spaces. A function f : X → Y is called continuous at a point x0 ∈ X if ∀ ε > 0 ∃ δ > 0 : f (BX (x0 , δ)) ⊆ BY (f (x0 ), ε). It is often convenient to describe continuity in terms of sequences. Theorem 1.16 (Sequential Characterization of Local Continuity). Let (X, ρ) and (Y, σ) be metric spaces. A function f : X → Y is continuous at a point x0 ∈ X iff, for each sequence (xn )n∈ℕ in X such that lim x n→∞ n = x0 in (X, ρ), we have: lim f (xn ) = f (x0 ) in (Y, σ). n→∞ Exercise 1.33. Prove. Hint. The necessity is proved directly. Prove the sufficiency by contrapositive. Theorem 1.17 (Properties of Numeric Continuous Functions). Let (X, ρ) be a metric space and Y = ℝ or Y = ℂ with the regular distance. If f and g are continuous at a point x0 ∈ X, then (1) ∀ c ∈ ℝ (or c ∈ ℂ), cf is continuous at x0 ; (2) f + g is continuous at x0 ; Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 1.4 Fundamentals of Metric Spaces  23 (3) f ⋅ g is continuous at x0 ; (4) provided g(x0 ) ≠ 0, gf is continuous at x0 . Theorem 1.18 (Continuity of Composition). Let (X, ρ), (Y, σ), and (Z, τ), f : X → Y and g : Y → Z. If for some x0 ∈ X f is continuous at x0 and g is continuous at y0 = f (x0 ), then the composition g(f (x)) is continuous at x0 . Exercise 1.34. Prove Theorems 1.17 and 1.18 using the sequential approach. Remark 1.22. The statements of Theorems 1.17 and 1.18 are naturally carried over to functions continuous on the whole space (X, ρ). The following characterization of continuity is sometimes given as its definition (see, e. g., [18, 21]). Theorem 1.19 (Characterization of Continuity). Let (X, ρ) and (Y, σ) be metric spaces. A function f : X → Y is continuous on X iff, for each open set A in (Y, σ) its preimage f −1 (A ) is open in (X, ρ), or equivalently, f −1 (G ) := {f −1 (A ) A ∈ G } ⊆ G , where G is the metric topology on Y generated by metric σ and G is the metric topology on X generated by metric ρ. Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 2 Basic Set Classes In this chapter, we introduce and discuss basic set classes needed for our subsequent discourse. Henceforth, X is regarded to be a nonempty set. 2.1 Semirings, Semialgebras Definition 2.1 (Semiring, Semialgebra). Let X be a nonempty set. A semiring (of sets) on X is a nonempty collection S of subsets of X such that: (1) 0 ∈ S ; (2) if A1 , . . . , An ∈ S (n ∈ ℕ), then ⋂ni=1 Ai ∈ S (closedness under finite intersections); (3) if A, B ∈ S , then n A \ B = ⋃ Ci i=1 with some n ∈ ℕ and pairwise disjoint Ci ∈ S , i = 1, . . . , n. A semialgebra S (of sets) on X is a semiring on X such that X ∈ S . Remarks 2.1. – Condition (2) can be replaced with the following equivalent one: if A, B ∈ R , then A ∩ B ∈ R . – Condition (3) makes condition (1) redundant. Examples 2.1. 1. On an arbitrary nonempty set X, its power set P (X) is a semialgebra. 2. On an arbitrary nonempty set X, a finite collection S := {A1 , . . . , An } (n ∈ ℕ) of pairwise disjoint subsets, which includes 0, is a semiring. In particular, {0, X} and {0} are semirings on X, the former being also a semialgebra. For X := {0, 1, 2} S := {0, {0}, {1, 2}} 3. is a semiring on X. On ℝ, S1 := {(a, b]  −∞ < a < b < ∞} ∪ {0} and S2 := {[a, b)  −∞ < a < b < ∞} ∪ {0} are semirings but not semialgebras. https://doi.org/10.1515/9783110600995002 Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 26  2 Basic Set Classes 4. The set collections S1 := {(a1 , b1 ] × (a2 , b2 ]  −∞ < ai < bi < ∞, i = 1, 2} ∪ {0} and S2 := {[a1 , b1 ) × [a2 , b2 )  −∞ < ai < bi < ∞, i = 1, 2} ∪ {0} 5. are semirings but not semialgebras on ℝ2 (see the Product Semiring Proposition (Proposition 2.1), Section 2.6, Problem 2). The set collections S1 := {(c, d]  a ≤ c < d ≤ b} ∪ {0} and S2 := {[c, d)  a ≤ c < d ≤ b} ∪ {0} 6. are semialgebras on (a, b] and [a, b) (−∞ < a < b < ∞), respectively. The set collections S1 := {(c1 , d1 ] × (c2 , d2 ]  ai ≤ ci < di ≤ bi , i = 1, 2} ∪ {0} and S2 := {[c1 , d1 ) × [c2 , d2 )  ai ≤ ci < di ≤ bi , i = 1, 2} ∪ {0} 7. are semialgebras on (a1 , b1 ] × (a2 , b2 ] and [a1 , b1 ) × [a2 , b2 ) (−∞ < a1 < b1 < ∞, −∞ < a2 < b2 < ∞), respectively (see the Product Semiring Proposition (Proposition 2.1), Section 2.6, Problem 2). On ℝ, C := {(a, b)  −∞ < a < b < ∞} ∪ {0} is not a semiring. Exercise 2.1. (a) Verify. (b) Give an example showing that a semiring need not be closed under finite unions. 2.2 Rings, Algebras Definition 2.2 (Ring, Algebra). Let X be a nonempty set. A ring (of sets) on X is a nonempty collection R of subsets of X such that Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 2.2 Rings, Algebras  27 (1) 0 ∈ R ; (2) if A1 , . . . , An ∈ R (n ∈ ℕ), then ⋃ni=1 Ai ∈ R (closedness under finite unions); (3) if A, B ∈ R , then A \ B ∈ R (closedness under set differences). An algebra A (of sets) on X is a ring on X such that X ∈ A . Remarks 2.2. – Condition (2) can be replaced with the following equivalent one: if A, B ∈ R , then A ∪ B ∈ R . – – Condition (3) makes condition (1) redundant (cf. Remark 2.1). As follows from conditions (2) and (3) by De Morgan’s laws, a ring R is closed under finite intersections, i. e., n if A1 , . . . , An ∈ R (n ∈ ℕ), then ⋂ Ai ∈ R . i=1 – As follows from condition (3), an algebra A is closed under complements, i. e., if A ∈ A , then Ac ∈ A . – Each ring/algebra is a semiring/semialgebra, respectively, but not vice versa. Exercise 2.2. Explain, verify. Examples 2.2. 1. On an arbitrary nonempty set X, (a) the power set P (X) and {0, X} are algebras and {0} is a ring but not an algebra; (b) for a finite collection C := {A1 , . . . , An } (n ∈ ℕ) of pairwise disjoint subsets, the collection R of all finite unions of Ai , i = 1, . . . , n, including 0 (⋃i∈0 Ai = 0), is a ring on X consisting of 2n elements. With C being a partition of X, i. e., in addition, n ⋃ Ai = X, i=1 R is an algebra on X. In particular, for X := {0, 1, 2} and the semiring S := {0, {0}, {1, 2}} (see Examples 2.1) forming a partition of X, A := {0, {0}, {1, 2}, X} Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 28  2 Basic Set Classes is an algebra on X; for X := [a, b] (−∞ < a < b < ∞) and C : {[a, (a + b)/2], ((a + b)/2, b]} forming a partition of X, A := {0, [a, (a + b)/2], ((a + b)/2, b], X} 2. is an algebra on X. The set collections S1 := {(a, b]  −∞ < a < b < ∞} ∪ {0} and S2 := {[a, b)  −∞ < a < b < ∞} ∪ {0} are semirings but not rings on ℝ and the set collections n R1 := {⋃(ai , bi ] n ∈ ℕ, −∞ < ai < bi < ∞, i = 1, . . . , n} ∪ {0} i=1 and n R2 := {⋃[ai , bi ) n ∈ ℕ, −∞ < ai < bi < ∞, i = 1, . . . , n} ∪ {0}, i=1 3. are rings but not algebras on ℝ. The set collections S1 := {(c, d]  a ≤ c < d ≤ b} ∪ {0} and S2 := {[c, d)  a ≤ c < d ≤ b} ∪ {0} are semialgebras but not algebras on (a, b] and [a, b) (−∞ < a < b < ∞), respectively, and the set collections n A1 := {⋃(ci , di ] n ∈ ℕ, a ≤ ci < di ≤ b, i = 1, . . . , n} ∪ {0} i=1 and n A2 := {⋃[ci , di ) n ∈ ℕ, a ≤ ci < di ≤ b, i = 1, . . . , n} ∪ {0} i=1 are algebras on (a, b] and [a, b) (−∞ < a < b < ∞), respectively. Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 2.3 σRings, σAlgebras  29 4. The set collections S1 := {(c1 , d1 ] × (c2 , d2 ]  ai ≤ ci < di ≤ bi , i = 1, 2} ∪ {0} and S2 := {[c1 , d1 ) × [c2 , d2 )  ai ≤ ci < di ≤ bi , i = 1, 2} ∪ {0} are semialgebras on (a1 , b1 ] × (a2 , b2 ] and [a1 , b1 ) × [a2 , b2 ) (−∞ < a1 < b1 < ∞, −∞ < a2 < b2 < ∞), respectively, and the collections n (i) (i) (i) (i) (i) (i) (i) (i) A1 := {⋃(c1 , d1 ] × (c2 , d2 ] a1 ≤ c1 < d1 ≤ b1 , a2 ≤ c2 < d2 ≤ b2 , i=1 i = 1, . . . , n} ∪ {0} and n (i) (i) (i) (i) (i) (i) (i) (i) A2 := {⋃[c1 , d1 ) × [c2 , d2 ) a1 ≤ c1 < d1 ≤ b1 , a2 ≤ c2 < d2 ≤ b2 , i=1 i = 1, . . . , n} ∪ {0} are algebras on (a1 , b1 ] × (a2 , b2 ] and [a1 , b1 ) × [a2 , b2 ) (−∞ < a1 < b1 < ∞, −∞ < a2 < b2 < ∞), respectively. An algebra can be equivalently defined as follows: Definition 2.3 (Algebra: Alternative Definition). Let X be a nonempty set. An algebra (of sets) on X is a nonempty collection A of subsets of X such that: (1) 0, X ∈ A ; (2) if A1 , . . . , An ∈ A (n ∈ ℕ), then ⋃ni=1 Ai ∈ A (closedness under finite unions); (3) if A ∈ A , then Ac ∈ A (closedness under complements). Exercise 2.3. Verify the equivalence of Definitions 2.2 and 2.3. 2.3 σRings, σAlgebras Definition 2.4 (σRing, σAlgebra). Let X be a nonempty set. A σring (of sets) on X is a nonempty collection R of subsets of X such that: (1) 0 ∈ R ; (2) if (An )n∈ℕ is a sequence in R , then ⋃∞ i=1 Ai ∈ R (closedness under countable unions); (3) if A, B ∈ R , then A \ B ∈ R (closedness under set differences). A σalgebra Σ (of sets) on X is a σring on X such that X ∈ Σ. Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 30  2 Basic Set Classes Remarks 2.3. – Condition (3) makes condition (1) redundant (cf. Remarks 2.2). – As follows from conditions (2) and (3) by De Morgan’s laws, a σring R is closed under countable intersections, i. e., ∞ if (An )n∈ℕ is a sequence in R , then ⋂ Ai ∈ R i=1 – – (cf. Remarks 2.2). Each σring/σalgebra is a ring/algebra, respectively, but not vice versa. Hence, in particular, each σalgebra Σ is closed under complements, i. e., if A ∈ Σ, then Ac ∈ Σ. – Each ring/algebra consisting of finitely many elements is a σring/σalgebra, respectively. Exercise 2.4. Verify. Examples 2.3. 1. All finite algebras from Examples 2.2 are σalgebras. 2. On an arbitrary nonempty set X, (a) the power set P (X) and {0, X} are σalgebras and {0} is a σring but not a σalgebra; (b) Σ := {A ⊆ X A or Ac is countable} is a σalgebra; (c) provided X is infinite, c A := {A ⊆ X A or A is finite} is an algebra, but not a σalgebra and R := {A ⊆ X  A is finite} is a ring, but not a σring on X; (d) provided X is countably infinite, if a countably infinite collection {An }n∈ℕ is a partition of X, i. e., the sets Ai , i ∈ ℕ, are pairwise disjoint and ∞ ⋃ Ai = X, i=1 the collection Σ of all countable unions of Ai , i ∈ ℕ, including 0 (⋃i∈0 Ai = 0), is a σalgebra of cardinality 2ℵ0 = c (cf. Section 2.6, Problem 8). Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 2.4 Monotone Classes  3. 31 An example of a σring, which is not a σalgebra, can be obtained by taking an arbitrary σalgebra Σ on a nonempty set X, e. g., P (X), and considering it on the set X := X ∪ {x }, where x ∉ X. For instance, P ([0, 1)) is a σalgebra on [0, 1) but only a σring on [0, 1]. Exercise 2.5. Verify. A σalgebra can be equivalently defined as follows. Definition 2.5 (σAlgebra: Alternative Definition). Let X be a nonempty set. A σalgebra (of sets) on X is a nonempty collection Σ of subsets of X such that: (1) 0, X ∈ Σ; (2) for any sequence (An )n∈ℕ in Σ, ⋃∞ i=1 Ai ∈ Σ (closedness under countable unions); (3) if A ∈ Σ, then Ac ∈ Σ (closedness under complements). Exercise 2.6. Verify the equivalence of Definitions 2.4 and 2.5. 2.4 Monotone Classes Definition 2.6 (Monotone Class). Let X be a nonempty set. A monotone class (of sets) on X is a nonempty collection M of subsets of X such that, if (An )n∈ℕ is a monotone sequence in M , then limn→∞ An ∈ M (closedness under the limits of monotone set sequences). Remarks 2.4. – As follows from the definition, a σring, in particular a σalgebra, is a monotone class. Exercise 2.7. Verify. – This fact readily provides many examples of monotone classes. However, as the following example shows, a monotone class need not even be a semiring. Examples 2.4. 1. On a nonempty finite set X, any nonempty set collection C is a monotone class. 2. On ℝ, M := {[m, n]  m, n ∈ ℤ, m < n} ∪ {(−∞, n]  n ∈ ℤ} ∪ {[n, ∞)  n ∈ ℤ} ∪ {0} ∪ {ℝ} is a monotone class. Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 32  2 Basic Set Classes Exercise 2.8. (a) Verify. (b) Show that M from the prior example is not a semiring. (c) Would M from the prior example be a monotone class if m, n ∈ ℝ? Theorem 2.1 (Monotone Ring Theorem). A monotone ring is a σring. Proof. Let R be simultaneously a ring and a monotone class on a nonempty set X. Then R immediately meets condition (3) of the definition of a σring (see Definition 2.4). It remains to verify condition (2) of Definition 2.4, i. e., the closedness of R under countable unions. For an arbitrary sequence (An )n∈ℕ in R , consider the set sequence n Bn := ⋃ Ai , n ∈ ℕ. i=1 Since R is a ring Bn ∈ R , n ∈ ℕ. The sequence {Bn }∞ n=1 is increasing since n n+1 i=1 i=1 Bn = ⋃ Ai ⊆ ⋃ Ai = Bn+1 , n ∈ ℕ, which, since R is a monotone class, implies that ∞ ∞ i=1 i=1 R ∋ lim Bn = ⋃ Bi = ⋃ Ai . n→∞ Hence, R is a σring. As an important particular case, we obtain the following. Corollary 2.1 (Monotone Algebra Theorem). A monotone algebra is a σalgebra. 2.5 Generated Set Classes Here, we introduce the important notion of the ring, algebra, σring, σalgebra, and monotone class “generated” by a nonempty collection C of subsets of a nonempty set X. To proceed, we need the following lemma. Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 2.5 Generated Set Classes  33 2.5.1 Intersection Lemma Lemma 2.1 (Intersection Lemma). The intersection of an arbitrary nonempty collection of rings, algebras, σrings, σalgebras, or monotone classes on a nonempty set X is a ring, algebra, σring, σalgebra, and monotone class on X, respectively. Proof. Let {Ri }i∈I be a nonempty collection of rings on X. Observe that ⋂ Ri = ̸ 0 i∈I since 0 ∈ Ri for each i ∈ I. For any A, B ∈ ⋂i∈I Ri , A, B ∈ Ri , i ∈ I, which, with Ri , i ∈ I, being a ring, implies that A ∪ B, A \ B ∈ Ri , i ∈ I, and hence, A ∪ B, A \ B ∈ ⋂ Ri . i∈I Thus, ⋂i∈I Ri is a ring on X. The statement for a nonempty collection {Ai }i∈I of algebras on X follows immediately with one more condition to verify: X ∈ ⋂ Ai , i∈I which is trivially true since X ∈ Ai , i ∈ I. Exercise 2.9. Prove the statement for σrings, σalgebras, and monotone classes. Remark 2.5. The analogue of the prior lemma does not hold for a nonempty collection of semirings. For example, on X := {0, 1, 2}, the following set collections: C1 := {0, {0}, {1, 2}, {0, 1, 2}} and C2 := {0, {0}, {1}, {2}, {0, 1, 2}} are semirings (C1 is even an algebra on X (see Examples 2.2)), but their intersection C1 ∩ C2 = {0, {0}, {0, 1, 2}} is not. Exercise 2.10. Verify (cf. Section 2.6, Problem 9). Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 34  2 Basic Set Classes 2.5.2 Generated Set Classes Definition 2.7 (Generated Set Classes). Let C be a nonempty collection of subsets of a nonempty set X. – r(C ) := ⋂R is a ring, C ⊆R R is the smallest ring containing C called the ring generated by C . – a(C ) := ⋂A is an algebra, C ⊆A A is the smallest algebra containing C called the algebra generated by C . – σr(C ) := ⋂R is a σring, C ⊆R R is the smallest σring containing C called the σring generated by C . – σa(C ) := ⋂Σ is a σalgebra, C ⊆Σ Σ is the smallest σalgebra containing C called the σalgebra generated by C . – m(C ) := ⋂M is a monotone class, C ⊆M M is the smallest monotone class containing C called the monotone class generated by C . Exercise 2.11. (a) Explain why the generated classes are welldefined. (b) If X ∈ C , r(C ) = a(C ) and σr(C ) = σa(C ). (c) If C1 ⊂ C2 , r(C1 ) ⊆ r(C2 ), a(C1 ) ⊆ a(C2 ), σr(C1 ) ⊆ σr(C2 ), σa(C1 ) ⊆ σa(C2 ). (d) Give an example showing that the notion of the semiring generated by a nonempty collection C of subsets of a nonempty set X, i. e., the smallest semiring on X containing C , is not welldefined (see Remark 2.5 and Section 2.6, Problem 9). Examples 2.5. For a nonempty set X, 1. r ({0}) = σr ({0}) = m ({0}) = {0}; 2. r ({X}) = a ({X}) = σr ({X}) = σa ({X}) = {0, X} and m ({X}) = {X}. Exercise 2.12. Verify. Theorem 2.2 (Generated Ring Theorem). Let S be a semiring on a nonempty set X. Then n (2.1) r(S ) := {⋃ Ai n ∈ ℕ, Ai ∈ S , i = 1, . . . , n} , i=1 i. e., the collection of all finite unions of sets of a semiring S on a nonempty set X is the ring generated by S . Proof. Let n C := {⋃ Ai n ∈ ℕ, Ai ∈ S , i = 1, . . . , n} . i=1 Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 2.5 Generated Set Classes  35 Then S ⊆ C ⊆ r(S ). (2.2) Exercise 2.13. Explain. Let us show that C is a ring on X. Indeed, for arbitrary m A = ⋃ Ai , m ∈ ℕ, Ai ∈ S , i = 1, . . . , m, i=1 and n B = ⋃ Bj ∈ C , n ∈ ℕ, Bj ∈ S , j = 1, . . . , n, j=1 by the very definition of C , A ∪ B ∈ C. Further, by De Morgan’s laws, m n m n m n i=1 j=1 i=1 j=1 i=1 j=1 A \ B = ⋃ Ai \ ⋃ Bj = ⋃ (Ai \ ⋃ Bj ) = ⋃ ⋂ Ai \ Bj . Since S is a semiring, without loss of generality, we can regard that the sets Ai , i = 1, . . . , m, are pairwise disjoint, and so are Bj , j = 1, . . . , n (see Section 2.6, Problem 10). Furthermore, for all i = 1, . . . , m, j = 1, . . . , n, l(i,j) (i,j) Ai \ Bj = ⋃ Ck k=1 (i,j) with some l(i, j) ∈ ℕ and pairwise disjoint Ck Hence, ∈ S , k = 1, . . . , l(i, j). m n m n l(i,j) i=1 j=1 i=1 j=1 k=1 (i,j) A \ B = ⋃ ⋂ Ai \ Bj = ⋃ ⋂ ⋃ Ck ∈ C. Exercise 2.14. Explain. Thus, we conclude that C is a ring on X, which in view of S ⊆ C implies that r(S ) ⊆ C . (2.3) Inclusions (2.2) and (2.3) jointly imply that C = r(S ), which completes the proof. Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 36  2 Basic Set Classes Remarks 2.6. – In particular, if X ∈ S , r(S ) = a(S ) – (see Exercise 2.11). As is noted in the above proof, without loss of generality, the sets A1 , . . . , An (n ∈ ℕ) in definition (2.1) of the Generated Ring Theorem (Theorem 2.2) can be regarded to be pairwise disjoint (see Section 2.6, Problem 10), i. e., the Generated Ring Theorem (Theorem 2.2) can be equivalently restated as follows. Theorem 2.3 (Generated Ring Theorem). The collection of all finite unions of pairwise disjoint sets of a semiring S on a nonempty set X is the ring generated by S . Examples 2.6. 1. On an arbitrary nonempty set X, a finite collection S := {A1 , . . . , An } (n ∈ ℕ) of pairwise disjoint subsets, which includes 0, is a semiring (see Examples 2.1), and hence, by the Generated Ring Theorem (Theorem 2.2), the collection of all finite unions of sets of S is the ring r(S ) generated by S . In particular, when S is a partition of X (see Examples 2.2), r(S ) is the algebra generated by S , i. e., in this case, r(S ) = a(S ). 2. By the Generated Ring Theorem (Theorem 2.2), n R1 := {⋃(ai , bi ] n ∈ ℕ, −∞ < ai < bi < ∞, i = 1, . . . , n} ∪ {0} = r(S1 ) i=1 and n R2 := {⋃[ai , bi ) n ∈ ℕ, −∞ < ai < bi < ∞, i = 1, . . . , n} ∪ {0} = r(S2 ), i=1 where S1 := {(a, b]  −∞ < a < b < ∞} ∪ {0} and S2 := {[a, b)  −∞ < a < b < ∞} ∪ {0} 3. (see Examples 2.2). By the Generated Ring Theorem (Theorem 2.2), n A1 := {⋃(ci , di ] n ∈ ℕ, a ≤ ci < di ≤ b, i = 1, . . . , n} ∪ {0} = a(S1 ) i=1 and n A2 := {⋃[ci , di ) n ∈ ℕ, a ≤ ci < di ≤ b, i = 1, . . . , n} ∪ {0} = a(S2 ), i=1 Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 2.5 Generated Set Classes  37 where S1 := {(c, d]  a ≤ c < d ≤ b} ∪ {0} and S2 := {[c, d)  a ≤ c < d ≤ b} ∪ {0} (−∞ < a < b < ∞) (see Examples 2.2). 4. By the Generated Ring Theorem (Theorem 2.2), n (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) A1 := {⋃(c1 , d1 ] × (c2 , d2 ] a1 ≤ c1 < d1 ≤ b1 , a2 ≤ c2 < d2 ≤ b2 , i=1 i = 1, . . . , n} ∪ {0} = a(S1 ) and n (i) (i) (i) (i) (i) (i) A2 := {⋃[c1 , d1 ) × [c2 , d2 ) a1 ≤ c1 < d1 ≤ b1 , a2 ≤ c2 < d2 ≤ b2 , i=1 i = 1, . . . , n} ∪ {0} = a(S2 ), where S1 := {(c1 , d1 ] × (c2 , d2 ]  ai ≤ ci < di ≤ bi , i = 1, 2} ∪ {0} and S2 := {[c1 , d1 ) × [c2 , d2 )  ai ≤ ci < di ≤ bi , i = 1, 2} ∪ {0} (−∞ < a1 < b1 < ∞, −∞ < a2 < b2 < ∞) (see Examples 2.2). Theorem 2.4 (Monotone Class Theorem). Let R be a ring on a nonempty set X. Then σr(R ) = m(R ). Proof. Since σr(R ) is a also monotone class containing R (see Remarks 2.4), we have the inclusion m(R ) ⊆ σr(R ). (2.4) Let us show that m(R ) is a ring. For a fixed A ∈ m(R ), consider L (A) := {C ⊆ X  A ∪ C, A \ C, C \ A ∈ m(R )} . Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 38  2 Basic Set Classes Whenever A ∈ R , (2.5) R ⊆ L (A). Indeed, since R is a ring, for each C ∈ R , A ∪ C, A \ C, C \ A ∈ R ⊆ m(R ). Further, for each A ∈ m(R ), L (A) is a monotone class. Indeed, for an arbitrary increasing sequence (Cn )n∈ℕ in L (A), A ∪ Cn , A \ Cn , Cn \ A ∈ m(R ), n ∈ ℕ, and, as is easily seen, the sequences (A∪Cn )n∈ℕ and (Cn \A)n∈ℕ are increasing, whereas the sequence (A \ Cn )n∈ℕ is decreasing in m(R ). Exercise 2.15. Verify. Since, m(R ) is a monotone class, ∞ ∞ m(R ) ∋ lim (A ∪ Cn ) = ⋃ (A ∪ Cn ) = A ∪ ⋃ Cn , n→∞ n=1 ∞ ∞ n=1 n=1 n=1 m(R ) ∋ lim (Cn \ A) = ⋃ (Cn \ A) = ⋃ Cn \ A, n→∞ and, in view of De Morgan’s laws, ∞ ∞ n=1 n=1 m(R ) ∋ lim (A \ Cn ) = ⋂ (A \ Cn ) = A \ ⋃ Cn . n→∞ Hence, by the definition of L (A) (see (2.5)), ∞ ⋃ Cn ∈ L (A). n=1 ion. The case of a decreasing sequence (Cn )n∈ℕ in L (A) is considered in the same fash Exercise 2.16. Consider. Thus, for each A ∈ R , L (A) is a monotone class containing R , and hence, we have the inclusion m(R ) ⊆ L (A), which implies that, for any A ∈ R and C ∈ m(R ), A ∪ C, A \ C, C \ A ∈ m(R ). Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 2.5 Generated Set Classes  39 Whence, we conclude that, for any C ∈ m(R ), R ⊆ L (C). With L (C) being a monotone class, the latter implies that, for each C ∈ m(R ), m(R ) ⊆ L (C). Hence, for any C1 , C2 ∈ m(R ), C1 ∪ C2 , C1 \ C2 , C2 \ C1 ∈ m(R ), which proves that m(R ) is a ring on X indeed. By the Monotone Ring Theorem (Theorem 2.1), we infer that m(R ) is a σring on X, and hence, σr(R ) ⊆ m(R ). (2.6) Inclusions (2.4) and (2.6) imply that σr(R ) = m(R ), which completes the proof. As an important particular case, we obtain the following. Corollary 2.2. Let A be an algebra on a nonempty set X. Then σa(A ) = m(A ). 2.5.3 Borel Sets Definition 2.8 (Borel σAlgebra). Let (X, ρ) be a metric space and G be the metric topology generated by metric ρ (see Definition 1.20). The σalgebra of Borel sets or the Borel σalgebra on X is the σalgebra generated by the metric topology G :1 B (X) := σa(G ). Exercise 2.17. (a) Let (X, ρ) be a metric space and F be the collection of all closed sets in (X, ρ). Prove that 1 Émile Borel (1871–1956). Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 40  2 Basic Set Classes (i) B (X) = σa(F ); (ii) provided (X, ρ) is separable, B (X) = σa ({B(x, r)  x ∈ X, r > 0}) = σa ({B(x, r) x ∈ X, r > 0}) (see Open Sets in Separable Spaces Proposition (Proposition 1.4)); (iii) each countable set, in particular each singleton, is a Borel set. (b) Prove that: (i) ℕ, ℤ, ℚ, ℝ \ ℚ ∈ B (ℝ); (ii) (a, b], [a, b) ∈ B (ℝ) (−∞ < a < b < ∞); (iii) B (ℝ) = σa ({(a, b)  −∞ < a < b < ∞}) = σa ({(a, b]  −∞ < a < b < ∞}) = σa ({[a, b)  −∞ < a < b < ∞}) = σa ({[a, b]  −∞ < a < b < ∞}) = σa ({(−∞, b)  b ∈ ℝ}) = σa ({(a, ∞)  a ∈ ℝ}) = σa ({(−∞, b]  b ∈ ℝ}) = σa ({[a, ∞)  a ∈ ℝ}) ; (iv) B (ℝ) = σa ({(a, b)  −∞ < a < b < ∞, a, b ∈ ℚ}) = σa ({(a, b]  −∞ < a < b < ∞, a, b ∈ ℚ}) = σa ({[a, b)  −∞ < a < b < ∞, a, b ∈ ℚ}) = σa ({[a, b]  −∞ < a < b < ∞, a, b ∈ ℚ}) = σa ({(−∞, b)  b ∈ ℚ}) = σa ({(a, ∞)  a ∈ ℚ}) = σa ({(−∞, b]  b ∈ ℚ}) = σa ({[a, ∞)  a ∈ ℚ}) . Remark 2.7. If (X, ρ) is a separable metric space, B (X) ≤ c, in particular, B (ℝn ) = c (n ∈ ℕ) (see, e. g., [3]). This immediately implies that there exist subsets of ℝn (n ∈ ℕ) that are not Borel. 2.6 Problems 1. (a) Show that, for any l > 0, Sl := {(a, b]  −∞ < a < b < ∞, b − a ≤ l} ∪ {0} is a semiring on ℝ. (b) Show that Sl2 ⊆ Sl1 , 0 < l2 < l1 . (c) Find ⋂l>0 Sl . Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 2.6 Problems  41 2. Prove Proposition 2.1 (Product Semiring). Let Si be a semiring on a nonempty set Xi , i = 1, 2. Then S1 × S2 := {A1 × A2  Ai ∈ Si , i = 1, 2} is a semiring on X1 × X2 . 3. Prove Proposition 2.2 (Ring Characterization). A nonempty collection C of subsets of a nonempty set X is a ring iff C is a semiring on X which is closed under finite unions. 4. Show that each ring R is closed under symmetric differences, i. e., if A, B ∈ R , then A △ B ∈ R . 5. Let Ri be a ring on a nonempty set Xi , i = 1, 2. As follows from the Product Semiring Proposition (Proposition 2.1) (see Problem 2), R1 × R2 := {A1 × A2  Ai ∈ Si , i = 1, 2} 6. 7. is a semiring on X1 × X2 . Give an example showing that R1 × R2 need not be a ring. If R is a σring on a nonempty set X, then, for any set sequence (An )n∈ℕ in R , (a) limn→∞ An ∈ R and limn→∞ An ∈ R ; (b) provided limn→∞ An exists (see Definition 1.3), limn→∞ An ∈ R . Prove that the collection of all symmetric sets in ℝ2 : Σ := {A ⊆ ℝ2 ∀ (x, y) ∈ A : (−x, −y) ∈ A} ∪ {0} is a σalgebra on ℝ2 . 8. * Does there exist a countably infinite σalgebra? 9. Let C := {(−∞, b]  b ∈ ℝ}. (a) Show that C is not a semiring on ℝ. (b) Show that S := {(a, b]  −∞ ≤ a < b < ∞} ∪ {0} is a semiring on ℝ containing C but not the smallest one. (c) Is there the smallest semiring on ℝ containing C (see Problem 1)? 10. Let S be a semiring on a nonempty set X. Show that, without loss of generality, in a finite union n ⋃ Ai , n ∈ ℕ, Ai ∈ S , i = 1, . . . , n, i=1 the sets Ai , i = 1, . . . , n, can be regarded to be pairwise disjoint (see the proof of the Generated Ring Theorem (Theorem 2.2) and Remarks 2.6). Hint. Use induction. Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 42  2 Basic Set Classes 11. Let C1 and C2 be nonempty collections of subsets of a nonempty set X. (a) Prove that, if C1 ⊆ C2 ⊆ r(C1 ), then r(C1 ) = r(C2 ). (b) Show that the analogous statements hold for the generated algebras, σrings, σalgebras, and monotone classes. 12. Let C be a nonempty collection of subsets of a nonempty set X and an arbitrary set B ⊆ X be fixed. Prove that (a) r(C ∩ B) = r(C ) ∩ B, where C ∩ B := {C ∩ B  C ∈ C } ; (b) σr(C ∩ B) = σr(C ) ∩ B. Hint. For (a), show that C ∩ B ⊆ r(C ) ∩ B and r(C ) ∩ B is a ring on X. 13. Let C = {A1 , . . . , An } (n ∈ ℕ) be a collection of n subsets of a nonempty set X. Prove that n (a) a(C ) consists of at most 22 sets; (b) σa(C ) = a(C ). Hint. For (a), consider all sets of the form Â 1 ∩ ⋅ ⋅ ⋅ ∩ Â n , where Â i = Ai or Â i = Aci , i = 1, . . . , n. Brought to you by  University of Warwick Authenticated Download Date  8/4/19 2:23 PM 3 Measures The concept of measure of a set as an estimate of its size generalizes the familiar and intuitive notions of conventional length, area, and volume and is focal for our entire discourse. 3.1 Set Functions Here, on a nonempty collection C of subsets of a nonempty set X, we consider set functions μ : C → R := [−∞, ∞] subject to certain conditions. Remarks 3.1. Before we proceed, we agree upon the following natural order and arithmetic rules involving infinity: – for any a ∈ ℝ, −∞ < a < ∞; – for any a ∈ ℝ, a + ∞ := ∞ =: ∞ + a, a + (−∞) := −∞ =: −∞ + a; – the expressions −∞ + ∞ and ∞ + (−∞) being indeterminate, we do not consider set functions that assume both −∞ and ∞ values. Thus, either μ : C → (−∞, ∞] or μ : C → [−∞, ∞). Definition 3.1 (Various Properties of Set Functions). Let C be a nonempty collection of subsets of a nonempty set X. A function μ : C → (−∞, ∞] is called: (1) nonnegative if μ : C → [0, ∞], i. e., for each A ∈ C , μ(A) ≥ 0; (2) finitely subadditive (or subadditive), if for any A1 , . . . , An ∈ C (n ∈ ℕ) with ⋃ni=1 Ai ∈ C , n n i=1 i=1 μ (⋃ Ai ) ≤ ∑ μ(Ai ); (3) countably subadditive (or σsubadditive), if for any sequence (An )n∈ℕ in C with ⋃∞ i=1 Ai ∈ C , https://doi.org/10.1515/9783110600995003 Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 44  3 Measures ∞ ∞ i=1 i=1 μ (⋃ Ai ) ≤ ∑ μ(Ai ); (4) finitely additive (or additive), if for any pairwise disjoint A1 , . . . , An ∈ C (n ∈ ℕ) with ⋃ni=1 Ai ∈ C , n n i=1 i=1 μ (⋃ Ai ) = ∑ μ(Ai ); (5) countably additive (or σadditive), if for any pairwise disjoint sequence (An )n∈ℕ in ∞ C with ⋃i=1 Ai ∈ C , ∞ ∞ i=1 i=1 μ (⋃ Ai ) = ∑ μ(Ai ); (6) monotone, if for any A, B ∈ C with A ⊆ B, μ(A) ≤ μ(B); (7) finite, if for each A ∈ C , μ(A) < ∞; (8) countably finite (or σfinite), if there is a sequence (An )n∈ℕ in C with μ(An ) < ∞, n ∈ ℕ, such that ∞ ⋃ Ai = X. i=1 Example 3.1. The length μ is a nonnegative, subadditive, additive, monotone, and finite function on the semirings S1 := {(a, b]  −∞ < a < b < ∞} ∪ {0} and S2 := {[a, b)  −∞ < a < b < ∞} ∪ {0} (see Examples 2.1) with μ([a, b)) := b − a =: μ((a, b]), μ(0) := 0 and is a nonnegative, subadditive, additive, monotone, and σfinite function on the set collections S1 ∪ {ℝ} and S2 ∪ {ℝ} with μ(ℝ) := ∞. Exercise 3.1. (a) Verify. (b) Let C be a nonempty collection of subsets of a nonempty set X and μ : C → (−∞, ∞]. Show that: (i) if 0 ∈ C , μ(A) < ∞ for at least one set A ∈ C , and μ is additive, then μ(0) = 0; (ii) if 0 ∈ C , μ(A) < ∞ for at least one set A ∈ C , and μ is σadditive, then μ(0) = 0 and μ is additive. Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 3.2 Measure  45 3.2 Measure 3.2.1 Definition and Examples Definition 3.2 (Measure). A measure is a nonnegative σadditive function on a semiring. Remark 3.2. In particular, a nonnegative σadditive function on a ring, an algebra, a σring, or a σalgebra is a measure (see Remarks 2.2 and 2.3). Examples 3.2. 1. Immediate trivial examples of a measure on an arbitrary semiring S on a nonempty set X are: (a) the zero measure S ∋ A → μ(A) := 0, (b) the infinite measure S ∋ A → μ(A) := ∞, (c) the almost infinite measure if A ≠ 0, ∞ S ∋ A → μ(A) := { 0 2. if A = 0. For an arbitrary nonempty set X, (a) number of elements in A if A is finite, P (X) ∋ A → μ(A) := { if A is infinite ∞ is a measure on P (X) called the counting measure; (b) for a fixed x ∈ X, 1 P (X) ∋ A → δx (A) := { 0 if x ∈ A, if x ∉ A is a measure on P (X), called the unit point mass measure at x; (c) for fixed distinct x1 , . . . , xn ∈ X and a1 , . . . , an ∈ [0, ∞) (n ∈ ℕ), n P (X) ∋ A → μ(A) := ∑ ai δxi (A) = i=1 ∑ i∈ℕ: xi ∈A ai is a measure on P (X), which can be called a mass distribution measure over the set {x1 , . . . , xn }; Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 46  3 Measures (d) provided X is infinite, for countably infinite {xn }n∈ℕ ⊆ X and a sequence (an )n∈ℕ ⊂ [0, ∞), ∞ P (X) ∋ A → μ(A) := ∑ ai δxi (A) = i=1 ∑ i∈ℕ: xi ∈A ai , is a measure on P (X), which can be called a mass distribution measure over the set {xn }n∈ℕ . Exercise 3.2. Verify. Remark 3.3. For X := ℕ, the counting measure on P (ℕ) coincides with the unit mass distribution measure over ℕ: ∞ P (ℕ) ∋ A → μ(A) := ∑ δxi (A) = i=1 ∑ i∈ℕ: xi ∈A ai with an = 1, n ∈ ℕ. Exercise 3.3. Verify. More meaningful examples of measures are forthcoming. 3.2.2 Properties of Measure The subsequent major theorem establishes certain immediate properties of a measure on a ring. Theorem 3.1 (Properties of Measure). Let μ be a measure on a ring R . Then: (1) provided μ is not infinite, μ(0) = 0; (2) μ is additive on R ; (3) μ is monotone on R ; (4) if A, B ∈ R with A ⊆ B and μ(A) < ∞, μ(B \ A) = μ(B) − μ(A); (5) if A, B ∈ R and μ(A) < ∞ or μ(B) < ∞, μ(A ∪ B) = μ(A) + μ(B) − μ(A ∩ B); (6) μ is subadditive; (7) μ is σsubadditive. Proof. Properties (1) and (2) (additivity) follow from a more general statement (see Exercise 3.1 (b)). Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 3.2 Measure  47 Property (3) (monotonicity) follows from the additivity and nonnegativity of μ. Indeed, for A, B ∈ R with A ⊆ B, since R is a ring, B\A∈R and, by the additivity and nonnegativity of μ, μ(B) = μ(A ∪ (B \ A)) = μ(A) + μ(B \ A) ≥ μ(A). Let A, B ∈ R with A ⊆ B and μ(A) < ∞, by the additivity of μ, μ(B) = μ(A) + μ(B \ A), and hence, the finite value μ(A) can be subtracted through, yielding μ(B \ A) = μ(B) − μ(A), which shows that property (4) holds as well. Let A, B ∈ R with μ(A) < ∞ or μ(B) < ∞. Considering that R is a ring, B \ (A ∩ B) ∈ R . Since A ∪ (B \ (A ∩ B)) = A ∪ B and A ∩ (B \ (A ∩ B)) = 0, by the additivity of μ, we have μ(A ∪ B) = μ (A ∪ (B \ (A ∩ B))) = μ(A) + μ (B \ (A ∩ B)) . (3.1) By the monotonicity of μ, μ(A ∩ B) ≤ min [μ(A), μ(B)] < ∞, and hence, by (4), we infer that μ (B \ (A ∩ B)) = μ(B) − μ(A ∩ B). (3.2) From (3.1) and (3.2), we infer that μ(A ∪ B) = μ(A) + μ(B) − μ(A ∩ B), which completes the proof of property (5). For any A1 , . . . , An ∈ R (n ∈ ℕ), let i−1 Bi := Ai \ ⋃ Aj , i = 1, . . . , n, j=1 with ⋃0j=1 Aj := 0, i. e., B1 := A1 . Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 48  3 Measures Since R is a ring, Bi ∈ R , i = 1, . . . , n. Also, n n i=1 i=1 ⋃ Bi = ⋃ Ai ∈ R and the sets Bi , i = 1, . . . , n, are pairwise disjoint. Exercise 3.4. Verify. Since Bi ⊆ Ai , i = 1, . . . , n, by the monotonicity of μ, μ(Bi ) ≤ μ(Ai ), i = 1, . . . , n. In view of this, by the additivity of μ, n n n n i=1 i=1 i=1 i=1 μ (⋃ Ai ) = μ (⋃ Bi ) = ∑ μ(Bi ) ≤ ∑ μ(Ai ), which completes the proof of property (6) (subadditivity). For any sequence (An )n∈ℕ in R with ⋃∞ i=1 Ai ∈ R , similarly to the proof of (6), we define the sequence n−1 Bn := An \ ⋃ Aj ⊆ An , n ∈ ℕ, j=1 of pairwise disjoint sets in R such that ∞ ∞ i=1 i=1 ⋃ Bi = ⋃ Ai ∈ R and by the σadditivity and monotonicity of μ, we infer that ∞ ∞ ∞ ∞ i=1 i=1 i=1 i=1 μ (⋃ Ai ) = μ (⋃ Bi ) = ∑ μ (Bi ) ≤ ∑ μ (Ai ) . Thus, property (7) (σsubadditivity) holds as well. Remarks 3.4. – In particular, the Properties of Measure (Theorem 3.1) hold for a measure on an algebra, a σring, or a σalgebra (see Remarks 2.2 and 2.3). Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 3.2 Measure  49 – – As follows from Exercise 3.1 (b) and the proof of the prior theorem, properties (1) and (3)–(6) hold for a noninfinite, nonnegative, and additive function μ on a ring R and do not require from μ the property of σadditivity. In particular, such a function μ is monotone and subadditive. By the property of monotonicity, the finiteness of a nonnegative additive function μ, in particular a measure, on an algebra A is equivalent to μ(X) < ∞. Exercise 3.5. Explain. 3.2.3 Continuity of Measure The following is another important property of a measure on a ring. Theorem 3.2 (Continuity of Measure). Let μ be a measure on a ring R . Then: (1) for any increasing sequence (An )n∈ℕ in R with limn→∞ An = ⋃∞ i=1 Ai ∈ R , μ( lim An ) = lim μ(An ) (continuity from below); n→∞ n→∞ (2) for any decreasing sequence (An )n∈ℕ in R with μ(A1 ) < ∞ and limn→∞ An = ⋂∞ i=1 Ai ∈ R , μ( lim An ) = lim μ(An ) (continuity from above). n→∞ n→∞ Proof. Let (An )n∈ℕ be an increasing sequence in R with limn→∞ An = ⋃∞ i=1 ∈ R . There are two possibilities: – ∃ N ∈ ℕ : μ(AN ) = ∞, or – ∀ n ∈ ℕ : μ(An ) < ∞. Suppose that ∃ N ∈ ℕ : μ(AN ) = ∞. Then, since, for any n ≥ N, ∞ AN ⊆ An ⊆ ⋃ Ai , i=1 by the monotonicity of μ, ∞ μ(An ) = μ (⋃ Ai ) = ∞, n ≥ N, i=1 and hence, ∞ lim μ(An ) = ∞ = μ (⋃ Ai ) . n→∞ i=1 Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 50  3 Measures Now, suppose that ∀ n ∈ ℕ : μ(An ) < ∞. (3.3) The sequence Bn := An \ An−1 ∈ R , n ∈ ℕ, with A0 := 0 is pairwise disjoint and ∞ ∞ i=1 i=1 ⋃ Bi = ⋃ Ai . Exercise 3.6. Verify both statements. Since Bn ⊆ An , n ∈ ℕ, in view of (3.3), by the Properties of Measure (Theorem 3.1), μ(Bn ) = μ(An ) − μ(An−1 ), n ∈ ℕ, and hence, by the σadditivity of μ, ∞ ∞ ∞ i=1 i=1 i=1 n μ (⋃ Ai ) = μ (⋃ Bi ) = ∑ μ (Bi ) = lim ∑ [μ(An ) − μ(An−1 )] = lim μ(An ), n→∞ n→∞ i=1 which completes the proof of the continuity from below. Let (An )n∈ℕ be a decreasing sequence in R with μ(A1 ) < ∞ and limn→∞ An = ⋂∞ i=1 Ai ∈ R . Then (A1 \ An )n∈ℕ is an increasing sequence in R , and hence, by De Morgan’s laws and the proven continuity from below, ∞ ∞ i=1 i=1 μ (A1 \ ⋂ Ai ) = μ (⋃ A1 \ Ai ) = lim μ(A1 \ An ). n→∞ Where, in view of μ(A1 ) < ∞, by the Properties of Measure (Theorem 3.1), ∞ μ(A1 ) − μ (⋂ Ai ) = lim [μ(A1 ) − μ(An )] , n ∈ ℕ, n→∞ i=1 which implies ∞ μ (⋂ Ai ) = lim μ(An ) i=1 n→∞ completing the proof of the continuity from above and of the theorem. Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 3.2 Measure  51 Remarks 3.5. – As is easily seen, the requirement of μ(A1 ) < ∞ in the continuity of measure from above can be replaced with a more general one: ∃ N ∈ ℕ : μ(AN ) < ∞, – i. e., the measure μ(An ), n ∈ ℕ, is eventually finite. The foregoing requirement is essential and cannot be dropped. Exercise 3.7. Explain and give a corresponding example. 3.2.4 More Examples of Measures 3.2.4.1 Jordan Measure The Jordan1 measure is an extension of the familiar notions of length, area, and volume to a lager collection of sets and to higher dimensions. For a set in ℝn (n ∈ ℕ) to be Jordan measurable, i. e., such, to which Jordan measure can be meaningfully assigned, it must, in a sense, be “permissible,” which, in particular, includes its being bounded. For instance, such subsets of ℝ as ℕ, ℚ, [0, ∞), and [0, 1] ∩ ℚ are not Jordan measurable. The Lebesgue2 measure further extends the concept of measurability of a set in ℝn to a still larger collection of sets making, in particular, all the foregoing sets to be Lebesgue measurable. As we see below, the Jordan measurable sets form a ring whereas the Lebesgue measurable sets constitute a σalgebra. Definition 3.3 (Partitions of ℝn ). The mthorder partition of the nspace ℝn (m ∈ ℤ+ , n ∈ ℕ) is the following collection: πn(m) := {Q(m) (k1 , . . . , kn ) ki ∈ ℤ, i = 1, . . . , n} of the pairwise disjoint blocks with k k +1 Q(m) (k1 , . . . , kn ) := {(x1 , . . . , xn ) ∈ ℝn ki ∈ ℤ, mi < xi ≤ i m , i = 1, . . . , n} 2 2 ℝn = ⋃ ki ∈ℤ, i=1,...,n Q(m) (k1 , . . . , kn ). Exercise 3.8. Describe the 0thorder and 1storder partitions of ℝ and ℝ2 . Proposition 3.1 (Properties of Partitions). Let n ∈ ℕ. (1) For each m ∈ ℤ+ , the (m + 1)thorder partition of ℝn is obtained from the blocks of the mthorder partition of ℝn via partitioning the latter into 2n congruent blocks; 1 Camille Jordan (1838–1922). 2 Henri Lebesgue (1875–1941). Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 52  3 Measures (2) The diameter of each block of the mthorder partition of ℝn in (ℝn , ρ2 ) is √n2−m . (3) The mthorder partition of ℝn naturally generates mthorder partitions of ℝk for k = 1, . . . , n − 1. (4) Let A be a bounded set in (ℝn , ρ2 ) (n ∈ ℝ). Then, for each m ∈ ℤ+ , the collection of blocks Q(m) (k1 , . . . , kn ) in the mthorder partition of ℝn , which have at least one point in common with A is finite. Exercise 3.9. Verify. Definition 3.4 (Measure of a Partition Block). Let m ∈ ℤ+ , n ∈ ℕ. The measure (or volume) of a partition block Q := ∏ni=1 (ai , bi ] ∈ πn(m) is n μ(Q) := ∏(bi − ai ) = 2−mn . i=1 Remark 3.6. For n = 1, 2, 3, the prior definition is consistent with that of length, area, and volume, respectively. Definition 3.5 (Measure of Finite Union of Partition Blocks). Let m ∈ ℤ+ , n ∈ ℕ, and Q1 , . . . , Qk ∈ πn(m) (k ∈ ℕ) be distinct partition blocks. The measure of their union ⋃ki=1 Qi is naturally defined as follows: k k i=1 i=1 μ (⋃ Qi ) := ∑ μ(Qi ). In order to define Jordan measurability and measure for a bounded set A in (ℝn , ρ2 ), some groundwork is to be done first. Let A be a bounded set in (ℝn , ρ2 ) (n ∈ ℕ). For each m ∈ ℤ+ , let A(m) := ⋃ Q∈πn(m) , Q⊆A Q, A(m) := ⋃ Q, Q∈πn(m) , Q∩A=0̸ and ΔA(m) := A(m) \ A(m) = ⋃ Q ̸ Q∩Ac =0̸ Q∈πn(m) , Q∩A=0, (see Figure 3.1). Remarks 3.7. – If there does not exist a single block Q ∈ πn(m) such that Q ⊆ A, e. g., when A is a singleton, A(m) := 0. – If there does not exist a single block Q ∈ πn(m) such that Q ∩ A ≠ 0, which happens iff A = 0, A(m) := 0. Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 3.2 Measure  53 Figure 3.1: Geometric illustration for n = 2. The measures of A(m) , A(m) , and ΔA(m) are defined in the sense of Definition 3.5, with μ(0) := 0, and, in view of A(m) ⊆ A(m) , m ∈ ℤ+ , we have μ(A(m) ) ≤ μ(A(m) ) and μ(ΔA(m) ) = μ(A(m) ) − μ(A(m) ), m ∈ ℤ+ . As follows from the Properties of Partitions (Proposition 3.1 (1)) and the definition, A(m) ⊆ A(m+1) ⊆ A(m+1) ⊆ A(m) , m ∈ ℤ+ , where 0 ≤ μ(A(m) ) ≤ μ(A(m+1) ) ≤ μ(A(m+1) ) ≤ μ(A(m) ), m ∈ ℤ+ . Hence, (μ(A(m) ))m∈ℤ is an increasing sequence bounded above by μ(A(0) ) and + (μ(A(m) ))m∈ℤ is a decreasing sequence bounded below by μ(A(0) ), which, by the + Monotone Convergence Theorem, implies that both sequences converge and lim μ(A(m) ) = sup μ(A(m) ) and lim μ(A(m) ) = inf μ(A(m) ) m→∞ m→∞ m∈ℤ+ m∈ℤ+ and makes the following concepts to be welldefined. Definition 3.6 (Inner and Outer Measures of a Bounded Set). The inner measure of a bounded set A in (ℝn , ρ2 ) (n ∈ ℕ) is μ∗ (A) := lim μ(A(m) ) = sup μ(A(m) ). m→∞ m∈ℤ+ The outer measure of a bounded set A in (ℝn , ρ2 ) is μ∗ (A) := lim μ(A(m) ) = inf μ(A(m) ). m→∞ m∈ℤ+ Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 54  3 Measures Remark 3.8. μ∗ (A) ≤ μ∗ (A). Exercise 3.10. Explain. Definition 3.7 (Jordan Measurability and Measure of a Bounded Set). A bounded set A in (ℝn , ρ2 ) (n ∈ ℕ) is called Jordan measurable if μ∗ (A) = μ∗ (A), in which case the number μ(A) := μ∗ (A) = μ∗ (A) is called the (ndimensional) Jordan measure of A. Examples 3.3. 1. A partition block Q ∈ πn(m) (m ∈ ℤ+ , n ∈ ℕ) is Jordan measurable and its Jordan measure is precisely the measure in the sense of Definition 3.4. 2. A block Q := ∏ni=1 (ai , bi ] ⊂ ℝn (n ∈ ℕ, −∞ < ai < bi < ∞, i = 1, . . . , n) is Jordan measurable and n n i=1 i=1 μ (∏(ai , bi ]) = ∏(bi − ai ). 3. A singleton {x} in ℝn (n ∈ ℕ) is Jordan measurable and μ({x}) = 0. 4. The set T := {(x1 , x2 ) ∈ ℝ2 x1 ≥ 0, x2 ≥ 0, x1 + x2 ≤ 1} is Jordan measurable and μ(T) = 1/2. 5. The sets F1 := [0, 1] ∩ ℚ and F2 := [0, 1]2 ∩ ℚ2 are not Jordan measurable since μ∗ (F1 ) = μ∗ (F2 ) = 0 ≠ 1 = μ∗ (F2 ) = μ∗ (F1 ). 6. Unbounded sets in (ℝn , ρ2 ) (n ∈ ℕ) are not Jordan measurable. Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 3.2 Measure  55 Exercise 3.11. Verify 1–5 (for 2, the case of n = 1 only). The following statement provides a characterization of Jordan measurability. Proposition 3.2 (Characterization of Jordan Measurability). A bounded set A in (ℝn , ρ2 ) (n ∈ ℕ) is Jordan measurable iff lim μ(ΔA(m) ) = 0. m→∞ Exercise 3.12. Prove. Theorem 3.3 (Jordan Ring). The collection Jn of all Jordan measurable sets in ℝn (n ∈ ℕ) is a ring on ℝn . Proof. For any A, B ∈ Jn . Let us show that, for each m ∈ ℤ+ , Δ(A ∪ B)(m) ⊆ ΔA(m) ∪ ΔB(m) (3.4) Δ(A \ B)(m) ⊆ ΔA(m) ∪ ΔB(m) . (3.5) and Indeed, for each m ∈ ℤ+ and any partition block Q ∈ πn(m) with Q ⊆ Δ(A ∪ B)(m) , there exist x ∈ Q ∩ (A ∪ B) = (Q ∩ A) ∪ (Q ∩ B) and, in view of De Morgan’s laws, y ∈ Q ∩ (A ∪ B)c = Q ∩ (Ac ∩ Bc ) = (Q ∩ Ac ) ∩ (Q ∩ Bc ). If x ∈ Q ∩ A, then since y ∈ Q ∩ Ac , Q ⊆ ΔA(m) . If x ∈ Q ∩ B, then since y ∈ Q ∩ Bc , Q ⊆ ΔB(m) . Thus, in both cases, Q ⊆ ΔA(m) ∪ ΔB(m) , which proves inclusion (3.4). Further, for each m ∈ ℤ+ and any partition block Q ∈ πn(m) with Q ⊆ Δ(A \ B)(m) , there exist x ∈ Q ∩ (A \ B) = Q ∩ (A ∩ Bc ) = (Q ∩ A) ∩ (Q ∩ Bc ) Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 56  3 Measures and, in view of De Morgan’s laws, y ∈ Q ∩ (A \ B)c = Q ∩ (A ∩ Bc )c = Q ∩ (Ac ∪ B) = (Q ∩ Ac ) ∪ (Q ∩ B). If y ∈ Q ∩ Ac , then since x ∈ Q ∩ A, Q ⊆ ΔA(m) . If y ∈ Q ∩ B, then since x ∈ Q ∩ Bc , Q ⊆ ΔB(m) . Thus, in both cases, Q ⊆ ΔA(m) ∪ ΔB(m) , which proves inclusion (3.5). By inclusions (3.4) and (3.5), for each m ∈ ℤ+ , we have 0 ≤ μ(Δ(A ∪ B)(m) ) ≤ μ(ΔA(m) ) + μ(ΔB(m) ) and 0 ≤ μ(Δ(A \ B)(m) ) ≤ μ(ΔA(m) ) + μ(ΔB(m) ). Where by the Characterization of Jordan Measurability (Proposition 3.2) applied to A and B and the Squeeze Theorem, we infer that lim μ(Δ(A ∪ B)(m) ) = lim μ(Δ(A \ B)(m) ) = 0. m→∞ m→∞ Now, applying the Characterization of Jordan Measurability (Proposition 3.2) to A ∪ B and A \ B, we conclude that A ∪ B, A \ B ∈ Jn , which completes the proof. Theorem 3.4 (Jordan Measure). Let n ∈ ℕ. The set function Jn ∋ A → μ(A) (3.6) is a measure (i. e., nonnegative and σadditive) on the ring Jn of Jordan measurable sets in ℝn called the Jordan measure. Proof. The set function μ defined by (3.6) is, obviously, nonnegative and finite on Jn (see Definition 3.7). Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 3.2 Measure  57 It is also subadditive. Indeed, for arbitrary A, B ∈ Jn and any m ∈ ℤ+ , (A ∪ B)(m) ⊆ A ∪ B ⊆ A(m) ∪ B(m) , and hence, μ((A ∪ B)(m) ) ≤ μ(A(m) ) + μ(B(m) ). that The latter, by the definition of the Jordan measure of a set (Definition 3.7), implies μ(A ∪ B) = lim μ((A ∪ B)(m) ) ≤ lim μ(A(m) ) + lim μ(B(m) ) = μ(A) + μ(B). m→∞ m→∞ m→∞ Now, to prove that μ is additive, it suffices to show that, for arbitrary disjoint A, B ∈ Jn , μ(A ∪ B) ≥ μ(A) + μ(B). Since, for any m ∈ ℤ+ , A(m) ⊆ A, B(m) ⊆ B, and A(m) ∪ B(m) ⊆ (A ∪ B)(m) , the disjointness of A and B implies disjointness for A(m) and B(m) , m ∈ ℤ+ , and hence, μ(A(m) ) + μ(B(m) ) = μ(A(m) ∪ B(m) ) ≤ μ((A ∪ B)(m) ), m ∈ ℤ+ . Where by the definition of the Jordan measure of a set (Definition 3.7), μ(A) + μ(B) = lim μ(A(m) ) + lim μ(B(m) ) ≤ lim μ((A ∪ B)(m) ) = μ(A ∪ B). m→∞ m→∞ m→∞ Thus, μ is additive on Jn . Being a finite, nonnegative, and additive function on a ring, the Jordan measure possesses properties (1) and (3)–(6) of the Properties of the Measure Theorem (Theorem 3.1) (see Remarks 3.4), and hence, in particular, is monotone. By the Characterization of Measure on a Ring (Proposition 3.3) (see Section 3.3, Problem 3) to prove the σadditivity for μ, it suffices to show that μ is σsubadditive. Let (Ak )k∈ℕ be a sequence in Jn such that ⋃∞ i=1 Ai ∈ Jn and let ε > 0 be arbitrary. By the construct of the Jordan measure of a set, for each k ∈ ℕ, there exist a closed set Fk ∈ Jn and open set Gk ∈ Jn and such that Fk ⊆ Ak ⊆ Gk and μ(Gk ) − μ(Fk ) < ε 2k (see the Approximation of Jordan Measurable Sets Proposition (Proposition 3.6), Section 3.3, Problem 11), and hence, by the monotonicity of μ, μ(Gk ) < μ(Fk ) + ε ε ≤ μ(Ak ) + k , k ∈ ℕ. 2k 2 (3.7) Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 58  3 Measures Similarly, there exist a closed set F ∈ Jn and open set G ∈ Jn and such that ∞ F ⊆ ⋃ Ai ⊆ G and μ(G) − μ(F) < ε, i=1 and hence, by the monotonicity of μ, ∞ μ (⋃ Ai ) ≤ μ(G) < μ(F) + ε. i=1 (3.8) Since ∞ ∞ i=1 i=1 F ⊆ ⋃ A i ⊆ ⋃ Gi , the sets of the sequence (Gn )n∈ℕ form an open cover of the closed and bounded, and hence, by the Heine–Borel Theorem (Theorem 1.13), compact in (ℝn , ρ2 ) set F. Therefore, there is a finite subcover, i. e., N ∃ N ∈ ℕ : F ⊆ ⋃ Gk . k=1 Where by the monotonicity and subadditivity of μ, we infer that N N k=1 k=1 μ(F) ≤ μ ( ⋃ Gk ) ≤ ∑ μ(Gk ), which, in view of (3.7), implies N μ(F) < ∑ [μ(Ak ) + k=1 ∞ ∞ ∞ ε ε ] < μ(A ) + = μ(Ak ) + ε. ∑ ∑ ∑ k k 2k k=1 k=1 2 k=1 By (3.8), ∞ μ(A) < μ(F) + ε < ∑ μ(Ak ) + 2ε. k=1 Since ε > 0 is arbitrary, passing to the limit as ε → 0+, we arrive at ∞ μ(A) ≤ ∑ μ(Ak ), k=1 and hence, μ is σsubadditive. Being nonnegative, additive, and σsubadditive on the ring Jn , by the Characterization of Measure on a Ring (Proposition 3.3) (see Section 3.3, Problem 3), μ is a measure on Jn . Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 3.2 Measure  59 Examples 3.4. 1. A finite set F in ℝn (n ∈ ℕ) is Jordan measurable and μ(F) = 0. 2. A countably infinite set C := {xn }n∈ℕ in ℝn (n ∈ ℕ) such that there exists lim x n→∞ n = x ∈ ℝn is Jordan measurable and μ(C) = 0. In particular, this applies to the set {1/n}n∈ℕ . Exercise 3.13. (a) Verify. (b) Give two examples (bounded and unbounded) showing that the countable union of Jordan measurable sets need not be Jordan measurable, i. e., that Jn is not a σring. Corollary 3.1. Let S1 := {(a, b]  −∞ < a < b < ∞} ∪ {0}. Then μ(0) := 0, μ((a, b]) := b − a, (a, b] ∈ S1 , is a measure on S1 . Corollary 3.2. Let S2 := {(a1 , b1 ] × (a2 , b2 ]  −∞ < ai < bi < ∞, i = 1, 2} ∪ {0}. Then μ(0) := 0, μ((a1 , b1 ] × (a2 , b2 ]) := (b1 − a1 )(b2 − a2 ), (a1 , b1 ] × (a2 , b2 ] ∈ S2 , is a measure on S2 . Exercise 3.14. Prove the two corollaries (cf. Examples 3.3 and Example 3.1). 3.2.4.2 Lebesgue–Stieltjes Measures on Interval Semiring Here, we introduce Lebesgue–Stieltjes3 measures, for the time being, on the interval semiring S1 := {(a, b]  −∞ < a < b < ∞} ∪ {0} 3 Thomas Joannes Stieltjes (1856–1894). Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 60  3 Measures (see Examples 2.1) as follows: λF (0) := 0, λF ((a, b]) := F(b) − F(a), (a, b] ∈ S1 , (3.9) where a function F : ℝ → ℝ is increasing: F(a) ≤ F(b), −∞ < a < b < ∞ and rightcontinuous: F(x0 ) = lim F(x), x0 ∈ ℝ, x→x0 + function. Exercise 3.15. Give three examples of such a function F. Theorem 3.5 (Lebesgue–Stieltjes Measures on Interval Semiring). For an increasing and rightcontinuous function F : ℝ → ℝ, the set function λF defined by (3.9) is a measure on the interval semiring S1 called the Lebesgue–Stieltjes measure on S1 associated with the function F. Proof. Since the function F is increasing, λF is nonnegative. As is easily verified, λF is also additive and subadditive. Exercise 3.16. Verify. To prove that λF is σadditive, consider an arbitrary pairwise disjoint sequence ((an , bn ])n∈ℕ in S1 such that ∞ ⋃(ai , bi ] = (a, b] ∈ S1 . i=1 Since S1 is a semiring, it can be shown inductively that, for each N ∈ ℕ, N l(N) i=1 k=1 (a, b] \ ⋃(ai , bi ] = ⋃ Ck(N) with some l(N) ∈ ℕ and pairwise disjoint Ck(N) ∈ S1 , k = 1, . . . , l(N). Exercise 3.17. Show. Where, by the additivity of λF , for each N ∈ ℕ, N l(N) i=1 k=1 λF ((a, b]) = ∑ λF ((ai , bi ]) + ∑ λF (Ck(N) ) , which implies that N λF ((a, b]) ≥ ∑ λF ((ai , bi ]) , N ∈ ℕ, i=1 Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 3.2 Measure  61 where, passing to the limit as N → ∞, we arrive at ∞ λF ((a, b]) ≥ ∑ λF ((ai , bi ]) . (3.10) i=1 Since the function F is rightcontinuous, ∀ ε > 0 ∃ a ∈ (a, b) : F(a ) − F(a) < ε, and hence, λF ((a, b]) − λF ((a , b]) = F(b) − F(a) − [F(b) − F(a )] = F(a ) − F(a) < ε; (3.11) also, ∀ n ∈ ℕ ∃ bn > bn : F(bn ) − F(bn ) < ε , 2n and hence, for each N ∈ ℕ, λF ((an , bn ]) − λF ((an , bn ]) = F(bn ) − F(an ) − [F(bn ) − F(an )] ε = F(bn ) − F(bn ) < n . 2 (3.12) Since ∞ ∞ i=1 i=1 [a , b] ⊂ (a, b] = ⋃(ai , bi ] ⊆ ⋃(ai , bi ), the collection {(an , bn )}n∈ℕ is an open cover of the compact in ℝ set [a , b], and hence, there is a finite subcover {(an , bn )}n=1,...,N (N ∈ ℕ), which implies that N N i=1 i=1 (a , b] ⊂ [a , b] ⊆ ⋃(ai , bi ) ⊆ ⋃(ai , bi ] (cf. the proof of Theorem 3.4). By the subadditivity of λF , N ∞ i=1 i=1 λF ((a , b]) ≤ ∑ λF ((ai , bi ]) ≤ ∑ λF ((ai , bi ]). Where, by (3.11) and (3.12), ∞ λF ((a, b]) ≤ λF ((a , b]) + ε ≤ ∑ [λF ((ai , bi ]) + i=1 ∞ ε λF ((ai , bi ]) + 2ε. ] + ε = ∑ 2i i=1 Brought to you by  University of Groningen Authenticated Download Date  8/4/19 2:25 PM 62  3 Measures Since ε > 0 is arbitrary, passing to the limit as ε → 0+, we arrive at ∞ λF ((a, b]) ≤ ∑ λF ((ai , bi ]) . i=1 (3.13) Inequalities (3.10) and (3.13) jointly imply that ∞ λF ((a, b]) = ∑ λF ((ai , bi ]), i=1 which shows that λF is σadditive on S1 and completes the proof. Remarks 3.9. – Of special importance is the p