Über die Autorin bzw. den Autor

Dean Corbae is the Rex A. and Dorothy B. Sebastian Centennial Professor in Business Administration at the University of Texas at Austin. Maxwell B. Stinchcombe is the E. C. McCarty Centennial Professor of Economics at the University of Texas at Austin. Juraj Zeman is researcher at the National Bank of Slovakia and lecturer in applied mathematics at Comenius University in Bratislava.

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"I've struggled in teaching a math for economics course for several years without an appropriate text. This book will remedy this problem and, more generally, will fill a gap that has existed in the profession for at least a decade."--L. Joe Moffitt, University of Massachusetts

"This book will prove extremely useful for anyone who wants to learn mathematical economics in an accessible and intuitive fashion, while still tackling advanced concepts. The range of topics is impressive, with many illuminating examples. An excellent text!"--Jaksa Cvitanic, California Institute of Technology

"This book makes accessible an extraordinary amount of mathematics used in economics and carries it to a high level. By means of illustrative examples, the authors succeed in explaining most of the main ideas of economic theory. This is an important resource for economists and an excellent text for mathematics courses for economic graduate students."--Truman F. Bewley, Yale University

"A much-needed textbook for graduate students and a useful desk reference for researchers, this book is of tremendous value to the economics profession because it bridges abstract mathematics and concrete economic applications. Given the current technical level required in research, knowledge of materials covered in this book is indispensable for graduate students."--Han Hong, Stanford University

"Without ever sacrificing rigor, the authors have a style that will help students trying to decipher arcane mathematical ideas. I recommend this book to students."--Richard P. McLean, Rutgers University

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"I've struggled in teaching a math for economics course for several years without an appropriate text. This book will remedy this problem and, more generally, will fill a gap that has existed in the profession for at least a decade."--L. Joe Moffitt, University of Massachusetts

"This book will prove extremely useful for anyone who wants to learn mathematical economics in an accessible and intuitive fashion, while still tackling advanced concepts. The range of topics is impressive, with many illuminating examples. An excellent text!"--Jaksa Cvitanic, California Institute of Technology

"This book makes accessible an extraordinary amount of mathematics used in economics and carries it to a high level. By means of illustrative examples, the authors succeed in explaining most of the main ideas of economic theory. This is an important resource for economists and an excellent text for mathematics courses for economic graduate students."--Truman F. Bewley, Yale University

"A much-needed textbook for graduate students and a useful desk reference for researchers, this book is of tremendous value to the economics profession because it bridges abstract mathematics and concrete economic applications. Given the current technical level required in research, knowledge of materials covered in this book is indispensable for graduate students."--Han Hong, Stanford University

"Without ever sacrificing rigor, the authors have a style that will help students trying to decipher arcane mathematical ideas. I recommend this book to students."--Richard P. McLean, Rutgers University

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AN INTRODUCTION TO MATHEMATICAL ANALYSIS FOR ECONOMIC THEORY AND ECONOMETRICS

PRINCETON UNIVERSITY PRESS

Contents

Preface...............................................................................................................xiUser's Guide..........................................................................................................xiiiNotation..............................................................................................................xixChapter 1 * Logic.....................................................................................................1Chapter 2 * Set Theory................................................................................................15Chapter 3 * The Space of Real Numbers.................................................................................72Chapter 4 * The Finite-Dimensional Metric Space of Real Vectors.......................................................106Chapter 5 * Finite-Dimensional Convex Analysis........................................................................172Chapter 6 * Metric Spaces.............................................................................................259Chapter 7 * Measure Spaces and Probability............................................................................355Chapter 8 * The Lp(Ω, F, P) and lp Spaces, p [member of] [1, ∞].....................452Chapter 9 * Probabilities on Metric Spaces............................................................................551Chapter 10 * Infinite-Dimensional Convex Analysis.....................................................................595Chapter 11 * Expanded Spaces..........................................................................................627Index.................................................................................................................655

Chapter One

The building blocks of modern economics are based on logical reasoning to prove the validity of a conclusion, B, from well-defined premises, A. In general, statements such as A and/or B can be represented using sets, and a "proof" is constructed by applying, sometimes ingeniously, a fixed set of rules to establish that the statement B is true whenever A is true. We begin with examples of how we represent statements as sets, then turn to the rules that allow us to form more and more complex statements, and then give a taxonomy of the major types of proofs that we use in this book.

1.1 * Statements, Sets, Subsets, and Implication

The idea of a set (of things), or group, or collection is a "primitive," one that we use without being able to clearly define it. The idea of belonging to a set (group, collection) is primitive in exactly the same sense. Our first step is to give the allowable rules by which we evaluate whether statements about sets are true.

We begin by fixing a set X of things that we might have an interest in. When talking about demand behavior, the set X has to include, at the very least, prices, incomes, affordable consumption sets, preference relations, and preference-optimal sets. The set X varies with the context, and is often not mentioned at all.

We express the primitive notion of membership by "[member of]," so that "x [member of] A" means that "x is an element of the set A" and "y [not member of] A" means that "y is not an element of A."

Notation Alert 1.1.A Capitalized letters are usually reserved for sets and smaller letters for points/things in the set that we are studying. Sometimes, several levels of analysis are present simultaneously, and we cannot do this. Consider the study of utility functions, u, on a set of options, X. A function u is a set of pairs of the form (x, u(x)), with x an option and u(x) the number representing its utility. However, in our study of demand behavior, we want to see what happens as u varies. From this perspective, u is a point in the set of possible utility functions.

Membership allows us to define subsets. We say "A is a subset of B," written "A [subset] B," if every x [member of] A satisfies x [member of] B. Thus, subsets are defined in terms of the primitive relation "[member of]." We write A = B if A [subset] B and B [subset] A, and A [not equal to] B otherwise.

We usually specify the elements of a set explicitly by saying "The set A is the set of all elements x in X such that each x has the property A, that is, that A(x) is true," and write "A = {x [member of] X : A(x)}" as a shorter version of this. For example, with X = R^l, the statement "x ≥ 0" is identified with the set R^l₊ = {x [member of] X : x ≥ 0}. In this way, we identify a statement with the set of elements of X for which the statement is true. There are deep issues in logic and the foundations of mathematics relating to the question of whether or not all sets can be identified by "properties." Fortunately, these issues rarely impinge on the mathematics that economists need. Chapter 2 is more explicit about these issues.

We are very often interested in establishing the truth of statements of the form "If A, then B." There are many equivalent ways of writing such a statement: "A -> B," "A implies B," "A only if B," "A is sufficient for B," or "B is necessary for A." To remember the sufficiency and necessity, it may help to subvocalize them as "A is sufficiently strong to guarantee B" and "B is necessarily true if A is true."

The logical relation of implication is a subset relation. If A = {x [member of] X : A(x)} and B = {x [member of] X :B(x)}, then "A -> B" is the same as "A [subset] B."

Example 1.1.1 Let X be the set of numbers, A(x) the statement "x² < 1," and B(x) the statement "|x| ≤ 1." Now, A -> B. In terms of sets, A = {x [member of] X :A(x)} is the set of numbers strictly between -1 and +1, B = {x [member of] X :B(x)} is the set of numbers greater than or equal to -1 and less than or equal to +1, and A [subset] B.

The statements of interest can be quite complex to write out in their entirety. If X is the set of allocations in a model E of an economy and A(x) is the statement "x is a Walrasian equilibrium allocated for the economy ε," then a complete specification of the statement takes a great deal of work. Presuming some familiarity with general equilibrium models, we offer the following.

Example 1.1.2 Let X be the set of allocations in a model ε of an economy; let A(x) be the statement "x is a Walrasian equilibrium allocation"; and B(x) be the statement "x is Pareto efficient for ε." The first fundamental theorem of welfare economics is A -> B. In terms of the definition of subsets, this is expressed as, "Every Walrasian equilibrium allocation is Pareto efficient."

In other cases, we are interested in...