Measure, Probability, and Mathematical Finance E-Book

Guojun Gan

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2014

Beschreibung

An introduction to the mathematical theory and financial models developed and used on Wall Street

Providing both a theoretical and practical approach to the underlying mathematical theory behind financial models, Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach presents important concepts and results in measure theory, probability theory, stochastic processes, and stochastic calculus. Measure theory is indispensable to the rigorous development of probability theory and is also necessary to properly address martingale measures, the change of numeraire theory, and LIBOR market models. In addition, probability theory is presented to facilitate the development of stochastic processes, including martingales and Brownian motions, while stochastic processes and stochastic calculus are discussed to model asset prices and develop derivative pricing models.

The authors promote a problem-solving approach when applying mathematics in real-world situations, and readers are encouraged to address theorems and problems with mathematical rigor. In addition, Measure, Probability, and Mathematical Finance features:

• A comprehensive list of concepts and theorems from measure theory, probability theory, stochastic processes, and stochastic calculus
• Over 500 problems with hints and select solutions to reinforce basic concepts and important theorems
• Classic derivative pricing models in mathematical finance that have been developed and published since the seminal work of Black and Scholes 

Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach is an ideal textbook for introductory quantitative courses in business, economics, and mathematical finance at the upper-undergraduate and graduate levels. The book is also a useful reference for readers who need to build their mathematical skills in order to better understand the mathematical theory of derivative pricing models.

Leseprobe

Contents

Cover

Half Title page

Title page

Copyright page

Dedication

Preface

Financial Glossary

Part I: Measure Theory

Chapter 1: Sets and Sequences

1.1 Basic Concepts and Facts

1.2 Problems

1.3 Hints

1.4 Solutions

1.5 Bibliographic Notes

Chapter 2: Measures

2.1 Basic Concepts and Facts

2.2 Problems

2.3 Hints

2.4 Solutions

2.5 Bibliographic Notes

Chapter 3: Extension of Measures

3.1 Basic Concepts and Facts

3.2 Problems

3.3 Hints

3.4 Solutions

3.5 Bibliographic Notes

Chapter 4: Lebesgue-Stieltjes Measures

4.1 Basic Concepts and Facts

4.2 Problems

4.3 Hints

4.4 Solutions

4.5 Bibliographic Notes

Chapter 5: Measurable Functions

5.1 Basic Concepts and Facts

5.2 Problems

5.3 Hints

5.4 Solutions

5.5 Bibliographic Notes

Chapter 6: Lebesgue Integration

6.1 Basic Concepts and Facts

6.2 Problems

6.3 Hints

6.4 Solutions

6.5 Bibliographic Notes

Chapter 7: The Radon-Nikodym Theorem

7.1 Basic Concepts and Facts

7.2 Problems

7.3 Hints

7.4 Solutions

7.5 Bibliographic Notes

Chapter 8: LP Spaces

8.1 Basic Concepts and Facts

8.2 Problems

8.3 Hints

8.4 Solutions

8.5 Bibliographic Notes

Chapter 9: Convergence

9.1 Basic Concepts and Facts

9.2 Problems

9.3 Hints

9.4 Solutions

9.5 Bibliographic Notes

Chapter 10: Product Measures

10.1 Basic Concepts and Facts

10.2 Problems

10.3 Hints

10.4 Solutions

10.5 Bibliographic Notes

Part II: Probability Theory

Chapter 11: Events and Random Variables

11.1 Basic Concepts and Facts

11.2 Problems

11.3 Hints

11.4 Solutions

11.5 Bibliographic Notes

Chapter 12: Independence

12.1 Basic Concepts and Facts

12.2 Problems

12.3 Hints

12.4 Solutions

12.5 Bibliographic Notes

Chapter 13: Expectation

13.1 Basic Concepts and Facts

13.2 Problems

13.3 Hints

13.4 Solutions

13.5 Bibliographic Notes

Chapter 14: Conditional Expectation

14.1 Basic Concepts and Facts

14.2 Problems

14.3 Hints

14.4 Solutions

14.5 Bibliographic Notes

Chapter 15: Inequalities

15.1 Basic Concepts and Facts

15.2 Problems

15.3 Hints

15.4 Solutions

15.5 Bibliographic Notes

Chapter 16: Law of Large Numbers

16.1 Basic Concepts and Facts

16.2 Problems

16.3 Hints

16.4 Solutions

16.5 Bibliographic Notes

Chapter 17: Characteristic Functions

17.1 Basic Concepts and Facts

17.2 Problems

17.3 Hints

17.4 Solutions

17.5 Bibliographic Notes

Chapter 18: Discrete Distributions

18.1 Basic Concepts and Facts

18.2 Problems

18.3 Hints

18.4 Solutions

18.5 Bibliographic Notes

Chapter 19: Continuous Distributions

19.1 Basic Concepts and Facts

19.2 Problems

19.3 Hints

19.4 Solutions

19.5 Bibliographic Notes

Chapter 20: Central Limit Theorems

20.1 Basic Concepts and Facts

20.2 Problems

20.3 Hints

20.4 Solutions

20.5 Bibliographic Notes

Part III: Stochastic Processes

Chapter 21: Stochastic Processes

21.1 Basic Concepts and Facts

21.2 Problems

21.3 Hints

21.4 Solutions

21.5 Bibliographic Notes

Chapter 22: Martingales

22.1 Basic Concepts and Facts

22.2 Problems

22.3 Hints

22.4 Solutions

22.5 Bibliographic Notes

Chapter 23: Stopping Times

23.1 Basic Concepts and Facts

23.2 Problems

23.3 Hints

23.4 Solutions

23.5 Bibliographic Notes

Chapter 24: Martingale Inequalities

24.1 Basic Concepts and Facts

24.2 Problems

24.3 Hints

24.4 Solutions

24.5 Bibliographic Notes

Chapter 25: Martingale Convergence Theorems

25.1 Basic Concepts and Facts

25.2 Problems

25.3 Hints

25.4 Solutions

25.5 Bibliographic Notes

Chapter 26: Random Walks

26.1 Basic Concepts and Facts

26.2 Problems

26.3 Hints

26.4 Solutions

26.5 Bibliographic Notes

Chapter 27: Poisson Processes

27.1 Basic Concepts and Facts

27.2 Problems

27.3 Hints

27.4 Solutions

27.5 Bibliographic Notes

Chapter 28: Brownian Motion

28.1 Basic Concepts and Facts

28.2 Problems

28.3 Hints

28.4 Solutions

28.5 Bibliographic Notes

Chapter 29: Markov Processes

29.1 Basic Concepts and Facts

29.2 Problems

29.3 Hints

29.4 Solutions

29.5 Bibliographic Notes

Chapter 30: Lévy Processes

30.1 Basic Concepts and Facts

30.2 Problems

30.3 Hints

30.4 Solutions

30.5 Bibliographic Notes

Part IV: Stochastic Calculus

Chapter 31: The Wiener Integral

31.1 Basic Concepts and Facts

31.2 Problems

31.3 Hints

31.4 Solutions

31.5 Bibliographic Notes

Chapter 32: The Itô Integral

32.1 Basic Concepts and Facts

32.2 Problems

32.3 Hints

32.4 Solutions

32.5 Bibliographic Notes

Chapter 33: Extension of the Itô Integral

33.1 Basic Concepts and Facts

33.2 Problems

33.3 Hints

33.4 Solutions

33.5 Bibliographic Notes

Chapter 34: Martingale Stochastic Integrals

34.1 Basic Concepts and Facts

34.2 Problems

34.3 Hints

34.4 Solutions

34.5 Bibliographic Notes

Chapter 35: The Itô Formula

35.1 Basic Concepts and Facts

35.2 Problems

35.3 Hints

35.4 Solutions

35.5 Bibliographic Notes

Chapter 36: Martingale Representation Theorem

36.1 Basic Concepts and Facts

36.2 Problems

36.3 Hints

36.4 Solutions

36.5 Bibliographic Notes

Chapter 37: Change of Measure

37.1 Basic Concepts and Facts

37.2 Problems

37.3 Hints

37.4 Solutions

37.5 Bibliographic Notes

Chapter 38: Stochastic Differential Equations

38.1 Basic Concepts and Facts

38.2 Problems

38.3 Hints

38.4 Solutions

38.5 Bibliographic Notes

Chapter 39: Diffusion

39.1 Basic Concepts and Facts

39.2 Problems

39.3 Hints

39.4 Solutions

39.5 Bibliographic Notes

Chapter 40: The Feynman-Kac Formula

40.1 Basic Concepts and Facts

40.2 Problems

40.3 Hints

40.4 Solutions

40.5 Bibliographic Notes

Part V: Stochastic Financial Models

Chapter 41: Discrete-Time Models

41.1 Basic Concepts and Facts

41.2 Problems

41.3 Hints

41.4 Solutions

41.5 Bibliographic Notes

Chapter 42: Black-Scholes Option Pricing Models

42.1 Basic Concepts and Facts

42.2 Problems

42.3 Hints

42.4 Solutions

42.5 Bibliographic Notes

Chapter 43: Path-Dependent Options

43.1 Basic Concepts and Facts

43.2 Problems

43.3 Hints

43.4 Solutions

43.5 Bibliographic Notes

Chapter 44: American Options

44.1 Basic Concepts and Facts

44.2 Problems

44.3 Hints

44.4 Solutions

44.5 Bibliographic Notes

Chapter 45: Short Rate Models

45.1 Basic Concepts and Facts

45.2 Problems

45.3 Hints

45.4 Solutions

45.5 Bibliographic Notes

Chapter 46: Instantaneous Forward Rate Models

46.1 Basic Concepts and Facts

46.2 Problems

46.3 Hints

46.4 Solutions

46.5 Bibliographic Notes

Chapter 47: Libor Market Models

47.1 Basic Concepts and Facts

47.2 Problems

47.3 Hints

47.4 Solutions

47.5 Bibliographic Notes

References

List of Symbols

Subject Index

MEASURE, PROBABILITY, AND MATHEMATICAL FINANCE

Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data is available.

Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

Guojun Gan, Chaoqun Ma, and Hong Xie

   ISBN 978-1-118-83196-0

To my parents–Guojun Gan

To my wife and my daughter–Chaoqun Ma

To my family and friends–Hong Xie

PREFACE

Mathematical finance, a new branch of mathematics concerned with financial markets, is experiencing rapid growth. During the last three decades, many books and papers in the area of mathematical finance have been published. However, understanding the literature requires that the reader have a good background in measure-theoretic probability, stochastic processes, and stochastic calculus. The purpose of this book is to provide the reader with an introduction to the mathematical theory underlying the financial models being used and developed on Wall Street. To this end, this book covers important concepts and results in measure theory, probability theory, stochastic processes, and stochastic calculus so that the reader will be in a position to understand these financial models. Problems as well as solutions are included to help the reader learn the concepts and results quickly.

In this book, we adopted the definitions and theorems from various books and presented them in a mathematically rigorous way. We tried to cover the most of the basic concepts and the important theorems. We selected the problems in this book in such a way that the problems will help readers understand and know how to apply the concepts and theorems. This book includes 516 problems, most of which are not difficult and can be solved by applying the definitions, theorems, and the results of previous problems.

This book is organized into five parts, each of which is further organized into several chapters. Each chapter is divided into five sections. The first section presents the definitions of important concepts and theorems. The second, third, and fourth sections present the problems, hints on how to solve the problems, and the full solutions to the problems, respectively. The last section contains bibliographic notes. Interdependencies between all chapters are shown in Table 0.1.

Table 0.1: Interdependencies between Chapters.

Chapter

Related to Chapter(s)

1. Sets and Sequences

2. Measures

1

3. Extension of Measures

1;2

4. Lebesgue-Stieltjes Measures

2;3

5. Measurable Functions

2

6. Lebesgue Integration

1;2;5

7. The Radon-Nikodym Theorem

2;6

8.

L

p

Spaces

2;6

9. Convergence

1;2;6;8

10. Product Measures

2;3;5;6

11. Events and Random Variables

1;2;4;5

12. Independence

2;3;5;11

13. Expectation

2;6;8;10;11;12

14. Conditional Expectation

1;2;5;6;7;8;10;11;12;13

15. Inequalities

8;11;14

16. Law of Large Numbers

2;8;9;10;12;13;15

17. Characteristic Functions

5;6;8;11;12;13;15

18. Discrete Distributions

12;14;17

19. Continuous Distributions

6;10;12;13;17

20. Central Limit Theorems

6;9;11

21. Stochastic Processes

2;5;10;11;12;19

22. Martingales

2;5;11;13;14;15

23. Stopping Times

2;5;9;11;14;21;22

24. Martingale Inequalities

2;6;8;13;14;15;23

25. Martingale Convergence Theorems

1;6;9;11;14;15;22

26. Random Walks

8;9;13;14;15;19;20;22;23;24

27. Poisson Processes

11;12;14;17;21;22

28. Brownian Motion

8;9;11;12;14;15;16;17;19

29. Markov Processes

2;6;11;14;21

30. Lévy Processes

1;5;6;11;12;14;17;19;22;27;28;29

31. The Wiener Integral

6;9;15;19;28

32. The Itô Integral

5;6;8;10;14;15;22;24;28

33. Extension of the Itô Integrals

9;10;14;22;23;32

34. Martingale Stochastic Integrals

14;15;19;27;32

35. The Itô Formula

6;8;9;22;24;32;34

36. Martingale Representation Theorem

9;14;25;28;32;33;35

37. Change of Measure

7;14;32;34;35

38. Stochastic Differential Equations

8;11;13;32;34;35

39. Diffusion

6;9;11;14;19;21;24;32;35;38

40. The Feynman-Kac Formula

6;14;32;35;38;39

41. Discrete-Time Models

7;12;14;22;23

42. Black-Scholes Option Pricing Models

9;14;19;24;32;33;35;36;37;38;41

43. Path-Dependent Options

10; 14;19;28;37;38;42

44. American Options

14; 15;21;22;23;32;35;36;37;42;43

45. Short Rate Models

11; 14;19;29;32;35;37;38;39;40

46. Instantaneous Forward Rate Models

10; 14;19;32;34;35;37;38;40;45

47. LIBOR Market Models

14; 32;37;45;46

In Part I, we present measure theory, which is indispensable to the rigorous development of probability theory. Measure theory is also necessary for us to discuss recently developed theories and models in finance, such as the martingale measures, the change of numeraire theory, and the London interbank offered rate (LIBOR) market models.

In Part II, we present probability theory in a measure-theoretic mathematical framework, which was introduced by A.N. Kolmogorov in 1937 in order to deal with David Hilbert’s sixth problem. The material presented in this part was selected to facilitate the development of stochastic processes in Part III.

In Part III, we present stochastic processes, which include martingales and Brownian motion. In Part IV, we discuss stochastic calculus. Both stochastic processes and stochastic calculus are important to modern mathematical finance as they are used to model asset prices and develop derivative pricing models.

In Part V, we present some classic models in mathematical finance. Many pricing models have been developed and published since the seminal work of Black and Scholes. This part covers only a small portion of many models.

In this book, we tried to use a uniform set of symbols and notation. For example, we used N, R, and to denote the set of natural numbers (i.e., nonnegative integers), the set of real numbers, and the empty set, respectively. A comprehensive list of symbols is also provided at the end of this book.

We have taken great pains to ensure the accuracy of the formulas and statements in this book. However, a few errors are inevitable in almost every book of this size. Please feel free to contact us if you spot errors or have any other constructive suggestions.

How to Use This Book

This book can be used by individuals in various ways:

(a) It can be used as a self-study book on mathematical finance. The prerequisite is linear algebra and calculus at the undergraduate level. This book will provide you with a series of concepts, facts, and problems. You should explore each problem and write out your solution in such a way that it can be shared with others. By doing this you will be able to actively develop an in-depth and comprehensive understanding of the concepts and principles that cannot be archived by passively reading or listening to comments of others.

(b) It can be used as a reference book. This book contains the most important concepts and theorems from mathematical finance. The reader can find the definition of a concept or the statement of a theorem in the book through the index at the end of this book.

(c) It can be used as a supplementary book for individuals who take advanced courses in mathematical finance. This book starts with measure theory and builds up to stochastic financial models. It provides necessary prerequisites for students who take advanced courses in mathematical finance without completing background courses.

Acknowledgments

We would like to thank all the academics and practitioners who have contributed to the knowledge of mathematical finance. In particular, we would like to thank the following academics and practitioners whose work constitutes the backbone of this book: Robert B. Ash, Krishna B. Athreya, Rabi Bhattacharya, Patrick Billingsley, Tomas Björk, Fischer Sheffey Black, Kai Lai Chung, Erhan Çinlar, Catherine A. Doléans-Dade, Darrell Duffie, Richard Durrett, Robert J. Elliott, Damir Filipović, Allan Gut, John Hull, Ioannis Karatzas, Fima C. Klebaner, P. Ekkehard Kopp, Hui-Hsiung Kuo, Soumendra N. Lahiri, Damien Lamberton, Bernard Lapeyre, Gregory F. Lawler, Robert C. Merton, Marek Musiela, Bernt Oksendal, Andrea Pascucci, Jeffrey S. Rosenthal, Sheldon M. Ross, Marek Rutkowski, Myron Scholes, Steven Shreve, J. Michael Steele, and Edward C. Waymire.

We are grateful to Roman Naryshkin and several anonymous reviewers for their helpful comments. Guojun Gan and Hong Xie would like to thank their friends and colleagues at the Global Variable Annuity Hedging Department of Manulife Financial for the pleasant cooperation over the last 4 years.

Guojun Gan gratefully acknowledges support from the CAIS (Canadian Academy of Independent Scholars) grant and thanks Simon Fraser University for giving him full access to its libraries. Guojun Gan wants to thank his parents and parents-in-law for all their love and support. He wants to thank his wife, Xiaoying, for taking care of their children.

This work was supported in part by the National Science Foundation for Distinguished Young Scholars of China (grant 70825006), Program for Changjiang Scholars and Innovative Research Team in University of Ministry of Education of China (grant IRT0916), the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (grant 71221001), and the Furong Scholar Program.

GUOJUN GAN, CHAOQUN MA, AND HONG XIE

Toronto, ON, Canada and Changsha, Hunan, P.R. China, February 28, 2014

Financial Glossary

American option An option that can be exercised at any time prior to the expiration date.

Asia option An option whose payoff is dependent on the average price of the underlying asset during a certain period.

barrier option An option whose payoff is dependent on whether the path of the underlying asset has reached a barrier, which is a certain predetermined level.

call option An option that gives the holder the right to buy an asset.

derivative A financial instrument whose price depends on the price of another asset (called the underlying asset); also referred to as derivative security or financial derivative.

down-and-in option A barrier option that comes into existence when the price of the underlying asset declines to the barrier.

down-and-out option A barrier option that ceases to exist when the price of the underlying asset declines to the barrier.

European option An option that can be exercised only on the expiration date. Let K be the strike price of an option. Let ST be the price of the underlying asset at maturity. The terminal payoff of a long position (the holder’s position) of a European call is given by max(ST − K, 0). The terminal payoff of a long position (the holder’s position) of a European put is given by max(K − ST, 0).

forward contract A nonstandardized agreement between two parties to buy or sell an asset at a certain future time for a certain price.

futures contract A standardized agreement between two parties to buy or sell an asset at a certain future time for a certain price.

LIBOR London interbank offered rate.

lookback option An option whose payoff is dependent on the maximum or minimum price of the underlying asset in a certain period.

option A derivative that gives the holder the right (not the obligation) to buy or sell an asset by a certain date for a predetermined price. The date is called the expiration date and the predetermined price is called the strike price or exercise price. An option is said to be exercised if the holder chooses to buy or sell the underlying asset.

put option An option that gives the holder the right to sell an asset.

term structure The relationship between interest rates and their maturities.

up-and-in option A barrier option that comes into existence when the price of the underlying asset increases to the barrier.

up-and-out option A barrier option that ceases to exist when the price of the underlying asset increases to the barrier.

zero-coupon bond A bond that does not pay coupons.

PART I

MEASURE THEORY

CHAPTER 1

SETS AND SEQUENCES

Sets are the most basic concepts in measure theory as well as in mathematics. In fact, set theory is a foundation of mathematics (Moschovakis, 2006). The algebra of sets develops the fundamental properties of set operations and relations. In this chapter, we shall introduce basic concepts about sets and some set operations such as union, intersection, and complementation. We will also introduce some set relations such as De Morgan’s laws.

1.1 Basic Concepts and Facts

Definition 1.1 (Set, Subset, and Empty Set). A set is a collection of objects, which are called elements. A set B is said to be a subset of a set A, written as B ⊆ A, if the elements of B are also elements of A. A set A is called an empty set, denoted by , if A contains no elements.

Definition 1.2 (Countable Set). A set A is said to be countable if either A contains a finite number of elements or every element of A appears in an infinite sequence x1, x2,…. A set A is said to be uncountable if it is not countable.

Definition 1.4 (Union, Intersection, and Complement of Sets). Let A and B be two subsets of a set S. The union of A and B is defined as

The intersection of A and B is defined as

The complement of A relative to S is defined as

Definition 1.5 (Difference and Symmetric Difference of Sets). Let A and B be two sets. The difference between A and B, denoted by A − B or A\B, is defined as

The symmetric difference between A and B is defined as

Definition 1.6 (Increasing and Decreasing Sequence of Sets). Let {An}n≥1 be a sequence of sets. We say that {An}n≥1 is an increasing sequence of sets with limit A, written as An ↑ A, if

We say that {An}n≥1 is a decreasing sequence of sets with limit A, written as An ↓ A, if

Definition 1.7 (Indicator Function). Let A be a set. Then the indicator function of A is defined as

Definition 1.8 (Upper Limit and Lower Limit of Sequences of Sets). Let {En}n≥1 be a sequence of subsets of S. Then lim sup En and lim inf En are defined as

and

respectively.

Definition 1.9 (Upper Limit and Lower Limit of Sequences of Real Numbers). Let {xn}n≥1 be a sequence of real numbers. Then lim sup xn and lim inf xn are defined as

and

respectively.

Definition 1.10 (Convergence of Sequences). A sequence {xn}nN of real numbers is said to be convergent if and only if

for all sS.

Definition 1.12 (Partial Ordering, Totally Ordered Sets, and Chains). A partial ordering ≤ on a set S is a relation that satisfies the following conditions, where a, b, and c are arbitrary elements of S:

Let S be a set with a partial ordering ≤. A subset C of S is said to be a totally ordered subset of S if and only if for all a, bC, we have either a ≤ b or b ≤ a. A chain in S is a totally ordered subset of S.

Theorem 1.1 (De Morgan’s Laws). Let {An}n≥1be a sequence of sets. Then

and

Theorem 1.2 (Zorn’s Lemma). Let S be a nonempty set with a partial ordering “≤”. Assume that every nonempty chain C in S has an upper bound, that is, there exists an element xS such that a ≤ x for all aC. Then S has a maximal element; in other words, there exists an element mS such that a ≤ m for all aS.

1.2 Problems

1.1. Let I be a countable set. For each iI, let Ai be a countable set. Show that the union

is again countable.

1.2. Let Q be the set of all rational numbers, which have the form of a/b, where a and b(b ≠ 0) are integers. Let R be the set of all real numbers. Show that

(a) Q is countable.

(b) R is uncountable.

1.3. Let {En}n≥1 be a sequence of sets. Show that

1.4. Let {En}n≥1 be a sequence of subsets of S. Show that

and

1.5. Let {En}n≥1 be a sequence of subsets of a set S. Show that

(1.1)

and

(1.2)

where I is the indicator function.

1.6. Let {An}n≥1 be a sequence of sets of real numbers defined as follows:

Calculate lim inf An and lim sup An.

1.7. Let {xn}n≥1 a sequence of real numbers. Show that xn converges (i.e., the limit limn→∞xn exists) in [−∞, ∞] if and only if

1.8. Let {xn}n≥1 and {yn}n≥1 be two sequences of real numbers. Let c be a constant in (−∞, ∞). Show that

1.10. Let {xn}n≥1 and {yn}n≥1 be two sequences of real numbers. Suppose that

exist. Show that limn→∞(xn + yn) exists and

1.3 Hints

1.1. Try to construct a sequence in which every element in A appears.

1.2. To prove part (a), show that all rational numbers can be written as a sequence. Part (b) can be proved by the method of contradiction, that is, by assuming that R is countable and can be written as a sequence (xn)n≥1. Then represent every xn as a decimal of finite digits and find a new number, which is not in the sequence.

1.3. This problem can be proved by using Definition 1.8.

1.4. This problem can be proved by using Definition 1.8 and Theorem 1.1.

1.5. An indicator function has only two possible values: 0 and 1. Hence the first equality of the problem can be proved by considering two cases: s lim sup En and s lim sup En. The second equality of the problem can be proved using the result of Problem 1.4.

1.6. The lower and upper limits of the sequence can be calculated by using Definition 1.8.

To prove part (c), try to establish the following inequality

Use parts (a) and (c) to prove part (d). Use parts (a) and (d) to prove part (e). Use part (c) to prove part (f). Use parts (a) and (f) to prove part (g).

1.9. Use Definition 1.11 and the results of Problems 1.5 and 1.7.

1.10. Use the results of Problems 1.7 and 1.8.

1.4 Solutions

1.1. Since I is countable, then {Ai : iI} is countable. Note that Ai is countable for each iI. There exists a sequence (Bn)n≥1 of countable sets such that every Ai appears in the sequence. For each integer n ≥ 1, as Bn is countable, there exists a sequence (xn,m)m≥1 such that every element of Bn appears in the sequence. Now let (yi)i≥1 be a sequence given by

Then every element in A appears in the sequence (yi)i≥1. Hence A is countable. This completes the proof.

1.2.

This completes the proof.

1.3. Let s lim inf En. Then by Definition 1.8, we have for some j0 ≥ 1. It follows that sEi for all i ≥ j0. Hence we have

for all j ≥ 1. Consequently, s lim sup En. Therefore, lim inf En ⊆ lim sup En.

1.4. By Definition 1.8 and Theorem 1.1, we have

1.5. To prove (1.1), we consider two cases: s lim sup En and s lim sup En. If s lim sup En, then

which implies

Thus sEn for infinitely many n. Hence we have

which gives

If s lim sup En, then sEn for only finitely many n. Thus we have

where M0 is a sufficient large number. Hence we have

Thus (1.2) holds.

and

1.8.

This finishes the proof.

Next, we prove the “only if” part. Suppose that limn→∞An exists. Then by definition, we have limn→∞IAn(s) exists for all sS. It follows that

1.10. Since limn→∞xn and limn→∞yn exist, it follows from Problem 1.7 that

and

Then by parts (c) and (d) of Problem 1.8, we have

which shows that

This completes the proof.

1.5 Bibliographic Notes

In this chapter, we introduced some concepts in set theory as well as some set operations and relations. For further information about these concepts, readers are referred to Papoulis (1991), Williams (1991), Ash and Doleans-Dade (1999), Jacod and Protter (2004), and Reitano (2010).

We also introduced some concepts related to sequences of real numbers, which are connected to sequences of sets via indicator functions. The properties of sequences of real numbers and sets are frequently used in later chapters.

Zorn’s lemma is an axiom of set theory and is equivalent to the axiom of choice. For a proof of the equivalence, readers are referred to Vaught (1995, p80), Dudley (2002, p20), and Moschovakis (2006, p114).

CHAPTER 2

MEASURES

Measurable sets are to measure theory as open sets are to topology (Williams, 1991). Measures are set functions defined on measurable sets. These concepts are used later to define integration. In this chapter, we shall introduce measurable sets, measures, and other relevant concepts such as algebras and σ-algebras.

2.1 Basic Concepts and Facts

Definition 2.1 (Algebra). An algebra or field ∑0 on S is a collection of subsets of that satisfies the following conditions: