Long Sequences and Neocompact Sets

Sergio Fajardo and H. Jerome Keisler

1 Introduction

Users of non-standard methods in mathematics have always been interested in the following question: what does non-standard analysis offer the mathematical community? The issues raised by this question are neverending and there is a whole spectrum of possible answers.

The paper [HK] offered an explanation from the point of view of logic, explaining how the principles used in the superstructure approach to nonstandard analysis are related to standard mathematical practice. One difficulty is that the mathematical community is not agreed on what “standard mathematical practice” is. [HK] used mathematical logic to provide a formal framework where these issues can be discussed.

The question posed above was approached from a different point of view in the series of papers beginning with [K1] and continuing with [FK1], [CK], [FK2],

[K2] and [K3]. This series of papers develops the notion of a neometric space, and the whole program is explained in the survey paper [K6] in this volume. The approach may be intuitively described as follows. Start from a part of mathematics, probability theory, where nonstandard methods have clearly offered new insights and enriched the field with new and interesting results. Then isolate and present in “standard terms” those features of nonstandard practice that have made this success possible. The results appeared in [FK1] and [FK2] where the notions of neocompact sets and neometric families were presented, and the basic mathematical theory around these new concepts was developed.

A few words about these two papers will help to explain our reason for writing the present paper. In [FK1], entitled “Existence Theorems in Probability Theory”, we developed a standard theory which captured the key elements from nonstandard analysis that made it possible to prove new existence theorems in stochastic analysis (see [AFHL], [K4] and [K5]). Using neocompact sets and neometric spaces we introduced a new class of probability spaces called “Rich Probability Spaces” and then proceeded to show that in those spaces the results obtained using nonstandard methods are true. The main new ingredient is that these results, in the new setting, are proved within standard mathematical practice. We just asked our readers to accept the existence of such spaces and then proceed to see what could be done with them.

In [FK2] we showed that rich spaces exist, a result that requires nonstandard analysis, and presented the theory of neometric spaces within the most general possible nonstandard framework, which we called the huge neometric family. There we explained how the properties of internal sets in nonstandard hulls give rise to neocompact sets and how the saturation property of the nonstandard universe translates into countable compactness for the neometric family.

A mathematician accustomed to working with nonstandard methods, and in particular within probability theory using liftings and standard parts, may be surprised by the way the results are presented in those papers. Our aim in this paper is to shed some light on the origins of our ideas. We are going to present “a nonstandard theory which explains the standard theory that came out of observing nonstandard practice.” Moreover, the results here can be used as a translation tool between traditional nonstandard arguments and the new theory of neometric spaces.

The idea centers around a fundamental fact from nonstandard analysis: sequences indexed by N, the natural numbers, can be extended to sequences indexed by ^{∗}N, the hyperintegers. This is the reason for the name long sequence. This elementary procedure allows us to capture many important facts from nonstandard practice.

We shall refer to the survey paper [K6] in this volume for the definitions and basic facts concerning the general notion of a neometric family, and in particular the huge neometric family.

Long sequences are introduced in Section 2 of this paper, and the theory is developed further and applied to the huge neometric family in Section 3. Needless to say, we assume the reader is familiar with the superstructure approach to nonstandard analysis (see [L], [AFHL] and [C]). Acquaintance with [FK1] and [FK2] is highly desirable to get the complete picture of the subject.

This research was partially financed by Colciencias, the University of los Andes, the National Science Foundation, and the Vilas Trust Fund at the University of Wisconsin-Madison.

2 Long Sequences

One way to bring nonstandard analysis to bear in proofs by convergence is to use sequences indexed by the hyperintegers rather than the integers. We shall call such sequences long sequences. The paper [K2] made extensive use of long sequences, without using this name. In this section we develop some relations between long sequences and neometric spaces.

We fix an ℵ_{1}-saturated nonstandard universe and let (H, B, C) be its huge neometric family as defined in [K6].

Definition 2.1 A function ⟨x_{n}⟩ mapping N into a set S will be called a sequence in S, and a (possibly external) function ⟨x_{J}⟩ mapping ^{∗}N into S will be called a long sequence in S.

As a warmup, before establishing the connection between long sequences and neocompact spaces, we prove some basic facts about sets of hyperintegers and long sequences. We frequently use the following consequences of ω_{1}-saturation (e.g. see [SB]):

Lemma 2.2 (i) The infinite hyperintegers have coinitiality ω_{1}, that is, every countable set of elements of ^{∗}N − N has an infinite lower bound.

(ii) For every internal set S and every sequence ⟨X_{n}⟩ in S, there exists an internal long sequence ⟨Y_{J}⟩ in S such that Y_{n} = X_{n} for all n ∈ N. □

Definition 2.3 We say that a statement ϕ(J) holds a.e., or that ϕ(J) holds for all sufficiently small infinite J, if there is an infinite hyperinteger K such that ϕ(J) is true for all infinite hyperintegers J ≤ K.

The following lemma is often used to verify that a statement holds a.e.

Lemma 2.4 (i) (Overspill principle) If S is an internal subset of ^{∗}N, then J ∈ S a.e. if and only if n ∈ S for all but finitely many n ∈ N.

(ii) (Countable completeness) The set of all S ⊂ ^{∗}N such that J ∈ S a.e. is a countably complete filter.

Proof: (i) is in any book on nonstandard analysis. (ii) If J ∈ S a.e. and S ⊂ T, then obviously J ∈ T a.e. Suppose J ∈ S_{n} a.e. for all n ∈ N, and let S = ⋂_{n} S_{n}. Then for each n ∈ N there is an infinite hyperinteger K_{n} such that J ∈ S_{n} for all infinite J ≤ K_{n}. By ω_{1}-saturation there is an infinite hyperinteger K such that K ≤ K_{n} for all n ∈ N. Then J ∈ S for all infinite J ≤ K, so J ∈ S a.e. □

The overspill principle will often be used in the following form.

Corollary 2.5 Let ⟨X_{J}⟩ be an internal long sequence in ^{∗}R and let b ∈ ^{∗}R. (i) X_{J} ≤ b a.e. if and only if X_{n} ≤ b for all but finitely many n ∈ N. (ii) X_{J} ≈ 0 a.e. if and only if

lim_{n→∞} st(X_{n}) = 0.

Proof: (i) Apply the overspill principle to the internal set S = {J ∈ ^{∗}N : X_{J} ≤ b}. (ii) follows from (i) and countable completeness. □

We now turn to long sequences in neometric spaces in the huge neometric family (H, B, C). For the remainder of this paper, M and N will always belong to H. The following is the key new concept we introduce in this paper.

Definition 2.6 If ⟨x_{J}⟩ is a long sequence in M and ⟨X_{J}⟩ is an internal long sequence in M̄ such that x_{J} = ^{o}X_{J} for all finite J and all sufficiently small infinite J, we say that ⟨X_{J}⟩ lifts ⟨x_{J}⟩. By an M-sequence we shall mean a long sequence ⟨x_{J}⟩ in M which has a lifting. A (short) sequence ⟨x_{n}⟩ of elements of M will be said to be M-extendible if it is the restriction to ^{∗}N of some M-sequence ⟨x_{J}⟩, and ⟨x_{J}⟩ will be called an M-extension of ⟨x_{n}⟩.

By ω_{1}-saturation, for every sequence ⟨x_{n}⟩ in M there is an internal long sequence ⟨X_{J}⟩ such that X_{n} lifts x_{n}, and hence X_{n} ∈ monad(M) for each n ∈ N. If in addition we have X_{J} ∈ monad(M) a.e., then the sequence ⟨x_{n}⟩ is M-extendible and its Mextension is the M-sequence given by x_{J} = ^{o}X_{J} a.e. If ⟨x_{n}⟩ is an M-extendible sequence, we use the convention that ⟨x_{J}⟩ denotes an M-extension of ⟨x_{n}⟩.

The next proposition shows that the notion of an M-extendible sequence is a generalization of the notion of a convergent sequence.

Proposition 2.7 (i) If

lim_{n→∞} x_{n} = b

in M, then ⟨x_{n}⟩ is M-extendible and ⟨x_{J}⟩ = b a.e.

(ii) If ⟨x_{n}⟩ is M-extendible and ⟨y_{n}⟩ is a sequence in M such that

lim_{n→∞} ρ(x_{n}, y_{n}) = 0,

then ⟨y_{n}⟩ is M-extendible.

Proof: (i) Let X_{n} lift x_{n}, let Y lift b, and extend ⟨X_{n}⟩ to a long sequence ⟨X_{J}⟩. By overspill, X_{J} ≈ Y and hence ^{o}X_{J} = b ∈ M a.e. Therefore ⟨x_{J}⟩ = ⟨^{o}X_{J}⟩ is an M-extension of ⟨x_{n}⟩ and x_{J} = b a.e.

(ii) Let ⟨x_{J}⟩ be an M-extension of ⟨x_{n}⟩ and let ⟨X_{J}⟩ lift ⟨x_{J}⟩. For each n ∈ N, let Y_{n} lift y_{n}, and by ω_{1}-saturation let ⟨Y_{J}⟩ be an internal long sequence in M̄ extending ⟨Y_{n}⟩. Then

lim_{n→∞} ^{o}ρ̄(X_{n}, Y_{n}) = 0,

so by overspill, X_{J} ≈ Y_{J} a.e. It follows that ⟨Y_{J}⟩ lifts an M-sequence ⟨y_{J}⟩ which extends ⟨y_{n}⟩, whence ⟨y_{n}⟩ is M-extendible. □

Given a product M × N of two spaces M, N ∈ H, ⟨z_{J}⟩ is an (M × N)-sequence if and only if there is an M-sequence ⟨x_{J}⟩ and an N -sequence ⟨y_{J}⟩ such that z_{J} = (x_{J}, y_{J}) a.e. Thus if ⟨x_{n}⟩ is M-extendible and ⟨y_{n}⟩ is N -extendible, then the sequence of pairs ⟨z_{n}⟩ = ⟨(x_{n}, y_{n})⟩ is M × N -extendible, and z_{J} = (x_{J}, y_{J}) a.e.

The following shows that the M-extension of a sequence is unique a.e.

Proposition 2.8 (i) Let ⟨x_{J}⟩ and ⟨y_{J}⟩ be M-sequences. Then

lim_{n→∞} ρ(x_{n}, y_{n}) = 0

if and only if x_{J} = y_{J} a.e.

(ii) (Uniqueness of the M-extension) Let ⟨x_{J}⟩ and ⟨y_{J}⟩ be two M-extensions of the same sequence ⟨x_{n}⟩ in M. Then x_{J} = y_{J} a.e.

Proof: (i) Let ⟨X_{J}⟩ lift ⟨x_{J}⟩ and ⟨Y_{J}⟩ lift ⟨y_{J}⟩. By overspill, the following are equivalent:

lim_{n→∞} ρ(x_{n}, y_{n}) = 0.

lim_{n→∞} ^{o}ρ̄(X_{n}, Y_{n}) = 0.

ρ̄(X_{J}, Y_{J}) ≈ 0 a.e.

x_{J} = y_{J} a.e.

(ii) is a special case of (i). □

Proposition 2.9 If ⟨x_{n}⟩ is M-extendible, then for each c ∈ M, the sequence ⟨ρ(x_{n}, c)⟩ is bounded in R.

Proof: Let ⟨X_{J}⟩ be a lifting of an M extension ⟨x_{J}⟩ of ⟨x_{n}⟩. Suppose ρ(x_{n}, c) is not bounded. Let c̄ lift c. Then for each k ∈ N there are arbitrarily large n ∈ N such that ρ(x_{n}, c) > k, and hence ρ̄(X_{n}, c̄) ≥ k. By overspill, for each infinite K ∈^{∗} N there is an infinite J ≤ K such that ρ̄(X_{J}, c̄) is infinite. This contradicts the hypothesis that ⟨X_{J}⟩ is a lifting of ⟨x_{J}⟩. □

3 The Huge Neometric Family

We now give conditions for neocompactness, neoclosedness, neocontinuity, and neoseparability in the huge family H in terms of long sequences. The first proposition is crucial.

Proposition 3.1 Let ⟨x_{J}⟩ be an M-sequence. Then for all sufficiently small infinite K, the set {x_{J} : J ≤ K} is basic in M and the set

{x_{J} : J ≤ K and J is infinite }

is neocompact in M.

Proof: Let ⟨X_{J}⟩ lift ⟨x_{J}⟩ with respect to M. Then for all sufficiently small infinite K, ^{o}X_{J} = x_{J} for all J ≤ K. For any such K, let

B = {X_{J} : J ≤ K}, C = {x_{J} : J ≤ K},
D = {x_{J} : J ≤ K and J is infinite }.

Then B is internal and C = ^{o}B, so C is basic. Moreover,

D = ^{o}(⋂_{n}(B − {X_{m} : m ≤ n})),

so D is neocompact. □

We need the notion of a countably determined set, which was introduced by Henson [He] and played an important role in [K2].

Definition 3.2 A set D ⊂ M̄ is countably determined if there is a countable sequence ⟨D_{n}⟩ of internal subsets of M̄ such that D is an infinite Boolean combination of the D_{n}’s. Equivalently, there is a countable sequence ⟨D_{n}⟩ of internal subsets of M̄ and a set S of subsets of N such that

D = ⋃_{F ∈S}( ⋂_{n∈F} D_{n}).  (1)

In fact, this representation can be chosen so that for any distinct F, G ∈ S, the intersections ⋂_{n∈F} D_{n} and ⋂_{n∈G} D_{n} are disjoint from each other.

Note that every internal, Π_{1}^{0}, and Σ_{1}^{0} set is countably determined.

Theorem 3.3 A set C ⊂ M is neocompact if and only if (a) The monad of C is countably determined, and

(b) Every (short) sequence ⟨x_{n}⟩ in C has an M-extension to a long sequence ⟨x_{J}⟩ in C.

Proof: First assume that C is neocompact. By Basic Fact 2.3 in [K6], there exists a sequence of internal sets ⟨C_{n}⟩ such that

monad(C) = ⋂_{m}((C_{m})^{1/m}).

So the monad of C is countably determined. Let ⟨x_{n}⟩ be a (short) sequence in C. ⟨x_{n}⟩ has a lifting ⟨X_{J}⟩. For each n and m, X_{n} ∈ (C_{m})^{1/m}. By overspill and countable completeness,

X_{J} ∈ ⋂_{m}((C_{m})^{1/m}) a.e.,

so ⟨X_{J}⟩ lifts an M-extension ⟨x_{J}⟩ in C.

Now assume that the monad of C is countably determined and that every (short) sequence ⟨x_{n}⟩ in C has an M-extension to a long sequence ⟨x_{J}⟩ in C. Then the monad of C can be represented in the form ( 1) with any two distinct intersections being disjoint. We claim that

monad(C) = ⋂{B : for some finite s ⊂ N, B = ( ⋃_{n∈s} C_{n}) ⊃ monad(C)}.

This will show that the monad of C is a Π_{1}^{0} set, and hence that C is neocompact. Clearly monad(C) is included in the right side. Suppose X belongs to the right side. Let G = {n ∈ N : X ∉ C_{n}}. Since X belongs to the right side, for each n there exists

Y_{n} ∈ monad(C) − ⋃{C_{k} : n ≥ k ∈ G}.

Let y_{n} = ^{o}Y_{n}. Then ⟨y_{n}⟩ is a (short) sequence in C, so by hypothesis it has an M-extension ⟨Y_{J}⟩ in C. We have Y_{J} ∈ monad(C) for all J. By overspill,

Y_{J} ∉ ⋃{C_{k} : k ∈ G} a.e.

Take an infinite J with this property. Then for some F ∈ S, Y_{J} ∈ ⋂_{n∈F} C_{n}. Moreover, for all n ∈ F, Y_{J} ∈ C_{n} and hence n ∉ G and X ∈ C_{n}. Therefore

X ∈ ⋂_{n∈F} C_{n} ⊂ monad(C).

This proves our claim and completes the proof. □

Just as Basic Fact 2.3 in [K6] shows that the monad of a neocompact set is countably determined, Basic Fact 2.4 in [K6] shows that the monad of a neoseparable set is countably determined.

The following corollary is a good illustration of how our neometric theory is closely related to the classical theory of metric spaces. Notice what happens if you replace “M-extendible” by “relatively compact”.

Corollary 3.4 A sequence ⟨x_{n}⟩ in M is M-extendible if and only if there is a neocompact set C ⊂ M such that x_{n} ∈ C for all n ∈ N.

Proof: If ⟨x_{n}⟩ has an M-extension ⟨x_{J}⟩, then by Proposition 3.1 the set C = {x_{J} : J ≤ K} is neocompact for some infinite K, and {x_{n} : n ∈ N} ⊂ C ⊂ M. If C is neocompact and {x_{n} : n ∈ N} ⊂ C ⊂ M, then ⟨x_{n}⟩ is M-extendible by Theorem 3.3. □

In applications of long sequences, it is important to know which sequences are M-extendible. We can use Corollary 3.4 to characterize the M-extendible sequences in various particular neometric spaces which have been studied in [K1], [FK1] and [FK2]. The next example characterizes the extendible sequences in a nonstandard hull.

Example 3.5 A sequence ⟨x_{n}⟩ is H(M̄, c)-extendible if and only if ρ(x_{n}, d) is bounded where d ∈ H(M̄, c).

Proof: By Proposition 2.9, for any H(M̄, c)-extendible sequence ⟨x_{n}⟩ and any d ∈ H(M̄, c), the sequence ⟨ρ(x_{n}, d)⟩ is bounded. Suppose ⟨x_{n}⟩ is a sequence in H(M̄, c) such that ρ(x_{n}, d) has a finite bound b. Each x_{n} belongs to the closed ball B = {y ∈ H(M̄, c) : ρ(y, d) ≤ b}. B is the standard part of an internal set and is therefore basic in H(M̄, c). Thus by Corollary 3.4, ⟨x_{n}⟩ is H(M̄, c)-extendible. □

Let’s now consider standard neometric spaces. We shall see that the only Mextendible sequences on a standard neometric space M ∈ S are the trivial ones, that is, the relatively compact sequences.

By definition, a sequence ⟨x_{n}⟩ in a complete metric space M is relatively compact if there is a compact set C ⊂ M which contains each x_{n}, or equivalently, every subsequence of ⟨x_{n}⟩ has a convergent subsequence. Thus a sequence in Euclidian space R^{m} is relatively compact if and only if it is bounded. By Corollary 3.4, in every neometric space M, every relatively compact sequence is M-extendible.

Example 3.6 Let M be a standard neometric space. Then a sequence ⟨x_{n}⟩ in M is M-extendible if and only if ⟨x_{n}⟩ is relatively compact.

Proof: This follows from Corollary 3.4 and Basic Fact 2.5 in [K6]. □

For the following examples let Ω be a Loeb probability space. The paper [K1] gave characterizations of the M-extendible sequences when M is a complete separable metric space and M is either the space L^{0}(Ω, M) of Loeb measurable functions with the metric of convergence of probability or the space L^{p}(Ω, M) where p ∈ [1, ∞). In fact, these results were the original inspiration for the long sequences approach to neometric spaces.

If x ∈ L^{0}(Ω, M), the Borel probability measure on M induced by x is denoted by law(x). The space of all Borel probability measures on M with the Prohorov metric is denoted by Meas(M). (See, for example [EK]).

Example 3.7 ([K1], Theorem 3.2 and Lemmas 7.2 and 7.4). Let Ω be a Loeb probability space and M be a complete separable metric space.

(i) A sequence ⟨x_{n}⟩ is L^{0}(Ω, M)-extendible if and only if ⟨law(x_{n})⟩ is relatively compact in Meas(M).

(ii) Let p ∈ [1, ∞). A sequence ⟨x_{n}⟩ is L^{p}(Ω, M)-extendible if and only if ⟨law(x_{n})⟩ is relatively compact in Meas(M) and (ρ(x_{n}(·), a))^{p} is uniformly integrable for each a ∈ M.

The next result gives another characterization of neocompact sets in the case that M is neoseparable.

Proposition 3.8 Suppose M is neoseparable. A set C ⊂ M is neocompact if and only if C is neoclosed in M and (b) of Theorem 3.3 holds, that is, every sequence ⟨x_{n}⟩ in C has an M-extension ⟨x_{J}⟩ in C.

Proof: Neocompactness implies neoclosed and (b) by Basic Fact 1.1 in [K6] and Theorem 3.3. Assume that (b) holds and that C is neoclosed but not neocompact. By Basic Fact 2.6 in [K6], C has a countable covering {O_{n} : n ∈ N} by neoopen sets in M which has no finite subcover. Let C_{n} = C − (⋃_{k≤n} O_{k}). Then ⟨C_{n}⟩ is a decreasing chain of nonempty neoclosed sets in M, and ⋂_{n} C_{n} is empty. Choose x_{n} ∈ C_{n}. Then x_{n} ∈ C, and by (b) we can extend ⟨x_{n}⟩ to a long sequence ⟨x_{J}⟩ in C. By Proposition 3.1 we may choose an infinite K so that the set S = {x_{J} : J ≤ K} is basic. Then ⟨S ∩C_{n}⟩ is a decreasing chain of neocompact subsets of C. By countable compactness of the huge neometric family, the intersection

⋂_{n}(S ∩ C_{n}) = S ∩ (⋂_{n} C_{n})

is nonempty, and this is a contradiction. □

Theorem 3.9 Let C ⊂ M. If C is neoclosed in M then

(c) For every M-sequence ⟨x_{J}⟩ such that x_{n} ∈ C for all n ∈ N, we have x_{J} ∈ C a.e.

If the monad of C is countably determined, then C is neoclosed in M if and only if this condition holds.

Proof: Suppose first that C is neoclosed in M. Let ⟨x_{n}⟩ be a sequence in C and ⟨x_{J}⟩ be an M-extension of ⟨x_{n}⟩. By Proposition 3.1 there is an infinite K such that the set D = {x_{J} : J ≤ K} is basic in M. Then C ∩ D is neocompact in M, and x_{n} ∈ C ∩ D for all n ∈ N. By Theorem 3.3, ⟨x_{n}⟩ has an M-extension to a long sequence ⟨y_{J}⟩ in C ∩ D. By uniqueness of the M-extension, y_{J} = x_{J} a.e. Then x_{J} ∈ C a.e., and (c) is proved.

Now suppose that the monad of C is countably determined and (c) holds. Let D be neocompact in M. Then the monad of D is countably determined. Since

monad(C ∩ D) = monad(C) ∩ monad(D),

monad(C ∩ D) is countably determined. Let ⟨x_{n}⟩ be a sequence in C ∩ D. By Theorem 3.3, ⟨x_{n}⟩ has an M-extension to a long sequence ⟨x_{J}⟩ in D. By condition (c), x_{J} ∈ C a.e. Then ⟨x_{J}⟩ is a long sequence in C ∩ D. We have shown that conditions (a) and (b) of Theorem 3.3 hold for C ∩ D. By Theorem 3.3, C ∩ D is neocompact in M, so C is neoclosed in M. □

Corollary 3.10 Suppose C ⊂ M, C is neoclosed, and ⟨x_{J}⟩ is an M-sequence such that

lim_{n→∞} ρ(x_{n}, C) = 0.

Then x_{J} ∈ C a.e.

Proof: For each n ∈ N we may choose y_{n} ∈ C such that ρ(x_{n}, y_{n}) ≤ 2ρ(x_{n}, C). Then

lim_{n→∞} ρ(x_{n}, y_{n}) = 0,

so by Propositions 2.7 and 2.8, ⟨y_{n}⟩ is M-extendible and x_{J} = y_{J} a.e. Theorem 3.9 shows that y_{J} ∈ C a.e., and therefore x_{J} ∈ C a.e. □

Observe that from the above theorem it follows right away that every neoclosed set in the huge neometric family is closed. Basic Fact 1.7 in [K6] says that this is true in all neometric families. Now, let’s take a look at a characterization of neocontinuity in terms of M-sequences.

Theorem 3.11 Let C ⊂ M be neoclosed, and let f : C → N. If f is neocontinuous from M to N, then

(d) For any M-sequence ⟨x_{J}⟩ in C, ⟨f(x_{J})⟩ is an N -sequence.

If the monad of the graph of f is countably determined, then f is neocontinuous from M to N if and only if this condition holds.

Proof: This generalizes a result from [K2]. Suppose first that f is neocontinuous from M to N. Let ⟨x_{J}⟩ be an M-sequence in C. By Proposition 3.1 there is an infinite K such that the set D = {x_{J} : J ≤ K} is basic in M. By Basic Fact 2.7 in [K6] there is an internal function F such that ^{o}F (X) = f(^{o}X) for all X ∈ monad(D). Since ⟨X_{J}⟩ and F are internal, ⟨F (X_{J})⟩ is internal. For all J ≤ K, oF (X_{J}) = f(^{o}X_{J}) = f(x_{J}). Therefore ⟨F (X_{J})⟩ is a lifting of ⟨f(x_{J})⟩ with respect to N, so ⟨f(x_{J})⟩ is an N -sequence. This proves the first half.

Now suppose the monad of the graph of f is countably determined, and assume (d). Let D ⊂ C be neocompact. Then monad(D) is countably determined, so

monad(f|D) = monad(f) ∩ (monad(D) × N̄)

is countably determined. Let ⟨x_{n}, f(x_{n})⟩ be a sequence in the graph of f|D. By Theorem 3.3, ⟨x_{n}⟩ has an M-extension ⟨x_{J}⟩ in D. By hypothesis, ⟨f(x_{J})⟩ is an N -sequence. Then ⟨x_{J}, f(x_{J})⟩ is an (M × N)-extension of ⟨x_{n}, f(x_{n})⟩ in f|D. Therefore by Theorem 3.9, f|D is neocompact in M × N, so f is neocontinuous from M to N. □

Corollary 3.12 Let C ⊂ M, and let f : C → N be neocontinuous from M to N. If a sequence ⟨x_{n}⟩ in C is M-extendible, then ⟨f(x_{n})⟩ is N -extendible to an N -sequence ⟨y_{J}⟩, and f(x_{J}) = y_{J} a.e. □

Finally, we give a necessary condition for neoseparability in terms of long sequences. An open question is whether this condition, together with the condition that the monad of the set is countably determined, is sufficient for neoseparability.

Proposition 3.13 Let C be neoseparable in M, and let ⟨x_{J}⟩ be an M-sequence such that x_{J} ∈ C a.e. Then for each k ∈ N, x_{n} ∈ C^{1/k} for all but finitely many n ∈ N.

Proof: Let monad(C) = ⋂_{n} ⋃_{m} (C_{m})^{1/n}. Suppose that there is a k ∈ N and an infinite subset p ⊂ N such that x_{n} ∉ C^{1/k} for all n ∈ p. By taking a subsequence, we may assume without loss of generality that x_{n} ∉ C^{1/k} for all n ∈ N. Let ⟨X_{J}⟩ lift ⟨x_{n}⟩. Then X_{J} ∈ ⋂_{n} ⋃_{m} (C_{m})^{1/n} a.e., and hence X_{J} ∈ ⋃_{m} (C_{m})^{1/k} a.e. However, for all n ∈ N we have X_{n} ∉ ⋃_{m} (C_{m})^{1/k} because ^{o}(⋃_{m} (C_{m})^{1/k}) ⊂ C^{1/k}. Then for each n, m ∈ N we have X_{n} ∉ (C_{m})^{1/k}, and by overspill, X_{J} ∉ (C_{m})^{1/k} a.e. By countable completeness, X_{J} ∉ ⋃_{m} (C_{m})^{1/k} a.e., contrary to our previous assumption. □

References

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[C] N. Cutland. Editor. Nonstandard Analysis and its Applications. LMSST 10. Cambridge University Press. 1988.

[CK] N. Cutland and H. J. Keisler. Applications of Neocompact Sets to Navier-Stokes Equations. To appear.

[EK] S. Ethier and T. Kurtz. Markov Processes. Wiley (1986).

[FK1] S. Fajardo and H. J. Keisler. Existence Theorems in Probability Theory. To appear.

[FK2] S. Fajardo and H. J. Keisler. Neometric Spaces. To appear.

[FK3] S. Fajardo and H. J. Keisler. Neometric Forcing. To appear.

[H] C. W. Henson, Analytic Sets, Baire Sets, and the Standard Part Map. Canadian J. Math. 31 (1979), pp. 663-672.

[HK] C. W. Henson and H. J. Keisler. On the strength of nonstandard analysis. J. Symb. Logic, vol. 51 (1986), pp. 377-386.

[K1] H. J. Keisler. From Discrete to Continuous Time. Ann. Pure and Applied Logic 52 (1991), 99-141.

[K2] H. J. Keisler. Quantifier Elimination for Neocompact Sets. To appear.

[K3] H. J. Keisler. Rich and Saturated Adapted Spaces. To appear.

[K4] H. J. Keisler. An Infinitesimal Approach to Stochastic Analysis. Memoirs Amer. Math. Soc. 297 (1984).

[K5] H. J. Keisler. Infinitesimals in Probability Theory. In [C] above.

[K6] H. J. Keisler. A Neometric Survey. This Volume.

[Li] T. Lindstrøm. An Invitation to Nonstandard Analysis; pp 1105 in Nonstandard Analysis and its Applications, ed. by N. Cutland, London Math. Soc. (1988).

Universidad de Los Andes and Universidad Nacional, Bogota, Colombia. sfajardo@cdcnet.uniandes.edu.co

University of Wisconsin, Madison WI, keisler@math.wisc.edu

Mathematical Subject Classification 1991: Primary 03H05 Secondary 26E35, 28E05, 54E35

Keywords: neocompact, neometric, nonstandard analysis, Loeb measure.