Real Analysis and Probability

R. M. Dudley

• Verlag: Chapman and Hall/CRC
• Jahr: 2018
• ISBN:
  9781351093095
• Auflage: 1
• Thema: Mathématiques
• Thema: Mathématiques générales

Written by one of the best-known probabilists in the world this text offers a clear and modern presentation of modern probability theory and an exposition of the interplay between the properties of metric spaces and those of probability measures. This text is the first at this level to include discussions of the subadditive ergodic theorems, metrics for convergence in laws and the Borel isomorphism theory. The proofs for the theorems are consistently brief and clear and each chapter concludes with a set of historical notes and references. This book should be of interest to students taking degree courses in real analysis and/or probability theory.

CHAPTER 1
Foundations; Set Theory

In constructing a building, the builders may well use different techniques and materials to lay the foundation than they use in the rest of the building. Likewise, almost every field of mathematics can be built on a foundation of axiomatic set theory. This foundation is accepted by most logicians and mathematicians con-cerned with foundations, but only a minority of mathematicians have the time or inclination to learn axiomatic set theory in detail.

To make another analogy, higher-level computer languages and programs written in them are built on a foundation of computer hardware and Systems programs. How much the people who write high-level programs need to know about the hardware and operating Systems will depend on the problem at hand.

In modern real analysis, set-theoretic questions are somewhat more to the fore than they are in most work in algebra, complex analysis, geometry, and applied mathematics. A relatively recent line of development in real analysis, “nonstandard analysis,” allows, for example, positive numbers that are infinitely small but not zero. Nonstandard analysis depends even more heavily on the specifics of set theory than earlier developments in real analysis did.

This chapter will give only enough of an introduction to set theory to define some notation and concepts used in the rest of the book. In other words, this chapter presents mainly “naive” (as opposed to axiomatic) set theory. Appendix A gives a more detailed development of set theory, including a listing of axioms, but even there, the book will not enter into nonstandard analysis or develop enough set theory for it.

Many of the concepts defined in this chapter are used throughout mathematics, and will, I hope, be familiär to most readers, at least those with the suggested background in intermediate real analysis.

1.1 Definitions for Set Theory

Definitions can serve at least two purposes. First, as in an ordinary dictionary, a definition can try to give insight, to convey an idea, or to explain a less familiar idea in terms of a more familiar one, but with no attempt to specify or exhaust completely the meaning of the word being defined. This kind of definition will be called informal. A formal definition, as in most of mathematics and parts of other sciences, may be quite precise, so that one can decide scientifically whether a Statement about the term being defined is true or not. In a formal definition, a familiar term, such as a common unit of length or a number, may be defined in terms of a less familiar one. Most definitions in set theory are formal. Moreover, set theory aims to provide a coherent logical structure not only for itself but for just about ali of mathematics. There is then a question of where to begin in giving definitions.

Informal dictionary definitions often consist of synonyms. Suppose, for example, that a dictionary simply defined “high” as “tali” and “tali” as “high.” One of these definitions would be helpful to someone who knew one of the two words but not the other. But to an alien from outer space who was trying to learn English just by reading the dictionary, these definitions would be useless. This Situation illustrates on the smallest scale the whole problem the alien would have, since all words in the dictionary are defined in terms of other words. To make a start, the alien would have to have some way of interpreting at least a few of the words in the dictionary other than by just looking them up.

In any case some words, such as the conjunctions “and,” “or,” and “but,” are very familiar but hard to define as separate words. Instead, we might have rules that define the meanings of phrases containing conjunctions given the meanings of the words or subphrases connected by them.

At first thought, the most important of ali definitions you might expect in set theory would be the definition of “set,” but quite the contrary, just because the entire logical structure of mathematics reduces to or is defined in terms of this notion, it cannot necessarily be given a formal, precise definition. Instead, there are rules (axioms, rules of inference, etc.) which in effect provide the meaning of “set.” A preliminary, informal definition of set would be “any collection of mathematical objects,” but this notion will have to be clarified and adjusted as we go along.

The problem of defming set is similar in some ways to the problem of defining number. After several years of school, students “know” about the numbers 0, 1, 2,…, in the sense that they know rules for operating with numbers. But many people might have a hard time saying exactly what a number is. Different people might give different definitions of the number 1, even though they completely agree on the rules of arithmetic.

In the late 19th century, mathematicians began to concern themselves with giving precise definitions of numbers. One approach is that beginning with 0, we can generate further integers by taking the “successor” or “next larger integer.”

If 0 is defined, and a successor Operation is defined, and the successor of any...