Table Of Content

Martin Väth Nonstandard Analysis Birkhäuser Verlag Basel· Boston · Berlin Contents Preface vii 1 Preliminaries 1 §1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Archimedean Fields and Infinitesimals . . . . . . . . . . . . 4 §2 Superstructures, Sentences, and Interpretations . . . . . . . . . . . 13 2.1 Superstructures . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Formal Language . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Nonstandard Models 23 §3 The Three Fundamental Principles . . . . . . . . . . . . . . . . . . 23 3.1 Elementary Embeddings and the Transfer Principle . . . . 23 3.2 The Standard Definition Principle . . . . . . . . . . . . . . 28 3.3 The Internal Definition Principle . . . . . . . . . . . . . . . 34 3.4 Existence of External Sets . . . . . . . . . . . . . . . . . . . 40 §4 Nonstandard Ultrapower Models . . . . . . . . . . . . . . . . . . . 44 4.1 Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Ultrapowers . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Embedding in a Superstructure . . . . . . . . . . . . . . . . 51 3 Nonstandard Real Analysis 59 §5 Hyperreal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.1 Hyperreal and Hypernatural Numbers . . . . . . . . . . . . 59 5.2 Interpretation of the Standard Part Homomorphism . . . . 70 §6 The Permanence Principle and ∗-finite Sets . . . . . . . . . . . . . 74 §7 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 vi Contents 7.2 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4 Enlargements and Saturated Models 103 §8 Enlargements, Saturation, and Concurrency . . . . . . . . . . . . . 103 §9 Saturated Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 9.1 Models for Enlargements . . . . . . . . . . . . . . . . . . . 114 9.2 Compact Enlargements . . . . . . . . . . . . . . . . . . . . 115 9.3 PolysaturatedModels . . . . . . . . . . . . . . . . . . . . . 122 5 Functionals, Generalized Limits, and Additive Measures 125 §10 Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 10.1 Linear Functionals and Operators . . . . . . . . . . . . . . 125 10.2 Hahn-Banach and Banach-Mazur Limits . . . . . . . . . . . 129 §11 Additive Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6 Nonstandard Topology and Functional Analysis 149 §12 Topologies and Filters . . . . . . . . . . . . . . . . . . . . . . . . . 149 12.1 TopologicalSpaces . . . . . . . . . . . . . . . . . . . . . . . 149 12.2 Filters in Nonstandard Analysis. . . . . . . . . . . . . . . . 150 12.3 Topologies in Nonstandard Analysis . . . . . . . . . . . . . 154 12.4 Functions in Nonstandard Topology . . . . . . . . . . . . . 167 §13 Uniform Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 13.1 Uniform Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 170 13.2 Nonstandard Hulls . . . . . . . . . . . . . . . . . . . . . . . 178 §14 Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . 185 7 Miscellaneous 197 §15 Loeb Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 §16 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 A Some Important ∗-Values 211 B Solutions to the Exercises 217 Bibliography 241 Index 245 Preface Historically, the idea of nonstandard analysis was to rigorously justify calcula- tions withinfinitesimal numbers.For example,formally,the chainrule of Leibniz’ calculus for the function F =f(g(x)) can be written as dF dF dg = , dx dg dx and for a formal proof, one may just divide numerator and denominator by the “infinitesimal small number” dg. Nowadays, nonstandard analysis has gone far beyond the realm of infinitesimals. In fact, it provides a machinery which enables one to describe “explicitly” mathematicalconceptswhichbystandardmethodscanonlybedescribed“implicitly” and in a cumbersome way. In the above example the “standard” notion of a limit is in a certain sense replaced by the “nonstandard” notion of an infinitesimal. If one applies a similar approach to other objects than the real numbers (like topologicalspacesorBanachspacesetc.), one has atoolwhichprovides“explicit” definitions for objects which can in principle not be described explicitly by standard methods. Examples of such objects are sets which are not Lebesgue measurable,orfunctionalswithcertainpropertieslikeso-calledHahn–Banachlim- its. Since it is possible in nonstandard analysis to simply “calculate” with such objects,onecanobtainresultsaboutthemwhichareextremelyhardto obtainby standard methods. This book is an introduction to nonstandard analysis. In contrast to some other textbooks on this topic, it is not meant as an introduction to basic calculus by nonstandard analysis. Instead, the above mentioned applications in analysis (which are not easily accessible by standard methods) are our main motivation. The infinitesimals are only described as an elementary example for the provided machinery. Consequently, the reader is supposed to be already familiar with (standard) basiccalculus.Fordeeperunderstanding,alsoexperiencewith(basic)topologyand Chapter 1 Preliminaries §§§1 Introduction 1.1 General Remarks Historically, the idea of nonstandard analysis was to find a rigorous justification forcalculationswith infinitesimalnumbers.However,inthe author’sopinion,this isnotthemostimportantpropertyofnonstandardanalysis.Instead,itappearsto theauthorthatitismoreessentialthatso-calledconcurrentrelationsaresatisfied. We will make this more precise later, but we already mention that this means, roughly speaking, the following. If there is a statement which holds for any finite subset of a given set, then it holds for the whole set in nonstandard analysis. Consider, for example, for any set M of positive real numbers the statement “there is some c > 0 with c < ε for all ε ∈ M”. Clearly, this statement is true for any finite set M of positive real numbers (in our later terminology, we denote suchafactby “concurrency”).This implies thatthe statementis alsotrue forthe set of all positive numbers in nonstandard analysis and so there indeed exists an infinitesimal c > 0 which is less than any positive real number. In other words, nonstandard analysis allows us to conclude that “true for each finite number” implies “true for all”. The formulation of the above considerations in precise mathematical terms is rather involved. For this reason (and to have a further motivation up to this point), we will first concentrate on the “classical” topic of nonstandard analysis: ThisisLeibniz’ideawhichmaybedescribedasfollows.Leibniz’programistojoin “infinitesimals” to the system of real numbers such that the enlarged system (cid:0) obeys the same “rules” as . As we shall see, this programcannot be carried out (cid:0) 2 Chapter 1. Preliminaries directly,becausethe systemofrealnumbersisuniquelydeterminedbytheserules (up to an isomorphism). The solution proposed by A. Robinson and W.A.J. Luxemburg out of this ∗ dilemma is the following: Consider together with a nonstandard real line (cid:0) (cid:0) whichcontains andalsoinfinitesimalnumbersaselementsandwhichsatisfiesthe (cid:0) following:Anyso-calledtransitivelyboundedsentenceabout canbetransferred ∗ (cid:0) into an analogous sentence about , and the latter sentence is true if and only (cid:0) if the sentence about was true. The crucial point in this concept is that more (cid:0) ∗ sentences can be formulated about than those transferred from a sentence (cid:0) about (andmanyoftheseadditionalsentencesaretrue).Later,theseadditional (cid:0) sentenceswillbecalledsentencesaboutnonstandardobjects.Afundamentalpoint is that true sentences about nonstandard objects can be combined to give a true sentence ∗α about ∗ which can be obtained by transferring some sentence α (cid:0) about . This allows us to conclude that α is true. (cid:0) To make this approachprecise, one has of course to define what is meant by a “sentence α about ”.Then one has to define what is meant by the transferred sentence ∗α. This is t(cid:0)he first problem we shall attack. Afterthisisdone,therearisesthefundamentalquestion:Doesthereactually ∗ exist an object with the required properties? Or does in contrastthe assump- (cid:0) ∗ tion that such an object exists even lead to a contradiction? (cid:0) The answer to the first question is “yes” if one assumes the axiom of choice (which we therefore do throughout). For this reason the answer to the second question is “no” (even if one rejects the axiom of choice). However, the axiom of choice really is essential. Applying theaboveideas,onecan“explicitlyconstruct”objects(inthenon- standard world) which in principle cannot be constructed in the standard world. Such objects are e.g. sets which are not Lebesgue-measurable or so-called Hahn- Banachlimits: Itis possible toprovetheexistence ofsuchobjectsinthe standard world by means of the axiom of choice, but it is not possible to give explicit for- mulas for them without the axiom of choice (even if one allows a weaker form of this axiom which allows countable recursive or nonrecursive choices). In fact, assuming the consistency of a so-calledinaccessible cardinal, this was first proved in the famous paper [Sol70]. Since in the nonstandardworldwe canreally “calculate” with such objects, it is easy to obtain results which cannot be obtained by standardmethods, or only with very abstractapplications of the axiom of choice. Thus, in a sense, nonstandard analysis might just be considered as a machinery to simplify such abstract applications of the axiomof choice by providing objects which implicitly contain this application. Of course,nonstandardanalysis means actually much more, but in the author’s opinion this is the most impor- tantadvantageofnonstandardanalysisoverstandardanalysis:Tohaveconvenient §1 Introduction 3 (almost “explicit”) representations of certain obects like Hahn–Banach limits for which by standard methods more or less only their mere existence can be proved with the axiom of choice. Of course,the above property means that the axiom of choice must actually ∗ be involvedinthe definition ofthe nonstandardworld(or ); we willsee that(a (cid:0) ratherstrong form of) this axiomcomes into play by the choice of an appropriate so-called filter. Due to this crucial role of the axiom of choice in nonstandard analysis, we will assume it throughout. Somebody who rejects the general axiom of choice always has to replace phrases like “... then ... is true” by a phrase like “... then it does not lead to a contradiction to assume that ... is true”. The study of nonstandard analysis naturally divides into two parts: One ∗ ∗ part is to define , and the other part is to “work” with by using the above (cid:0) (cid:0) describedtransferringofsentences.Thefirstpartbelongstotherealmofso-called model theory while the second part can be considered as the actual nonstandard analysis(oralsojustasanapplicationofthefirstpart).Itturnsoutthatthesecond partcanbe done toa largeextentwithoutappealingto the firstpart,i.e.without ∗ explicitly knowing how is defined. There even is an approach to nonstandard (cid:0) analysis (Nelson’s internal set theory [Nel77]; see also e.g. [vdB87, LG81, Ric82, ∗ Rob88]) which completely hides the definition of , and only uses some axioms ∗ (cid:0) to describe a new set theory for . However, we shall use Robinson’s approach (cid:0) ([Rob70],seealso[AFHKL86,Cut88,Dav77,Gol98,HL85,LR94,Lux73,Lux69b, ∗ SL76, SB86]) which is also concerned with the definition of . (cid:0) This hasnotonlythe advantagethatwe workwith“moreconcrete”objects. ∗ Butthishasalsoanimportantpracticaladvantage:Inthedefinitionof onehas (cid:0) manychoices,morethancanbedescribedbyany axiomaticsystem(thisisrelated withthe axiomofchoicewhichweuse for the construction:Roughlyspeaking,we can fix any finite number of choices in a way that we like). Thus, by choosing an ∗ “appropriate”definitionof ,wecangetsomeadditionalproperties.Thisiswhy (cid:0) theRobinson/LuxemburgapproachisactuallymorepowerfulthantheNelsonap- proachtononstandardanalysis.Therearesomeapplicationswherethis difference really plays a role. A comparison of the two approaches can be found in [DS88] (seealso[CK90,LR94]).Thereisanothermorealgebraicapproachtononstandard analysis via the so-called “Ω-calculus” (see [Lau86]) which, however, is in essence contained in the Robinson/Luxemburg approach. For another approach due to Hrbacek(whichis similartoNelson’sapproach)andseveralotherapproachesand comparisons,we refer the reader to the monograph [RK04]. The crucialpointforfurther applicationsofnonstandardanalysisisthatthe ∗ definitionofanonstandardobject X is notonlypossible forthe caseX = but (cid:0) also for any other objects X, for example a topological space. 4 Chapter 1. Preliminaries 1.2 Archimedean Fields and Infinitesimals ∗ As mentioned in Section 1.1, the idea that the set (containing infinitesimals) (cid:0) should have the same properties as soon leads to severe difficulties. We shall (cid:0) discuss these difficulties now in more detail. Recall that a relation ≤ on a set X is called an order, if 1. a≤a. 2. a≤b and b≤a implies a=b. 3. a≤b and b≤c implies a≤c. The order is called total, if for each two elements a,b of the set we have either a≤b orb≤a.We write a<b to denote thata≤b anda(cid:4)=b.The orderis called well-order if each nonempty set has a smallest element. Each well-order is a total order (consider the set {a,b} to see this). A set X with two operations + and · is called a (commutative) field, if X is a commutative group with respect to + (we denote the neutral element by 0 ), X and X \{0 } is a commutative group with respect to · (we denote the neutral X element by 1 ), 0 ·a = a·0 = 0 , and if furthermore the distributive law X X X X a(b+c)=ab+ac holds. X is a totally ordered field if it is equipped with a total order such that the relations a ≤ b and c ≥ 0 imply a+c ≤ b+c and ac ≤ bc. X Note that this implies a2 ≥ 0 for each a ∈X: For a≥ 0 , this is clear, and for X X a < 0 , we have 0 = a−a ≤ 0 −a = −a, and so −a ≥ 0 which implies X X X X a2 =(−a)2 ≥0 , as claimed. X EachtotallyorderedfieldX containsa“canonicalcopy”oftheset ,namely {1 ,1 +1 ,1 +1 +1 ,...} (we write :={1 ,2 ,...}). Note(cid:0)that X X X X X X X X X X is indeed infinite, since 1 <2 < 3 < ···(cid:0)because 12 =1 >0 , and so(cid:0)e.g. X X X X X X 2 < 2 +1 = 3 , and so on. Similarly, X contains a “canonical copy” of the X X X X sets and ofintegerandrationalnumbers.By“canonicalcopy”,wemeanthat there(cid:1)is an(cid:2)isomorphism, i.e. a bijection f : → which preserves the order X (cid:2) (cid:2) and the arithmetic operations (e.g. f(x+y)=f(x)+f(y)). Theorem1.1. In a totally ordered field X, the following statements are equivalent: 1. X has the Archimedean property. For each x ∈ X there is some n ∈ X such that n>x. (cid:0) 2. X has the Eudoxosproperty: For each ε∈X, ε>0 , there is some n∈ X X such that n−1 <ε. (cid:0) 3. is dense in X, i.e. for each x<y there is some q ∈ with x<q <y. X X (cid:2) (cid:2) Proof. Assume that X has the Archimedean property, and x < y. Then we find some n ∈ such that n > (y −x)−1 and some m ∈ with m > nx and X X m > −nx.(cid:0)Let z be the smallest number of {−m,...,m}(cid:0)which satisfies z > nx,