Principal Real Analysis: Measure and Integration (De Gruyter Textbook)

Real Analysis: Measure and Integration (De Gruyter Textbook)

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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 well-designed text is ideal for a one-semester Master's level graduate course on real analysis with emphasis on the measure and integration theory for students majoring in mathematics, physics, computer science, and engineering. A concise but profound and detailed presentation of the basics of real analysis with emphasis on the measure and integration theory. Designed for a one-semester graduate course, with plethora of examples, problems, and exercises. Is of interest to students and instructors in mathematics, physics, computer science, and engineering. Prepares the students for more advanced courses in functional analysis and operator theory. ContentsPreliminariesBasic Set ClassesMeasuresExtension of MeasuresMeasurable FunctionsAbstract Lebesgue IntegralLp SpacesDifferentiation and IntegrationSigned MeasuresThe Axiom of Choice and Equivalents
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Marat V. Markin
Real Analysis

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Marat V. Markin

Real Analysis
|

Measure and Integration

Mathematics Subject Classification 2010
28-01, 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 978-3-11-060097-1
e-ISBN (PDF) 978-3-11-060099-5
e-ISBN (EPUB) 978-3-11-059882-7
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 one-semester 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 seventy-five-minute 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
proof-based 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 p-norm, Lp spaces, and the basics of normed vector spaces. It further demonstrates the deficiencies of the Riemann
https://doi.org/10.1515/9783110600995-201

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 one-semester 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 Well-Ordering Principle),
and ordered sets, which is fundamental for proving the famed Vitali Theorem on the
existence of a non-Lebesgue 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, Semi-algebras | 25
Rings, Algebras | 26
σ-Rings, σ-Algebras | 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 Non-Lebesgue 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 Σ-Σ󸀠 -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
p-Norms | 215
Hölder’s Inequality | 217
Minkowski’s Inequality | 219
Convergence in p-Norm | 221
Definitions and Examples | 221
Convergence in ∞-Norm | 224
Uniqueness A. E. of Limit in p-Norm | 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 n-spaces of all ordered n-tuples 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/10.1515/9783110600995-001

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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 .

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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 well-defined 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

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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
one-to-one 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.

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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).

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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.

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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.

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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→∞

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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 n-space, ℝn or ℂn , is a metric space
relative to p-metric,
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).

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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

(p-summable/bounded sequences, respectively) is a metric space relative to p-metric
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/complex-valued functions continuous on [a, b] (−∞ <
a < b < ∞) is a metric space relative to p-metric
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.

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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).

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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 Augustin-Louis Cauchy (1789–1857).

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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).

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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 ε-ball 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.

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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.

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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.

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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.

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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.

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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 n-tuples
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

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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 ε-balls
{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).

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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 1/2-balls
{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).

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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 ;

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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.

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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, Semi-algebras
Definition 2.1 (Semiring, Semi-algebra). 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 semi-algebra 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 semi-algebra.
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 semi-algebras.
https://doi.org/10.1515/9783110600995-002

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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 semi-algebras 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 semi-algebras 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 semi-algebras 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

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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/semi-algebra, 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}

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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 semi-algebras 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.

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2.3 σ-Rings, σ-Algebras

| 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 semi-algebras 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 σ-Rings, σ-Algebras
Definition 2.4 (σ-Ring, σ-Algebra). Let X be a nonempty set. A σ-ring (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 σ-algebra Σ (of sets) on X is a σ-ring on X such that X ∈ Σ.

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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 σ-ring 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 σ-ring/σ-algebra is a ring/algebra, respectively, but not vice versa.
Hence, in particular, each σ-algebra Σ is closed under complements, i. e.,
if A ∈ Σ, then Ac ∈ Σ.

–

Each ring/algebra consisting of finitely many elements is a σ-ring/σ-algebra, respectively.

Exercise 2.4. Verify.
Examples 2.3.
1. All finite algebras from Examples 2.2 are σ-algebras.
2. On an arbitrary nonempty set X,
(a) the power set P (X) and {0, X} are σ-algebras and {0} is a σ-ring but not a
σ-algebra;
(b)
󵄨
Σ := {A ⊆ X 󵄨󵄨󵄨 A or Ac is countable}
is a σ-algebra;
(c) provided X is infinite,
c

A := {A ⊆ X 󵄨󵄨󵄨 A or A is finite}

󵄨

is an algebra, but not a σ-algebra and
R := {A ⊆ X | A is finite}

is a ring, but not a σ-ring 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 σ-algebra of cardinality 2ℵ0 = c (cf. Section 2.6, Problem 8).

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2.4 Monotone Classes |

3.

31

An example of a σ-ring, which is not a σ-algebra, can be obtained by taking an
arbitrary σ-algebra 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 σ-algebra on [0, 1) but
only a σ-ring on [0, 1].

Exercise 2.5. Verify.
A σ-algebra can be equivalently defined as follows.
Definition 2.5 (σ-Algebra: Alternative Definition). Let X be a nonempty set. A σ-algebra
(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 σ-ring, in particular a σ-algebra, 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.

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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 σ-ring.
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 σ-ring (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 σ-ring.
As an important particular case, we obtain the following.
Corollary 2.1 (Monotone Algebra Theorem). A monotone algebra is a σ-algebra.

2.5 Generated Set Classes
Here, we introduce the important notion of the ring, algebra, σ-ring, σ-algebra, and
monotone class “generated” by a nonempty collection C of subsets of a nonempty
set X. To proceed, we need the following lemma.

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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, σ-rings, σ-algebras, or monotone classes on a nonempty set X is a
ring, algebra, σ-ring, σ-algebra, 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 σ-rings, σ-algebras, 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).

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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 σ-ring, C ⊆R R is the smallest σ-ring containing C called the σ-ring
generated by C .
– σa(C ) := ⋂Σ is a σ-algebra, C ⊆Σ Σ is the smallest σ-algebra containing C called the
σ-algebra 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 well-defined.
(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 well-defined (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

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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.

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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

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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 )} .

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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 ).

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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 σ-ring 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 σ-Algebra). Let (X, ρ) be a metric space and G be the metric topology generated by metric ρ (see Definition 1.20).
The σ-algebra of Borel sets or the Borel σ-algebra on X is the σ-algebra 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).

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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 .

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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 σ-ring 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 σ-algebra on ℝ2 .
8. * Does there exist a countably infinite σ-algebra?
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.

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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, σ-rings,
σ-algebras, 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
 1 ∩ ⋅ ⋅ ⋅ ∩  n ,
where  i = Ai or  i = Aci , i = 1, . . . , n.

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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 σ-subadditive), if for any sequence (An )n∈ℕ in C with
⋃∞
i=1 Ai ∈ C ,
https://doi.org/10.1515/9783110600995-003

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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 σ-additive), 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 σ-finite), 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 σ-finite 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 σ-additive, then μ(0) = 0
and μ is additive.

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3.2 Measure | 45

3.2 Measure
3.2.1 Definition and Examples
Definition 3.2 (Measure). A measure is a nonnegative σ-additive function on a semiring.
Remark 3.2. In particular, a nonnegative σ-additive function on a ring, an algebra, a
σ-ring, or a σ-algebra 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 };

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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 σ-subadditive.
Proof. Properties (1) and (2) (additivity) follow from a more general statement (see Exercise 3.1 (b)).

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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 .

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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 σ-additivity 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) (σ-subadditivity) holds as well.
Remarks 3.4.
– In particular, the Properties of Measure (Theorem 3.1) hold for a measure on an
algebra, a σ-ring, or a σ-algebra (see Remarks 2.2 and 2.3).

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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 σ-additivity. 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

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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 σ-additivity 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.

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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 σ-algebra.
Definition 3.3 (Partitions of ℝn ). The mth-order partition of the n-space 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 0th-order and 1st-order partitions of ℝ and ℝ2 .
Proposition 3.1 (Properties of Partitions). Let n ∈ ℕ.
(1) For each m ∈ ℤ+ , the (m + 1)th-order partition of ℝn is obtained from the blocks of
the mth-order partition of ℝn via partitioning the latter into 2n congruent blocks;
1 Camille Jordan (1838–1922).
2 Henri Lebesgue (1875–1941).

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52 | 3 Measures
(2) The diameter of each block of the mth-order partition of ℝn in (ℝn , ρ2 ) is
√n2−m .
(3) The mth-order partition of ℝn naturally generates mth-order 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 mth-order 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.

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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 well-defined.
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+

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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 (n-dimensional) 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.

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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 )

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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 σ-additive) 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).

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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 σ-additivity for μ, it suffices to show that μ is σ-subadditive.
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)

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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 σ-subadditive.
Being nonnegative, additive, and σ-subadditive 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 .

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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
σ-ring.
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).

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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 right-continuous:
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 right-continuous 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 σ-additive, 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

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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 right-continuous,
∀ 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 ∃ b󸀠n > bn : F(b󸀠n ) − F(bn ) <

ε
,
2n

and hence, for each N ∈ ℕ,
λF ((an , b󸀠n ]) − λF ((an , bn ]) = F(b󸀠n ) − F(an ) − [F(bn ) − F(an )]
ε
= F(b󸀠n ) − F(bn ) < n .
2

(3.12)

Since
∞

∞

i=1

i=1

[a󸀠 , b] ⊂ (a, b] = ⋃(ai , bi ] ⊆ ⋃(ai , b󸀠i ),
the collection {(an , b󸀠n )}n∈ℕ is an open cover of the compact in ℝ set [a󸀠 , b], and hence,
there is a finite subcover {(an , b󸀠n )}n=1,...,N (N ∈ ℕ), which implies that
N

N

i=1

i=1

(a󸀠 , b] ⊂ [a󸀠 , b] ⊆ ⋃(ai , b󸀠i ) ⊆ ⋃(ai , b󸀠i ]
(cf. the proof of Theorem 3.4).
By the subadditivity of λF ,
N

∞

i=1

i=1

λF ((a󸀠 , b]) ≤ ∑ λF ((ai , b󸀠i ]) ≤ ∑ λF ((ai , b󸀠i ]).
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

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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 σ-additive on S1 and completes the proof.
Remarks 3.9.
– Of special importance is the p