An Introduction to Nonstandard Real Analysis

238 Pages·2014·4.4 MB·English

by  Albert E. Hurd, Peter A. Loeb

An Introduction to
Nonstandard
Real Analysis
ALBERT E. HURD
Department of Mathematics
University of Victoria
Victoria. British Columbia
Canada
PETER A. LOEB
Department of Mathematics
University of Illinois
Urbana, Illinois
1985
ACADEMIC PRESS, INC.
(Harcourt Brace Jovannvich, Publishers)
Orlando San Diego New York London
Toronto Montreal Sydney Tokyo
COPYRIGHTo 1985 BY ACADEMICPR ESS, INC.
ALL RIGHTS RESERVED.
NO PARTOFTHIS PUBLICATION MAY BE REPRODUCEDOR
TRANSMITTED IN ANY FORMOR BY ANY MEANS. ELECTRONIC
OR MECHANICAL, INCLUDING PHOTOCOPY. RECORDING.OR ANY
INFORMATION STORAGE AND RETRIEVAL SYSTEM. WITHOUT
PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS, INC.
Orlando, Florida 32887
United Kingdom Edition published by
ACADEMIC PRESS INC. (LONDON) LTD.
24/28 Oval Road, London NWI .7DX
Library of Congress Cataloging in Publication Data
Main entry under title:
An introduction to nonstandard real analysis.
Includes bibliographical references and index.
1. Mathematical analysis, .N onstandard. I. Hurd. A. E.
(Albert Emerson), DATE 11. Loeb, P. A.
P A299.82.158 1985 515 84-24563
SBN 0-12-362440-1 (alk. paper)
PRINTED INTHE UNITEDSTATkS OFAMtRlCA
85868788 9 8 7 6 5 4 3 2 1
Preface
The notion of an infinitesimal has appeared off and on in mathematics since the
time of Archimedes. In his formulation of the calculus in the 1670s. the German
mathematician Wilhelm Gottfried Leibniz treated infinitesimals as ideal num-
bers, rather like imaginary numbers, which were smaller in absolute value than
any ordinary real number but which nevertheless obeyed all of the usual laws of
arithmetic. Leibniz regarded infinitesimals as a useful fiction which facilitated
mathematical computation and invention. Although it gained rapid acceptance on
the continent of Europe, Leibniz’s method was not without its detractors. In
commenting on the foundations of calculus as developed both by Leibniz and
Newton, Bishop George Berkeley wrote, “And what are these same evanescent
increments? They are neither finite quantities, nor quantities infinitely small, nor
yet nothing. May we not call them the ghosts of departed quantities’?’’T he ques-
tion was, How can there be a positive number which is smaller than any real
number without being zero? Despite this unanswered question, the infinitesimal
calculus was developed by Euler and others during the eighteenth and nineteenth
centuries into an impressive body of work. It was not until the late nineteenth cen-
tury that an adequate definition of limit replaced the calculus of infinitesimals and
provided a rigorous foundation for analysis. Following this development. the use
of infinitesimals gradually faded, persisting only as an intuitive aid to con-
ceptualization.
There the matter stood until 1960 when Abraham Robinson gave a rigorous
foundation for the use of infinitesimals in analysis. More specifically, Robinson
showed that the set of real numbers can be regarded as a subset of a larger set of
“numbers” (called hyperreal numbers) which contains infinitesimals and also,
with appropriately defined artithmetic operations, satisfies all of the arithmetic
rules obeyed by the ordinary real numbers. Even more, he demonstrated that the
relational structure over the reals (sets, relations. etc.) can be extended to a sim-
ilar structure over the hyperreals in such a way that all statements true in the real
structure remain true, with a suitable interpretation, in the hyperreal structure.
This latter property, known as the transfer principle. is the pivotal result of Robin-
son’s discovery.
ix
X Preface
Robinson’s invention, called nonstandard analysis, is more than a justification
of the method of infinitesimals. It is a powerful new tool for mathematical re-
search. Rather quickly it became apparent that every mathematical structure has a
nonstandard model from which knowledge of the original structure can be gained
by applications of the appropriate transfer principle. In the twenty-five years
since Robinson’s discovery, the use of nonstandard models has led to many new
insights into traditional mathematics, and to solutions of unsolved problems in
areas as diverse as functional analysis, probability theory, complex function
theory, potential theory, number theory, mathematical physics, and mathematical
economics.
Robinson’s first proof of the existence of hyperreal structures was based on a
result in mathematical logic (the compactness theorem). It was perhaps this
aspect of his work, more than any other, which made it difficult to understand for
those not adept at mathematical logic. At present, the most common demonstra-
tion of the existence of nonstandard models uses an “ultrapower” construction.
But the use of ultrapowers is not restricted to nonstandard analysis. Indeed, the
construction of ultrapower extensions of the real numbers dates back to the 1940s
with the work of Edwin Hewitt [ 171 and others, and the use of ultrapowers to study
Banach spaces [ 10,161 has become an important tool in modem functional analy-
sis. Nonstandard analysis is a far-reaching generalization of these applications of
ultrapowers. One essential difference between the method of ultrapowers and the
method of nonstandard analysis is the consistent use of the transfer principle in
the latter. To present this principle one needs a certain amount of mathematical
logic, but the logic is used in an essential way only in stating and proving the
transfer principle, and not in applying nonstandard analysis. We hope to dem-
onstrate that the amount of logic needed is minimal, and that the advantages
gained in the use of the transfer principle are substantial.
The aim of this book is to make Robinson’s discovery, and some of the subse-
quent research, available to students with a background in undergraduate math-
ematics. In its various forms, the manuscript was used by the second author in
several graduate courses at the University of Illinois at Urbana-Champaign. The
first chapter and parts of the rest of the book can be used in an advanced under-
graduate course. Research mathematicians who want a quick introduction to
nonstandard analysis will also find it useful. The main addition of this book to the
contributions of previous textbooks on nonstandard analysis [ 12, 37, 42, 461 is
the first chapter, which eases the reader into the subject with an elementary model
suitable for the calculus, and the fourth chapter on measure theory in nonstandard
models.
A more complete discussion of this book’s four chapters must begin by noting
H. Jerome Keisler’s major contribution to nonstandard analysis in the form of his
1976 textbook, “Elementary Calculus” [23] together with the instructor’s vol-
ume, “Foundations of Infinitesimal Calculus” [24]. Keisler’s book is an excellent
Preface xi
calculus text (see the second author’s review [30])w hich makes that part of non-
standard analysis needed for the calculus available to freshman students. Keis-
ler’s approach uses equalities and inequalities to transfer properties from the real
number system to the hyperreal numbers. In our first chapter, we have modified
that approach to an equivalent one by formulating a simple transfer principle
based on a restricted language.
The first chapter begins by using ultrafilters on the set of natural numbers to
construct a simple ultrapower model of the hyperreal numbers. A formal lan-
guage is then developed in which only two kinds of sentences are used to transfer
properties from the real number system to the larger, hyperreal number system.
The rest of the chapter is devoted to extensive applications of this simple transfer
principle to the calculus and to more-advanced real analysis including differential
equations. By working through these applications, the reader should acquire a
good feeling for the basics of nonstandard analysis by the end of the chapter. Any-
one who begins this book with no background in mathematical logic should have
no problem with the logic in the first chapter and hence should easily pick up the
background needed to proceed. Indeed, it is our hope that such a reader will grow
quite impatient with the restrictions on the language we impose in the first chap-
ter, and thus be more than ready for the general language introduced in Chapter 11
and used in the rest of the book. We will not comment on what might be in the
mind of a logician at that point.
Chapter I1 extends the context of Chapter I to “higher-order’’ models ap-
propriate to the discussion of sets of sets, sets of functions, etc., and covers the
notions of internal and external sets and saturation. These topics, together with a
general language and transfer principle, are held in abeyance until the second
chapter so that the beginner can master the subject in reasonably easy steps. They
are, however, essential to the applications of nonstandard analysis in modem
mathematics. External constructions, such as the nonstandard hulls discussed in
Chapter I11 and the standard measure spaces on nonstandard models described in
Chapter IV, have been the principal tools through which new results in standard
mathematics have been obtained using nonstandard analysis.
The general theory of Chapter I1 is applied in Chapter I11 to topological spaces.
These are sets with an additional structure giving the notion of nearness. The
presentation assumes no familiarity with topology but is rather brisk, so that ac-
quaintance with elementary topological ideas would be useful. The chapter in-
cludes discussions of compactness and of metric, normed, and Hilbert spaces.
We present a brief discussion of nonstandard hulls of metric spaces, which are
important in nonstandard technique. Some of the more advanced topics in Kelley ’s
“General Topology,” such as function spaces and compactifications, are also
included.
Finally, in Chapter IV, we introduce the reader to nonstandard measure theory,
certainly one of the most active and fruitful areas of present-day research in non-
xii Preface
standard analysis. With measure theory one extends the notion of the Riemann in-
tegral. We shall take a "functional" approach to the integral on nonstandard
spaces. This approach will produce both classical results in standard integration
theory and some new results which have already proved quite useful in prob-
ability theory, mathematical physics, and mathematical economics. The de-
velopment in this chapter does not assume familiarity with measure theory be-
yond the Riemann integral. Most of the results in [27, 29, 32, 331 are presented
without further reference. We note here that the measures and measure spaces
constructed on nonstandard models in Chapter IV are often referred to in the lit-
erature as Loeb measures and Loeb spaces.
With one exception (Section 1.15). every section of the book has exercises. In
designing the text, we have assumed the active participation of the reader, so
some of the exercises are details of proofs in the text. At the back of the book
there is a list of the notation used, together with the page where the notation is
introduced. Of course, we freely use the symbols E. u, and n for set member-
ship, union, and intersection. We have starred sections that can be skipped at the
first reading. Every item in the book has three numbers, the number of the chapter
(I. II, 111 or IV). the number of the section, and the number of the item in the sec-
tion. Thus. Theorem IV.2.3 is the third item in the second section of the fourth
chapter. In referring to an item, we shall omit the chapter number for items in the
same chapter as the reference, and the section number for items in the same sec-
tion as the reference.
CHAPTER I
In$nitesimals and
The Calculus
Our aim in this chapter is to introduce the reader to nonstandard analysis
in the familiar context of the calculus. It was in this context that the concept
of an infinitesimal was used by Leibniz and his followers to define the deriv-
ative, thus launching the infinitesimal calculus on its spectacular develop-
ment. The notion of an infinitesimal is a cornerstone in all applications of
nonstandard methods to analysis, and so an understanding of this chapter is
basic to the rest of the book. Moreover, such an understanding will make
the technical elaborations of the later chapters easier to appreciate.
In spite of the many technical advantages attending the use of infinites-
imals as developed by Leibniz, the notion of infinitesimal was always con-
troversial. The main question was whether infinitesimals actually existed.
Since an infinitesimal real number was supposed to be smaller in absolute
value than any ordinary positive one, it was clear that all infinitesimals
other than zero were not ordinary real numbers. Leibniz regarded them as
“numbers” in some ideal world. Further, he implicitly made the important
but somewhat vague hypothesis that the infinitesimals satisjed the same rules
as the ordinary real numbers. Consider how this hypothesis would work in
the calculation of the derivative of the functiron exi. L eibniz would write
’),
d ex+dx - ex
-ex = = ex
dx dx ~
where dx is an infinitesimal. A separate calculation (Example 11.3.2) would
show that (edx- l)/dx = 1. We will learn in this chapter that the foregoing
calculation is correct as long as the equality signs are replaced by N, where
a N- b means that a and b are infinitesimally close. Two facts should be noted:
(a) We need to be able to add infinitesimals to ordinary real numbers.
This implies that both infinitesimals and ordinary reals are contained in a
larger set of “numbers” for which the operations of arithmetic are defined.
1
2 I. lnfinitesimals and The Calculus
(b) The function ex needs to be extended to this larger set of numbers in
such a way that the law of exponents is satisfied.
The example of the previous para~aphsh ows that to make Leibniz’s
approach to the calculus rigorous we must
A. construct a set *R of “numbers” and define operations of addition,
mu~tiplicat~oann, d linear ordering on *R so that (i) the field R of real num-
bers (or an isomorphic copy of R) is embedded as a subfield of *Ra nd (ii) the
laws of ordinary arithmetic are valid in *R,
B. show how functions and relations on R are extended to functions and
relations on *R, thus extending the “relational” structure on R to one on *R,
C. ensure that statements true in the relational structure on R are “ex-
tended to statements true in the relational structure on *R.
A set *R having the properties men ti one^ in A is develo~din 81.1 using
ultrafilters. We show in the Appendix that the existence of ultrafilters follows
from Zorn’s lemma, a form of the axiom of choice. In 51.2 we show how rela-
tions and functions on R are extended to relations and functions on *R.
To deal with C we must develop a very modest amount of mathematical
logic (@1.3 and 1.4) in order to make precise what is meant by the words
“statement” and “true.” The sense in which true statements for R “extend”
to true statements for *R is made precise in the transfer principle, which is
stated in 51.5. This principle is at the heart of nonstandard methods as de-
veloped by Abraham Robinson. Its proof is deferred to $1.15 since it is not
necessary to know the proof in order to apply the transfer principle. In the
intervening sections we show how to use the transfer principle to prove re-
sults in the calculus. The proofs are usually similar to those developed in
the early days of the calculus except for the role played by mathematical
logic.
As noted in the Preface, we have used a very simple formal language in
this chapter in order to facilitate the initiation of readers not familiar with
formal languages. Consideration of a more elaborate language and nonstan-
dard model is deferred until Chapter 11.
1.1 The Hyperreal Number System
as an Ultrapower
We assume that anyone reading this book is familiar with the real number
+
system as a complete linearly ordered field 41, = (R, , <), where R denotes
+, a,
the set of real numbers and and < denote the usual algebraic opera-
*,
1.1 The Hyperreal Nurrber System as an Ultrapower 3
tions and relations of addition, multiplication, and linear ordering on R.
Our object in this section is to construct another linearly ordered field $t =
+,
(R, <) which contains an isomorphic copy of W but is strictly larger
a,
than 9. W will be called a nonstandard or hyperreal number system.
The construction of W is reminiscent of the construction of the reals from
the rationals by means of equivalence classes of Cauchy sequences. To begin
the construction, let N denote the natural numbers and d denote the set of
fi
all sequences of real numbers (indexed by N); i.e., each element in is of the
form r = (rl,r2, r3,.. .). For convenience we denote (rl,r2,r3,. . .) by
(ri:i E N) or simply (ri). Operations of addition, $, and multiplication, 0,
can be defined on d in the following way: If r = (ri) and s = (si) are ele-
d,
ments of we define
+
r @ s = (ri si>
and
r 0 s = (ri si).
It is easy to check (Exercise 1) that (A, @,0is) a commutative ring with an
identity (1,1,. . .) and a zero (O,O, . . .) (where 1 and 0 are the unit and zero
in R). However, the ring is not a field; for example,
(l,O,l,O,l,. . .) 0 (O,l,O,l,O,. . .) = (O,O,O,. . .),
so the product of nonzero elements can be zero. We remedy the situation by
d
introducing an equivalence relation on and defining operations and rela-
tions +, ., and < on the set R of equivalence classes which make (R,+ , ., <)
into a linearly ordered field.
To introduce the equivalence relation we need the notion of an ultrafilter
(for more on ultrafilters see the Appendix).
1.1 Definition Let I be a nonempty set. A jlter on I is a nonempty collec-
tion 9 of subsets of I having the following properties:
(i) The empty set 0 $%.
(ii) If A, B E 4, then A n B E 9.
(iii) If A E 42 and I 2 B 2 A, then B E 9.
A filter 4 is an ultrajlter if
(iv) for any subset A of I either A E 9 or its complement A’ = I - A E 9
[but not both by (i) and (ii)].
For each x E I there is a $xed ultrafilter 4, = {B G I:xE S}. If I is an
infinite set the collection 9:= {A E I:I - A is finite} is a filter called the
cojnite or Frichet filter on I. An ultrafilter 9 on I isfree if it contains 9:.
4 I. lnfinitesimals and The Calculus
A free ultrafilter 9 cannot contain any finite set F, since otherwise F‘
is cofinite and hence in 4 and so F n F‘ = 521, contradicting l.l(i). Intu-
itively an ultrafilter is a very large collection of subsets, but not too large,
since, for example, it cannot contain two disjoint sets by l.l(i) and (ii). Note
that if 4 is an ultrafilter on I, then I E 4. Note also that if Al, A,, . . . , A,
are a finite number of sets in I with Ai n Aj = 521 for i # j and
UAXl s i 5 n) = I, then one and only one of the sets Ai is in 42 (Exer-
cise 2).
It is not at all obvious that free ultrafilters exist. These, however, are the
important ultrafilters for our construction. Therefore we take as a basic as-
sumption the following axiom.
1.2 Ultrafilter Axiom If 9 is a filter on I, then there is an ultrafilter 9 on I
which contains 9.
We show in the Appendix that the ultrafilter axiom follows from Zorn’s
lemma (that is, the axiom of choice).
Now assume that we have chosen a free ultrafilter 9 on N. We define a
relation = on fi as follows (= will depend on 9,b ut this dependence will
not be indicated explicitly).
1.3 Definition If r = (Ti) and s = (si) are in fi, then r = s if and only if
{i E N:ri = si} E 4. We then say that (r,) = (si) almost everywhere (a.e.).
1.4 Lemma The relation = is an equivalence relation on fi.
Proof: The relation = is reflexive (r = r) because N E 9,sy mmetric (r = s
implies s = I) because = is a symmetric relation on R, and transitive (r = s
and s = t imply r = t) because of conditions l.l(ii) and (iii) for a filter. The
details are left to the reader (Exercise 3). 0
Note that two sequences can have the same limit as n + co and not be
equivalent. For example, (l,i,f., . . ) f (O,O,O, . . . ) since 0 $9;seq uences
. .
like (l,i,$, .) will later be used to define “infinitesimal numbers” different
from zero. We will see shortly that the equivalence relation also eliminates
the problem that the product of nonzero elements can be zero. For example,
consider again the sequences (l,O, 1,0,1,. . .) and (0,1,0,1,0,. . .). By
. .
l.l(iv), one of these two sequences is equivalent to (40, .); which one
depends on the particular ultrafilter 4 used to define = (there are many
such).