1. Loeb Measures 1.1 Introduction Loeb measures, discovered by Peter Loeb in 1975 [71], are very rich stan- dard measure spaces, constructed using nonstandard analysis (NSA). The range of fields in which they have found significant applications is vast, in- cludingmeasureandprobabilitytheory,stochasticanalysis,differentialequa- tions (ordinary, partial and stochastic), functional analysis, control theory, mathematical physics, economics and mathematical finance theory. The richness of Loeb measures makes them good for -constructing measureswithspecialproperties(forexampletherichprob- ability spaces of Fajardo & Keisler [49, 50, 62]); - representing complex measures in ways that make them more manage- able (for example Wiener measure) – see section 1.3.3 below; - modelling physical and other phenomena; - proving existence results in analysis – for example solving differential equations(DEs)ofallkinds(includingpartialDEs,stochasticDEsandeven stochastic partial DEs) and showing the existence of attractors. LaterlectureswilldescribesomerecentusesofLoebspacesthatillustrate these themes – in fluid mechanics, in stochastic calculus of variations and related topics, and in mathematical finance theory. This lecture will outline the basic Loeb measure construction and give some simple applications, with a little of the theory of Loeb integration. Fromonepointofview,Loebmeasuresaresimplyultraproductsofpreviously givenmeasurespaces,suchaswereconsideredinanearlypaperofDacunha- Castelle & Krivine [46]. The roˆle of NSA in their construction is to provide a systematic way to understand their properties, which opens the way for efficient and powerful applications; without this we would have a supply of very rich measure spaces but only ad hoc means to comprehend them. Necessarily these lectures will be somewhat informal and lacking in a great deal of detail. The aim is to convey some of the basic ideas and flavour of Loeb measures and how they work, as well as pointing to the literature where the topics can be pursued in depth. We must begin with a brief and informal look at NSA itself. N.J.Cutland:LNM1751,pp.1–28,2000. (cid:1)c Springer-VerlagBerlinHeidelberg2000 2 1 Loeb Measures 1.2 Nonstandard Analysis 1.2.1 The hyperreals Nonstandardanalysis(discoveredbyAbrahamRobinsonin1960[83])begins with the construction of a richer real line ∗R called the hyperreals or non- standard reals. This is an ordered field that extends the (standard) reals R in two main ways: (1) ∗R contains non-zero infinitesimal numbers; and (2) ∗R contains positive and negative infinite numbers. This is made precise by the following definitions (where |·| is the exten- sion1 of the modulus function to ∗R). Definition 1.1 Let x∈∗R. We say that (i) x is infinitesimal if |x|<ε for all ε>0, ε∈R; (ii) x is finite if |x|<r for some r ∈R; (iii) x is infinite if |x|>r for all r ∈R. (iv) We say that x and y are infinitely close, denoted by x≈y, if x−y is infinitesimal. So x≈0 is another way to say that x is infinitesimal. (v) The monad of a real number r is the set monad(r)={x:x≈r} of hyperreals that are infinitely close to r. Thus monad(0) is the set of in- finitesimals, and monad(r)=r+monad(0). Of course, once a field has non-zero infinitesimals, then there must be infinite elements also – these are the reciprocals of infinitesimals. It follows also that R is enriched in having, for each r ∈R, new elements x with x≈r (taking x=r+δ where δ is infinitesimal). One way to construct ∗R is as an ultrapower of the reals ∗R=RN/U where U is a nonprincipal ultrafilter2 (or maximal filter) on N. That is, ∗R consists of equivalence classes of sequences of reals under the equivalence relation ≡U, defined by (an)≡U (bn) ⇐⇒ {n:an =bn}∈U. Sets in U should be thought of as big sets, or more strictly U-big, with thosenotinU designatedU−small.Theultrafilterpropertymeansthatevery 1 This takes its values in ∗R, and is defined just as in R, so that |x|=x if x≥0 and |x|=−x if x<0. 2 A nonprincipal ultrafilter U on N is a collection of subsets of N that is closed underintersectionsandsupersets,containsnofinitesets,andforeverysetA⊆N has either A∈U or N\A∈U. 1.2 Nonstandard Analysis 3 set is either U-big (those in U) or U-small (those not in U). It is convenient tousetheterminologyU-almost all tomean“forasetAofnaturalnumbers with A∈U”. Using this terminology we can say that the equivalence relation ≡U iden- tifies sequences (an) and (bn) that agree on a U-big set of indices n, or that agree U-almost always. We denote the equivalence class of a sequence (an) by (an)U (sometimes thenotation[(an)]isusedinstead).TherealsRareidentifiedwiththeequiv- alence classes of constant sequences, so that ∗R is then an extension of R. The algebraic operations +,× and the order relation < are extended to ∗R pointwise (after checking that this is safe); strictly the extensions should be denoted ∗+,∗×,∗<, but there is usually no ambiguity if the ∗ is dropped. Itisalmostimmediatethatanexampleofanon-zeroinfinitesimalisgiven by (1,1,1,...)/U. 2 3 Thewaytopicture∗Risasfollows(notethatsomefeaturesinthisdiagram are yet to be explained). Infinitesimal microscope monad(r)={x∈∗R:x≈r} (cid:1) r (cid:3) (cid:3) (cid:2) (cid:4) infinite elements ∗R (cid:1) (cid:2) r 0 1 2 NN+1 standard part mapping (cid:4) r 0 1 2 R The Hyperreals With the above construction of ∗R it is easy to prove: Theorem 1.2 (∗R,+,×,<) is an ordered field. Exercise Prove this! Most of the field axioms follow easily from the fact that they hold at each co-ordinate of the representing sequences. The axiom 4 1 Loeb Measures of inverses is not quite so obvious. If x = (an)U (cid:9)= 0 then we could have an =0 for some indices n. As a hint, note that nevertheless we have an (cid:9)=0 for U-almost all n, so we can define y =(bn)U with bn =a−n1 for those n. For the remaining U-small set of n define bn =0. Now show that xy =1. The axioms for an ordered field are proved in somewhat similar fashion. To see what else can be said about ∗R, first note that all functions f and relations R on R (including unary relations – that is, subsets of R) can be extended to ∗R pointwise – with the extensions denoted by ∗f and ∗R say.3 Asanexercisethereadermightliketoshowthattheextension∗|·|ofthe modulus function defined in this way is the same as that used in Definition 1.1; that is, if x = (an)U and y = ∗|x| = (|an|)U then y = x if x∗ ≥ 0 and y =−xotherwise.Further,showthatxisfinite(accordingtoDefinition1.1) if there is some r ∈R with |an|<r for U-almost all n, and x is infinitesimal if, for every real ε>0 we have |an|<ε for U-almost all n. Important examples of extensions of relations include ∗N,∗Z and ∗Q, the sets of hypernatural numbers, hyperintegers and hyperrationals respectively. A hyperrational number is thus an element x = (an)U with an ∈ Q for U-almost all n. Itisnothardtoseethatthepropertiesoffunctionsf andrelationsRare transferred to (or inherited by) ∗f and ∗R – for example, if f is an injection, so is ∗f, and if R is an equivalence relation then so is ∗R. If f : A → B then ∗f :∗A→∗B. Moreover, connections between functions and relations are also transferred–forexample∗sin2x+∗cos2x=1forallx∈∗R.Thefullextentof this idea is described neatly by the Transfer Principle discussed below. First let us write R=(R,(f)f∈F,(R)R∈R) for the full structure with domain R together with every possible function and relation on it, and then write ∗R=(∗R,(∗f)f∈F,(∗R)R∈R). The following fundamental result gives the complete picture as to which properties of R are inherited by (or transferred to) ∗R. Theorem 1.3 (Transfer Principle) Let ϕ be any first order statement. Then ϕ holds in R ⇐⇒ ∗ϕ holds in ∗R A first order statement ϕ (respectively ∗ϕ) is one that refers to elements of R (respectively ∗R), both fixed and variable, and to fixed relations and functions f,R (respectively ∗f,∗R). First order statements can use the usual 3 By the pointwise extension of a binary relation R ⊂ R×R, say, we mean that ((an)U,(bn)U)∈∗R⇔(an,bn)∈RforU-almostalln;so∗R⊂∗R×∗R.Itiseasy to see that this is equivalent to defining ∗R using a pointwise extension of the characteristic function – i.e. χ∗R((an)U,(bn)U)=(χR(an,bn))U. 1.2 Nonstandard Analysis 5 logical connectives of mathematics, namely and (symbolically ∧), or (∨), implies (→) and not (¬). Moreover, we can quantify over elements (∀x, ∃y) butnotoverrelationsorfunctions(so∀f,∃Rarenotallowed).Herearesome illustrations of this. 1. Density of the rationals in the reals. The density of the rationals in the reals can be expressed by a first order statement ϕ that is a formal version of the following. Between every two distinct reals there is a rational. We could for example take ϕ as the statement ∀x∀y(x<y →∃z(z ∈Q∧(x<z <y))) The transfer principle tells us that ∗ϕ holds in ∗R which means that the hyperrationals are dense in the hyperreals. 2. Discreteness of the ordering of the integers. This can be expressed by a first order statement ψ which is a formal version of the following. Every n∈Z has an immediate predecessor and successor in Z . The Transfer Principle tells us that ∗ψ holds in ∗R which means that Every n∈∗Z has an immediate predecessor and successor in ∗Z . Thus the discreteness of Z is inherited by ∗Z. The reader is invited to check that the immediate predecessor and suc- cessor of a hyperinteger (mn)U ∈ ∗Z are given by (mn ∓1)U. Likewise the density of the hyperrationals can be established quite easily from first prin- ciples (if x = (an)U and y = (bn)U then take z = (cn)U with an < cn < bn where possible). The proof of the Transfer Principle is just a generalisation of the procedure involved in a direct verification of these two examples. The Transfer Principle itself avoids the need to verify properties of ∗R on an ad hoc basis, and instead gives us all properties from the beginning. AkeyresultthatallowsustogetbacktoRfrom∗R(andextendstomore general topological situations) is the following (recall the definition 1.1 of a finite hyperreal). Theorem 1.4 (Standard Part Theorem) If x ∈ ∗R is finite, then there is a unique r ∈ R such that x ≈ r; i.e. any finite hyperreal x is uniquely expressible as x=r+δ with r a standard real and δ infinitesimal. Proof Put r = sup{a ∈ R : a ≤ x} = supA, say. The set A is nonempty and bounded above (in R) since x is finite, and so the least upper bound r exists. It is routine to check that |x−r|<ε for every real ε>0. (cid:18)(cid:19) Definition 1.5 (Standard Part) If x is a finite hyperreal the unique real r ≈x is called the standard part of x. 6 1 Loeb Measures For a finite hyperrealx∈∗R there are two notations (both useful) for the standard part of x: ◦ x=st(x)=the standard part of x. On occasions, when considering extended real valued functions (with val- ues in R=R∪{−∞,∞}), it is convenient to write ◦x=±∞ if x is positive (resp. negative) infinite. Remark The Standard Part Theorem is equivalent to the completeness of R. The next two theorems illustrate the way in which real analysis develops usingtheadditionalstructureof∗R.Forthesakeofcompletenesswegivebrief proofsthatprovideaflavourofthenonstandardmethodology,andespecially the use of the Transfer Principle. Forafullaccountofthedevelopmentofrealanalysisusinginfinitesimals, see any of the references [30, 54, 47, 56, 58, 60, 69]. For the following, note that since a sequence s = (sn)n∈N of reals is just a function s : N → R, its nonstandard extension ∗s = (sn)n∈∗N is simply a function ∗s:∗N→∗R. Theorem 1.6 Let (sn) be a sequence of real numbers and let l∈R. Then sn →l as n→∞ ⇐⇒ ∗sK ≈l for all infinite K ∈∗N. Proof Suppose first that sn →l, and fix infinite K ∈∗N. We have to show that |∗sK −l|<ε for all real ε>0. For any such ε there is a number n0 ∈N such that the following holds in R: ∀n∈N[n≥n0 →|sn−l|<ε] The Transfer Principle now tells us that ∀N ∈∗N[N ≥n0 →|∗sN −l|<ε] istruein∗R.InparticulartakingN =K weseethat|∗sK−l|<εasrequired. Conversely, suppose that ∗sK ≈ l for all infinite K ∈ ∗N. Then, for any given real ε>0, we have ∃K ∈∗N ∀N ∈∗N[N ≥K →|∗sN −l|<ε] The Transfer Principle applied to this statement shows that in R: ∃k ∈N∀n∈N[n≥k →|sn−l|<ε] Taking n0 to be any such k proves that sn →l. (cid:18)(cid:19) For the next result note that if f is a real function defined on the open interval ]a,b[ then∗f is defined on the hyperreal interval∗]a,b[={x∈∗R:a< x<b}, and takes values in ∗R. 1.2 Nonstandard Analysis 7 Theorem 1.7 Let c∈]a,b[ (where a,b,c∈R) and f :]a,b[→R. Then f is continuous at c ⇐⇒ ∗f(z)≈f(c) whenever z ≈c in ∗R. Proof The proof is very similar to that of Theorem 1.6. Suppose first that f is continuous at c, and fix a hyperreal z ≈ c. We have to show that |∗f(z)−f(c)|<ε for all real ε>0. For any such ε there is a number 0<δ ∈R such that the following holds in R: ∀x[|x−c|<δ →|f(x)−f(c)|<ε] The Transfer Principle now tells us that ∀X[|X−c|<δ →|∗f(X)−f(c)|<ε] is true in ∗R. In particular taking X = z we see that |∗f(z)−f(c)| < ε as required. Conversely, suppose that |∗f(z)−f(c)|≈0 for all z ≈c in ∗R. Let a real ε>0 be given. Then taking Y to be any positive infinitesimal the following holds in ∗R: ∃Y∀X[|X−c|<Y →|∗f(X)−f(c)|<ε] The Transfer Principle applied to this statement gives, in R: ∃y∀x[|x−c|<y →|f(x)−f(c)|<ε] Taking δ to be any such y shows that f is continuous at c as required. (cid:18)(cid:19) Beforemovingtothenextsection,itshouldbepointedoutthatthereare several other ways to construct the hyperreals. Moreover, the conventional terminology is misleading in that different constructions do not necessarily give isomorphic structures. All versions of the hyperreals however obey the Transfer Principle, and this is all that is needed to do basic nonstandard real analysis. Indeed, one perfectly workable approach to the subject is an axiomaticone,whichmerelyspecifiesthat∗RisanextensionofRthatobeys the Transfer Principle. (This approach would be parallel to a development of real analysis that proceeds without being concerned with any particular construction of R, using only the assumption that R is a complete ordered field.) 1.2.2 The nonstandard universe To use Robinson’s ideas beyond the realm of real analysis, it is necessary to repeat the construction of ∗R for any mathematical object M that might be needed, giving a nonstandard version ∗M of M that contains ideal elements (such as infinitesimals in the case of ∗R). M could be a group, ring, measure space, metric space or any mathematical object, however complicated. 8 1 Loeb Measures Rather than construct each nonstandard extension ∗M as required, it is more economical to construct at the outset a nonstandard version ∗V of a working portion of the mathematical universe V that contains each object M that might be needed. Then ∗V will contain ∗M for every M ∈ V. Such a construction has the additional advantage that the corresponding Transfer Principle preserves connections between structures as well as their intrinsic properties. Here, briefly, is the way it works. First, for most mathematical practice, an adequate portion of the mathematical universe is the superstructure over R, denoted by V=V(R), defined as follows. V0(R)=R Vn+1(R)=Vn(R)∪P(Vn(R)), n∈N and (cid:1) V=V(R)= Vn(R). n∈N (If V(R) is not big enough to contain all the objects4 required, simply replace the starting set R by a suitable larger set S, giving V=V(S).) The next step is to construct a mapping ∗ : V(R) → V(∗R) which asso- ciates to each object M∈V a nonstandard extension ∗M∈V(∗R). Roughly, we have M ⊂ ∗M with ∗M\M consisting of “ideal” or “nonstandard” el- ements. For example ∗N\N consists of infinite (hyper)natural numbers; if M is an infinite dimensional Hilbert space H together with its finite dimen- sional subspaces then ∗M will contain some infinite hyperfinite dimensional subspaces. The way to visualise the resulting nonstandard universe is as follows. Externalobjects (cid:5) (cid:4) Standard objects (cid:6) A A∗ V=V(R) V(∗R) ∗V(R) (=internal objects) R ∗R The Nonstandard Universe The nonstandard universe is in fact the collection 4 We are now taking the view that all mathematical objects are sets. 1.2 Nonstandard Analysis 9 ∗V={x:x∈∗M for some M∈V} consisting of all new and old members of sets in V. Although ∗V⊂V(∗R), it is crucial to realise that ∗V is not the same as V(∗R). Sets in ∗V are known as internal sets. One way5 to construct ∗V is by means of an ultrapower VN/U although there is a little more work to do (compared to the corresponding construction of ∗R). The set membership relation ∈ that gives the structure (V,∈), when extended pointwise to the ultrapower VN/U, gives a “pseudo- membership” relation E, say, resulting in the structure (VN/U,E). Itisthennecessarytotakethe“Mostowskicollapse”ofthisstructure,which constructs simultaneously the collection ∗V and an injection i:(∗V,∈)→(VN/U,E). Although i is not surjective, its range includes the equivalence class of each constant sequence, and then ∗M is defined by ∗M=i−1((M,M,M,...M)/U). The key property of the nonstandard universe ∗V is a Transfer Principle which again indicates precisely which properties of the superstructure V are inherited by ∗V. Theorem 1.8 (The Transfer Principle) Supposethatϕisaboundedquan- tifier statement. Then ϕ holds in V if and only if ∗ϕ holds in ∗V. A bounded quantifier statement (bqs) is simply a statement of mathe- matics that can be written in such a way that all quantifiers range over a prescribed set. That is, we have subclauses such as ∀x ∈ A and ∃y ∈ B but notunboundedquantifierssuchas∀xand∃y.Mostquantifiersinmathemat- ical practice are bounded (often only implicitly in exposition). A bqs ϕ may also contain fixed sets M from V, which will be replaced in ∗ϕ by ∗M. Membersofinternalsetsareinternal(thisfollowseasilyfromtheconstruc- tion) and since the sets ∗M are also internal, it follows that the information weobtainfromtheTransferPrincipleisentirelyaboutinternal sets.Toillus- trate, the Transfer Principle tells us that any internal bounded subset of ∗R 5 Thissketchofaconstructionof∗Vcanbeskippedwithoutanyloss–itisincluded toshowthatanonstandarduniverseisaverydown-to-earthandnon-mysterious mathematical construct. 10 1 Loeb Measures hasaleastupperbound,whereasthiscanfailforexternal6 sets.Forexample, thesetNisasubsetof∗Rthatisbounded(byanyinfinitehyperreal)buthas noleastupperbound–fromwhichwededucethatNisexternal.Incidentally this demonstrates that there actually are external sets – i.e. V(∗R)\∗V(cid:9)=Ω. An easy application of the Transfer Principle gives the following very useful properties. Proposition 1.9 Let A⊆∗R be an internal set. (a) (Overflow) If A contains arbitrarily large finite numbers then it also contains an infinite number; (b) (Underflow) If A contains arbitrarily small positive infinite7 numbers then it contains a positive finite number. Taking reciprocals gives a corresponding pair of principles for the set of in- finitesimals. As with ∗R it is possible (and quite convenient) to take an axiomatic approach to ∗V, which simply postulates the existence of a set ∗V and a mapping ∗ : V → ∗V that obeys the Transfer Principle. For most purposes (and certainly the construction of Loeb measures) one further assumption is needed, which we now discuss. 1.2.3 ℵ1-saturation A nonstandard universe constructed as a countable ultrapower has an addi- tional property called ℵ1-saturation, which we highlight here because of its importance. Definition 1.10 Anonstandarduniverse ∗V issaidtobe ℵ1-saturated ifthe following holds: if (Am)m∈N(cid:2)isacountabledecreasingsequenceof internal sets witheach Am (cid:9)=Ω, then m∈NAm (cid:9)=Ω. Theorem 1.11 A nonstandard universe ∗V constructed as a countable ultra- power is ℵ1-saturated. Proof (Sketch) Each set Am is represented by a sequence of standard sets (Xm,n)n∈N. Since each Am is nonempty and the sequence is decreasing, then forU-almostall8 nwehaveXm+1,n ⊆Xm,n andXm,n (cid:9)=Ω.Byasystematic modification of the sets Xm,n on a U-small9 set of indices n we may assume that Xm+1,n ⊆ Xm,n and Xm,n (cid:9)= Ω for all n and m. Now pick xn ∈ Xn,n 6 an external set is one that is not internal 7 thatis,foreverypositiveinfinitex∈∗Rthereisanelementa∈Awithainfinite and a<x 8 that is, the set {n:Xm+1,n ⊆Xm,n} belongs to U. 9 i.e. a set that is not in the ultrafilter U.