Foundations of Infinitesimal Stochastic Analysis

STUDIES IN LOGIC AND

THE FOUNDATIONS OF MATHEMATICS VOLUME 119

Editors

J. BARWISE, Stanford D. KAPLAN, LosAngeles H. J. KEISLER, Madison P. SUPPES,Stanford A. S . TROELSTRA, Amsterdam

NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD

FOUNDATIONS OF INFINHESIMAL STOCHASTIC ANKYSIS

K. D. STROYAN Mathematics Department The University of Iowa lo wa City, l owa 52242 U .S.A. and

Jose Manuel BAYOD Facultad de Ciencias Universidad de Santander Santander, Spain

1986

NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD

0 ELSEVIER

SCIENCE PUBLISHERS B.V., 1986

AN rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the priorpermission of the copyright owner.

ISBN: 0 444 87927 7

Published by: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. Box 1991 1000 BZ Amsterdam The Netherlands Sole distributors for the U.S.A.and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52Vanderbilt Avenue NewYork, N.Y. 10017 U.S.A.

Library of Congress Cataloging-inqublicstionh t r Stroyan, K. 0. Foundations of infinitesimal stochastic analysis. (Studies in logic and the foundations of mathematics ; V.

119)

Bibliography: p. Includes index. 1. Stochastic analysis. 2. Mathematical analysis, Nonstandard. .I. Bayod, Jose Manuel. 11. Title. 111. Series.

~ ~ 2 7 4 . 2 1986 . ~ ~ ISBN 0-444-87927-7

519.2

PRINTED IN THE NETHERLANDS

85-28540

viii

ACKNOWLEDGEMENTS This project has taken much longer than expected. Our final worry is that we will forget to thank one of the many people who offered us their help during the many years! We appreciate even the smallest suggestions because we know that a sum of infinitesimals can be infinite. Most of all we thank H. Jerome Keisler for his seminar notes, ideas, examples, criticism, preprints and encouragement. This book would not exist without his help. C. Ward Henson also gave us a great deal of help and an example. Jorg Flum, L. C.r Moore, Jr.. Robert M. Anderson, and Tom L. Lindstrom generously gave us detailed criticism of parts of early drafts. Douglas N. Hoover, Edwin Perkins. L . L. Helms. Nigel Cutland, J. E. Fenstad and Peter A. Loeb sent us their preprints and discussed the K. Jon Barwise, Juan Gatica. project with us at meetings. Eugene Madison, Robert H. Oehmke, John Birch, Constantin Drossos. Gonzalo Mendieta. David Ross. Vitor Neves, Anna Roque. Lee Panetta and others participated in seminars on various parts of the book. We thank all these people for their help and encouragement. It seems to us that the combined effort of many people is what has made this branch of Robinson's Theory of Infinitesimals blossom. Bayod thanks the Fulbright Foundation for partial support during the first part of the project and his colleagues at the University of Santander. who, by increasing their work load. Stroyan allowed him to take a two-semester leave in Iowa. gratefully acknowledges support, years ago, of the National Science Foundation of The United States for summer research that appears in parts of the book. Stroyan thanks The University of Iowa for its tolerance and sometimes generous support of his peculiar research interests. We thank Ada Burns for her superb typing of infinitely many drafts, revisions and corrections of author errors. We also thank Laurie Estrem f o r excellent typing of part of the next-tolast draft. We thank the staff of North-Holland in advance for the production task they are about to undertake. The final draft of this book was prepared with the excel lent new technical word processor T3 from TCI Software Research, Inc. and printed by them on an HP LaserJet+ printer. The series editors, Arjen Sevenster and others at NorthHolland have been patient and helpful in making arrangements for this book to appear in The Studies in Logic and The Foundations o f Mathematics. We are delighted that this book will appear in the same series as Abraham Robinson's classic book on infinitesimal s .

ix

W e dedicate this book to Jerry Keisler f o r h i s professional help and to our wives f o r their emotional support during the project.

to

Jev, Carol and Cristina

X

FOREWORD Bayod obtained support from the Fulbright Foundation to visit The University of Iowa for the 78-79 academic year in order to learn about Abraham Robinson's Theory of Infinitesimals (so-called "Non-Standard Analysis"). We agreed to focus our seminar on infinitesimal analysis of probability and measures because of exciting work of Keisler and Perkins then in progress and with hopes of further applications. We made careful notes, while, unknown to us, Keisler was doing the same thing with his students. The second draft of this book corpbined both sets of notes and comprised roughly the present chapters 0 to 4 . Chapter 0 contains all the 'nonstandard stuff' that our reader needs in order to learn about the applications in this book. The reader who is familiar with the basic principles of infinitesimal analysis can go directly to Chapter 1. Chapter 0 tries to give the beginner in infinitesimal analysis the working tools of the trade without proof. We feel that the logical principles such as Leibniz' (transfer) Principle and the Internal Definition Principle, together with Continuity Principles such as Robinson's Sequential Lemma, saturation and comprehension are the things our beginning reader should focus his attention on. Section (0.1) gives the definition of "all of classical Section (0.2) analysis" in the form of a "superstructure." explains the meaning of Leibniz' Principle and begins to show its usefulness. We believe that our reader can get a working knowledge of these tools of logic by working several of the exercises. Section (0.3) contains more explanation of some basic notions of infinitesimal analysis that are used throughout the book. Section ( 0 . 4 ) contains the important saturation property that we base our measure constructions upon. The approach to measures in this book was initiated by Peter A. Loeb [1975]. Most of the basic results are due to him. but we have given a more elementary new exposition based on inner and outer internally generated measures. This approach replaces the use of Caratheodory's extension theorem by direct elementary arguments. We have also added some fine points and examples not found in the literature. Section (1.1) deals with probability measures, while section (1.2) treats infinite measures. Since infinite measures cause extra technicalities, we have given a short outline for the reader who is anxious to (It appears right apply Loeb's construction to probability. after the Table of Contents.) C. Ward Henson [1979a.b] discovered the connections between Loeb. Bore1 and Souslin sets and first proved uniqueness of hyperfinite extension and the unbounded case. The remainder of Chapter 1 explains the relationship between *finite sums and integrals against

xi

Foreword

hyperfinite measures. Robert M. Anderson [1976. 19821 systematically investigated Radon measures, filled in some of the basics on S-integrability and studied product measures. S-integrability is the main ingredient needed to relate sums and integrals. In Chapter 2 we study the relation between Borel and hyperfinite measures. The basic idea there is to 'pull back with the standard part map'. The case of a completed Borel measure is technically easier to treat, so section (2.1) treats Lebesgue measure independently, while sections (2.2) and (2.3) treat naked sigma algebras and measures. This draws on the works of Anderson and Henson cited above. Anderson and Salim Rashid [1978] and Loeb [1979a] investigated weak standard parts of measures: section (2.4) presents a simple case of those results. Chapter 3 contains a Fubini-type theorem due to H. Jerome Keisler [1977] as well as a mixed Fubini-type theorem. Anderson's work above showed that a hyperfinite product measure extends the product of hyperfinite measures, while Douglas N. Hoover [1978] showed that that extension is strict. The Fubini theorem holds anyway. Chapter 4 has a basic treatment of distributions, laws and independence from the point of view of infinitesimal analysis. Even the "foundations" of stochastic analysis consist of more than measures. Anderson [1976] discussed Brownian motion, including path continuity and Ito's lemma, using Loeb's techniques. Loeb [1975] treated infinite coin tossing and a different approach to the Poisson process. Keisler's [1984] preprint investigated more general processes. (P. Greenwood and R. Hersh [1975] and Edward Nelson [1977] have some infinitesimal analysis of stochastic processes using different techniques.) We discussed these things in the spring of 1979. but only wrote rough notes including some extensions of this work. In the meantime we learned of other fundamental work of Tom L. Lindstrom [1980] and of Hoover and Edwin Perkins [1983]. Our treatment of Chapters 5, 6 and 7 relies most heavily on the work of Keisler. Hoover-Perkins and Lindstrom. Chapter 5 is devoted to 'path properties' of processes. Our treatment of paths with only jump discontinuities is a little different from Lindstrom-Hoover-Perkins'. We precede that with the less technical case of continuous paths and include a lot of details in both cases. Section (5.4) contains results and extensions of results from Keisler [1984] relating Loeb. Borel and Souslin sets on the product space [O.l] x R . Section (5.5) sketches how one makes the extension of these results to C 0 . m ) . The infintesimal analysis is very similar to [O,l], but since the classical metrics are rather technical we avoided a complete account. Chapter 6 gives the basic theory of how events evolve We measure-theoretically in time on a hyperfinite scheme. follow Keisler [1984] again, but extend his results to include "pre-visible measurability" as well as "progressive measurabi 1 i ty " A hyperfinite evolution scheme has

.

xii

Foreword

'measurability' and 'completeness' properties that make i t richer than an arbitrary 'filtered (or adapted) probability space' of the "general theory of processes." We prove that the error "adapted implies progressively measurable" is actually true. We show that previsibility arises from a left filtration. Hence, while we borrowed extensively from Lindstrom [1980] and especially Hoover and Perkins [1983] in later chapters, we adapted their results to Keisler's more specific combinatorial framework. We feel that this would be justified simply because of the concrete "liftings" we obtain. However, Hoover and Keisler [1982] show that this results in no loss of generality in a certain specific logical sense described briefly in the Afterword. The a i m of all our chapters is to give fundamental results needed to apply infinitesimal analysis to the study of stochastic processes. We drew the line of what we call "foundations" at "measure theory" but our reader should not take this seriously. We hope many people will work to extend the foundations of infinitesimal stochastic analysis as well as to give new applications of these methods in solving problems about stochastic processes. We doggedly adhered to a hyperfinite bias. That made certain things 'nice' and should not hinder our reader from learining and using other 'Loeb-space' techniques. Chapter 7 gives a hyperfinite treatment of semimartingale integrals. We have written i t at two levels. Sections (7.1). (7.2). (7.3) and the beginning of (7.7) treat the easily integrated case in complete detail. Section ( 7 . 4 ) shows how the more general local theory at least partly parallels the square integrable case. The remainder o f the chapter only outlines the main ideas of the best known contemporary theory. We hope that the statement of results and examples will act as a guide to further study. The survey article of Cutland [1983b] could be read by a beginner to get an overview of hyperfinite measure theory.

1

CHAPTER 0 :

PRELIMINARY CONSTRUCTIONS

(0.0) Motivation with a Finite Probability Experiment

Consider the probability experiment of tossing a f a i r c o i n n

times. We can represent each possible outcome as a sequence

of

H’s

T’s

and

.

These outcomes can be viewed as the

elements of the set:

nl= {H.T)~. Rolling a f a i r d i e

n

times could be viewed as elements of the

set:

R2= {I, 11,111.IV,V.VI}”. Coin tossing can be modeled in the die experiment by considering an even

outcome as

Sampling

n

“heads“ and

an

odd

outcome as

”tails“.

times from an urn with r e p l a c e m e n t a n d c o m p l e t e

m i x i n g after each sample can be modeled on the set

R3=

W

where the

is a set with

A

urn.

m

particular

w”.

elements representing the balls in

sequence

of

draws

from

the urn

is

W

on

represented by a function u E

Q3

u : (1 .*.*.n} +

This means that the first draw,

u(1) u(2)

W.

is the particular ball in the urn

is the ball on the second draw, and s o

on.

If

m

is even, coin tossing can be modeled inside

considering even outcomes as ”heads” and odd outcomes as “tails“.

This can be coded by any function

Rg by

Chapter

2

c: W + {H,T} #

(We use

#

with

,

[c

0: Preliminary Constructions

-1

=

(H)]

#

[c

-1

(T)].

for the finite cardinality function.)

[a]

The number

of sample sequences that have exactly half heads is n # [ u € R 3 : #[j : c(u(j)) = HI = 2 1 . (There will not be any unless m

that

If

is even.)

n

6

is even and we already assumed

divides

m

then the fair die

R 3 in a similar way.

experiment can be modeled inside

to have lots of divisibility we may as well assume that n m = n , for a single integer

and let

h

€

*IN

h

>

In order n = h

(Eventually we wi 1

1.

be an infinite integer.)

We wish to think of the index in the component functions (such as

above) as a time, s o that i t is more convenient to

u

represent our random experiment a slightly different way. means that. when we make

h

(and hence

n

and

m)

This

larger,

more sampling takes place in the same elapsed time. We first take a set of times

U = { A t , 2 A t , 3 A t . . . . , nAt}.

where

At

=

1 n .

Then we

take a finite set

W

with

m

elements and define our space of sample sequences to be the set of all functions from

(0.0.1)

Now i f

t=5

U

into

R = { w I

w

E R,

then

w ( 51)

W

w:U+W}=W

U

is the ball selected at the time

for the particular sequence of samples represented by

The idea of a m i x e d urn mentioned above means that each

w.

Section

ball

0.0: A Finite Experiment

in

W

sampling. sequence

is equally Thus

the

likely to be drawn at each

of

probability

each

AP(w) =

1 n m

The set

&

those sample sequences Ah€ E

Ah =

[RI

R

is called the a l g e b r a o f

For example, the event consisting of that contain exactly half heads is

w

given by

{a E R

:

#[t

:

n = H] = -} 2

c(o(t))

The p r o b a b i l i t y o f an e v e n t

A

€

8

is given by the finite A:

sum of the individual probabilities from

P[A]

=

Z[AP(A)

:

h E A]

For example, the probability o f the event the binomial coefficient

I,;,[

Ah

above is given by

over the number of ways that

one can assign heads or tails to a sample along

[ $1 2” n

(0.0.4)

sample

1 #

of all subsets o f

e v e n t s of the experiment.

(0.0.3)

individual

time o f

w E R , is the uniform amount

(0.0.2)

the set

3

=

T.

1

’

We could attach values to heads or tails corresponding to winning or losing a step in a game, for example,

Chauter

4

In

this

case

the

running

0: Preliminary Constructions

total

of

signed

winnings

is

the

a)

stochastic process (a random walk of step size

We have written very much about a simple example, because

we want to point out what sorts of mathematical involved.

First. we have an

element set into

W.

W.

and

the

R

U

e ement set R

Next, we have the set

Then we have a function

all subsets of

AP

n

P

entities are and an

defined on the set

summation

function

I[-

*I.

:

Moreover,

B

Finally, we have a function

UxR

:

.

by the summation function and another function. AB let

At

be an infinitesimal

6t

formulas to analyse Brownian motion. will

require

more

of

E

in terms of the (constant uniform) function

computing an event defined in terms of the function

{H,T} .

U

of functions from

P

associate certain simple combinatorial formulas with

into

m

than

just

c +

we

when

from

R

W

given

We want to

and use the same kinds of Our infinitesimal analysis

extending

numbers

to

include

infinitesimals. but will also require extensions of functions, functions of functions (including summation), sets of these, and combinatorial formulas relating to them all. Abraham allows us

Robinson's

to enlarge

infinitesimals

and

contemporary

Theory of

Infinitesimals

"all of classical analysis" infinite

numbers

as

functions. sets, sets of functions, and s o on.

well

to include as

certain

This extension

0.0: A Finite Experiment

Section

procedure

satisfies

a

precise

5

transfer

principle

akin

to

Leibniz’ old idea that what holds for ordinary numbers, curves, etc.,

also

holds

of

formulation

for

the

the

ideal

principle

extensions.

uses

formal

a

Robinson’s language

for

precision, but in practice we need only care with quantifiers and some training in the limitations of the formal transfer.

LEIBNIZ’ TRANSFER PRINCIPLE (heuristic form) A property only

Q

is true in cLassicaL anaLysis i f and

* transform

if its

*Q

is

true

in

infinitesimal

anaLysis.

The precise section

formulation of Appendix

(0.2).

constructions

needed

for

the transfer principle

2

the

sketches

some

formulation.

chapter is t o show the reader how to

use

is in

set-theoretic

The aim

of

this

Leibniz’ Principle-

proofs can be found in the references given in the appendices.

For now you should think of

*

as a mapping defined on all the

objects of classical analysis (in an informal sense).

For example.

*IR

is a set extending the set of real numbers,

in the following sense. fixed number

r E IR.

*

restriction of 0

R

=

{r

property

€

*IRI of

Nevertheless,

(3

to

for example,

R

to the set s €

two

IR)[r =

numbers

*

is defined on each

*0. * 1.

* A , *r.

+

binary functions

and

*+

because

being

unequal

and the relation and

**

The

maps bijectively onto the set

*s ] } ,

*

preserves

to

each

is a proper subset of the set

OR

binary functions

The mapping

IR




0

62

>

6m

>

6" > * * *

>

0

the

IR,

in

x, 1

with

a

x # 0 there

the

x+r # r ,

implies are

unlimited

of

inf initesimals

w i t h maximal order ideal to

Ei

a.

a r e a totally o r d e r e d ring

a n d the homomorphism

c a l l e d the standard part.

01%

is

-& IR

is

T h e quotient field o C 0

the

clustered

Here is a fancy way to express this idea.

0,

so

* real

limited numbers are just

"monad"

T h e limited numbers,

an

This is because we may transfer

course, but

distinct points

(0.3.2) PROPOSITION:

isomorphic

0

same

x E IR.

the statement that for all

numbers

>

€ R)[E

B

the

(At

infinitely many

infinitesimal distance away.

r + 6 # r+ti2 # r

(V

Leibniz' Principle says that around each

there are

real,

field

Archimedes' axiom

infinitely many distinct infinitesimals. for E

ordered

infinitesimals,"

Thus, Leibniz'

infinitesimals

there are

proper

a

must be non-archimedean.

"There

*archimedean

that

Section

In other words, every s

E IR

25

0.3: Basic Infinitesimals

infinitely nearby,

infinitesimal

limited st(r) =

The fact

L .

r E or

s

that

*Ut

has a standard r =

for some

S+L

the infinitesimals

form a

maximal order ideal means that i f a number has magnitude less than an

infinitesimal, then

is

it

also

infinitesimal, only

infinitesimals have unlimited reciprocals, and a limited number times an infinitesimal is infinitesimal ('moderate size times very small is very small'). found

in one of

The proof of this assertion can be

the references

such as Stroyan & Luxemburg

[1976. (4.4.4)-(4.4.7)].

For our work in measure theory i t is very convenient to define the extended standard part into the standard two point compactification of

IR,

[-m.+m].

(0.3.3) DEFINITION: The map

st :

st(r) -m

*IR

a[-m,m]

-B

,

if

r

,

if

r

,

if

r

The love knot symbols

fm

is g i u e n b y

i s limited.

>
5). , for

infinitesimal"

*IR

where we might like to

(no such number

[

exists

No countable nested sequence of intervals

6,,,% 0. will have empty intersection; there will

always be an infinitesimal

11

>

6m

for all

m E

An extremely useful consequence of saturation is a function extension property called "comprehension."

The model can

0.4:

Section

37

Saturation 81 ComDrehension

comprehend "small" infinite sets by extending them (we can even

* finite).

make the extension

(0.4.3) THE COMPREHENSION PRINCIPLE: fE 9

Let

f,

the d o m a i n o f

D

R

and

:


L!lS=l

7 .

3 s q

G(l,[n/2])

is

not

sequentially compact, since no function in i t is almost equal to N

hn

(to prove

this, take

before, inverting the roles of

hq q

€

H(l,[n/2]) and

n).

(1.6.6) QUESTION: What is the closure of the sequence

{h,}?

and

proceed

as

115

CHAPTER 2:

MEASURES AND THE STANDARD PART MAP

T h i s chapter makes several connections between hyperfinite measures and classical constructions in Euclidean spaces. of

the

results

noted below.

have generalizations to

topological

Many

spaces as

T h e idea of the chapter is to measure all points

with a n appropriate hyperfinite

measure

that

being measured by the classical measure. the inverse of the standard part map.

lie near

points

T h i s means that w e use We begin with Lebesgue

measure because a complete measure is easier to treat than the naked Bore1 algebra which follows. Let

0d = {r

€

*Rd : r = ( r l . - * * . r d ) with each r

. i

denote the Cartesian product of the limited scalars.

limited}

Recall the

definition of the standard part map f r o m section (0.3).

(2.1) Lebesgue Measure Let d-fold

Ud

be a

* finite

Cartesian product.)

every standard point of

Rd

subset of We

*Rd.

say that

Hd

( I t need not be a

is

S-dense

is close to some point of

W e say that a n internal weight function approximates volumes o f standard rectangLes in bounded standard rectangle, for example.

if

Ud,

6a : Ud + *[O,-)

Rd

i f for every

1 = {r

the

2:

ChaDter

116

E IRd

d-dimensional volume o f

: aj

I




are

bounded

liftings

of

s

convergence

says

The

m.

(hence

f (x) = m

an

says

Let

gm

for

form

is

f-.

and

Using part (a), we may choose a sequence

S-continuous

s

the

f 2 0

where

S-Cauchy im(t)da(t)

=

is

a

fm

L' (dh(x) )-sequence.

convergent Extend

gm( t )

= gm(t)

for

formula

gn(t)

of

SL'((r)

gn(t)

= g(t)

g E SL'(a)

n

>

to a n internal m 2 It1

is and

there

g

for infinite is

an

SL1(a)

the

gn(t)

and maintain the internal truncation

= min[n.gn(t)]

says

sequence satisfying

S-completeness

infinite

limit

is a n almost

n.

of

n

such

g,(t).

S-continuous

that

Certainly lifting of

f.

N

I t remains to show that w

JTr and

N

gm + g

in

g(t)da(t)

L1(a)

\o

= 0.

but we k n o w that

N

each

gm

satisfies

this

integral

^. .formula so the integral formula follows for

g.

T h e rest is a

consequence of (2.1.2).

slf(x)

Conversely, suppose such a

(2*1*2)

= [odlg(t)

g

Ida(t)

exists.




for

Icp(x)-+(x)l

all

x

f o r all

in

x

*IRd, in

*IRd.

there is an Then

Section

147

2.4: Weak Standard Parts

The

S-continuity of

means that we can associate

a

associate saw

in

a

with a countably additive measure

the

measure

last

can be

section how

represented

every

by

We can also

BC(Rd).

with a n element of the continuous dual of

a

Td. We

on

countably additive

such

and

a ' s

a

this

Bore1

section

explores one aspect of the opposite direction.

(2.4.3) EXAMPLE: Let

6 a : Td + [O.l]

let

Consider

p

:

the

BC(IRd)

+

standard

9.

IR

be

then

Jqdp

p(lim

the indicator

externally

defined

given by

p(q)

The norm of

p

c o u n t a b l y additiue m e a s u r e o n p(9) =

Hd

be an unlimited or infinite element of

to

Rd

= st(a(

function of

standard

*9 ) ) =

is clearly

{to}.

functional

* st( p(tO)) 1.

that r e p r e s e n t s

and

for

There

p.

is n o

since if

and'we pick the monotone decreasing sequence

9 ) = lim p(p )

by the monotone convergence theorem n n *vn(tO) = 1 for finite This is not possible because

for

p.

n.

while

lim 9,

E 0.

The "problem" with the

a

example Is that i t is carried on the unlimited points.

in this

148

Chapter

2:

Measures 81 Standard Parts

M(Wd)

Let

denote the space of finitely or countably d additive measures on IR . Every continuous linear functional on BC(IRd)

can be represented by a finitely additiue measure and [1972]

Loeb

* finite

shows how

sets.

internal

of

those on certain

We will content ourselves with studying positive

a's

results to

to represent all

on our set

+ a -a7)

Ud

(we can always apply

these

since our modest aim is construction of some

interesting processes, not representation of all possible ones.

(2.4.4) DEFINITION:

Ud

Let

be

6a : U d + * [ O . m ) say

We

a

a[A]

has

near-standard

A

It follows that if

and

let

carrier prouided

that

for

a

i s an

S-tite measure i f

a[Ud]

has near-standard carrier i f and only

a

This is a'more compact way of saying i t , but

we can do even better: Then by (1.2.25).

*IRd

has near-standard carrier.

a

a(Ud\O d ) = 0.

of

containing only unlimited points o f

S 0. We say

i s finite and

subset

be a positiue internal weight function.

each internal set

Ud,

* finite

a

a

Od

is Loeb. s o

Ud\Od

is Loeb as well.

has near-standard carrier i f and only i f

a(U d\O d ) = 0. The generalization of spaces can be

found

this notion

to arbitrary Tychonoff

in Anderson and Rashid

[1978]

and Loeb

[1979a] as well as in the special extension treated in chapter 5 below for the path spaces

C[O,l]

and

DCO.11.

In non-locally

compact spaces a distinction must be made between "limited" and "near-standard'' or "unlimited" and "remote," and i t is worse

than

that

enters.

because

the

"Baire"

vs.

"Borel"

distinction

also

(Loeb [1979a] has a nice universal measurability result

in Borel sets.) from

149

2.4: Weak Standard Parts

Section

the

These technicalities distract the uninitiated

central

infinitesimal

analysis

substantially harder for the initiated).

(and

aren't

Moreover, this form is

useful to us in the description of the law of a process. The next result says that

S-tite measures are weakly near See Stroyan 8r Luxemburg

countably additive standard measures.

lo]

[1976. chapts. 8 &

for the infinitesimal functional analysis

jargon; we will show that the infinitesimal relation holds.

PROPOSITION:

(2.4.5)

If

ud c

*Rd,

pa = a

0

S-tite

measure

the

countabLy

is

the

(weak-star)

part

of

that

a.

*finite

on a

then

st -1

standard cp €

is a n

a

additive

measure

a(M(Wd) ,BC(Wd))for

is,

set

each

standard

BC(Rd) c(t)Wt)

%

J r(x)dlJa(x).

PROOF : First of all, what does near-standard carrier have to do with it?

Clearly,

a

0

st-'

is always a Borel measure-but

notice that i t is zero for example (2.4.3). whereas does not "see it" as zero.

1

uBC(!Rd)

N

We always have of Chapter 1 , since Since .[It1

>

such

that

a(Ud)

n] zz 0 . n

Xp( t)a( t)

and

9

is

a

%

q( t)da( t)

are finite.

limited

and

for standard positive

>

implies

by general results

a[ltl

for B




a(T d ) - B .

m

Thus,

2:

Chapter

150

n udl

a[od

Measures & Standard Parts

=: a[adl.

= a[st-'(~~)l

so

N

= Jod cp(t)da(t)A

standard

bounded

continuous

function

is

S-continuous at each (near-standard or) limited N

finite

and

this means

t;

N

= cp(st(t)).

cp(t)

or

neighborhood of

t.

is constant

cp

on

the

infinitesimal

Combining this with the last remark. we see

N

that =

= cp(st(-)).

q(*)

J cp(st(t))da(t). =:

(2.4.6)

cp(x)dpa(x).

of

Variables

1

=

cp(st(t))da(t)

Now we know

a].

The Change

shows that Zcp(t)a(t)

[a.e.

q(t)da(t)

(2.3.5)

Theorem

cp(x)dpa(x),

(Notice that

r

we

so

st-'(Borel)

see

that

is Loeb.)

PROPOSITION:

Td C *IRd

Let

a sequence

be

* finite

o f positive

{ak : k

and let

internal

€

alN}

be

T.

functions on

The

following are equivalent: ak

a) additive cp E

converges weakly

finite Bore1

to a

measure

p,

standard that

is,

countably f o r each

BC(Rd). S-lim Zcp( t)ak( t) = Jcp(x)dp(x).

b)

For each internal extension

Td.

sequence o f functions on

such that f o r all infintte am (st-'(

1)

= a (st-l(

1)

{ak : k

€

*a}

to a

there exists a n infinite

m


- N

be a decreasing 2 r[Wm] < l/m , for

with

By summation we see that

because

N E Wm,

Since

the outer measure m.

T h i s means

-~ {

{u : u[NU]

that

u:

>

l/m} C {u

-u[NU]

>

l/m}

:

u[Wi]

>




so that

finite

0) = 0.

Hence almost all sections of a set of measure zero themselves have measure z e r o , so this concludes the proof of part c). Part d) follows easily from part c).

= {v

:

f(u.v)




x

=

*min[k

two

difficuit

as

internal the

reader

infinitesimal amount. S-lim F(y) Y 1X whenever

exists, X" Z

x

distributions can

see

only

for

:

%

.

x,

m in

standard ,

x.

.

G = F

with

by

x'

with

G(-m) = G(x),

N

Comparing

The l i m i t

is standard and right continuous by

I f we take

G(x).

%

Distributions

that we have just used.

G

above is pointwise and

we

4:

Chapter

176

is

translating

more

jumps

an

I t is easy to show that i f

then

and

there

x" 2

exists then

X I .

x'

F(x")

Z

x

such

F(x')

Z

These difficulties come up in a more serious way

Z

that

S-limit.

for paths in

Chapter 5 and for Stieltjes integrals in Chapter 7 where we show

[XI,

(roughly) that the be a small enough

trick always works provided we let

infinite number

(so you miss

m

the whole gap

between translated jumps).

(4.1.6) PROPOSITION: Let Suppose

(n,P)

be a u n i f o r m

that internal random

*f i n t t e

probability space.

X

uartables

f t n t t e a.s. a n d haue dtstrtbutton functions N

respecttueLy.

Y

F

and

=

G.

Then

there N

internal bijection

(I

:

R

are

G,

N

F

sattsfytng

and

R

such that

X(o)

is

an

N

= Y(o(o))

0.s.

PROOF : An internal bijection

(I

preserves

P

because all points

have the same weight. . . . N

Since

F = G.

for each finite natural number

m

there is

xh

a sequence -m2




t 6t

zeros"

t].

Finally, by (5.3.9). for sufficiently large infinitesimal

1

(5.3.11)

P[T~ i At]

%

At,

a

or equivalently,

S-lim

P[T~




is

The

p(At),

independent of

= 0

is approximately

J(l-At)


_ {I - max ~ [ I ~ ( t ) - ~ ( s ) l > e l } ~ [ m a x l ~ ( s )>l 2 e l . s>

t

(4.3.4)

f(t,u)

X(t)

S-continuous and

Proposition

0,

is finite

is the converse of

the

These processes are infinitely close to stationary

independent

increment

standard

processes.

We

have

seen

the

examples of Brownian motion. the Poisson process and the Cauchy process

(4.3.6-7)

X(t)

moments, e.g., arises

this

above.

way

by

Observe

SL1(R).

C

that

the

for

has

no

X(t) = ct

Deterministic drift 6X(s) = c6t

taking

latter

all

s

with

probability one.

Also, a deterministic process is independent iuct of anything else and f(t,u) = e in this case.

The decent

following

paths.

Skohorod's

We

results show only

prove

lemma works.

More

that the

internal

easy

part

'versions' have to

indicate how

'general' sampling results with

weaker hypotheses and conclusions are proved later.

(5.3.20) LEMMA:

be

a

: 0 < s < t , step at]. * independent identically

X(t) = B[6X(s)

Let

sum

of

infinttesimal

increments

finite a.s. f o r

0


b]} Z 0 whenever b O 1 >r

n %(u). u>>r

PROOF : (a)

I wt = ut)

{t

is internal and contains all

an external set [cf. (0.3.8)]. (b)

I wt = ut}

{t

Thus i t contains a

for all

Thus i t contains a

>>

r.

E

Since

r.

t t


>

is internal and contains all

an external set [cf. (0.3.8)]. (c)

t

Z

t

by (b) and

If u E

(D), = D.

D

or

w E

[D] t = D.

D

and

t ut = w ,

ChaDter

270

Strictness of second inclusion: u u

for

= o

-

0




+

m.

For

Ym

S-Cauchy

there exists an

E[lYJ-Ykl]

to an

is an




P[lX-P1

a]




r

: p(w)

time and

for some and


1 P

or ~Y(T~)-Y(cJ)~> -3 1 P

Since the definition of

{T~)

< -. 1 P

is internal, there exists an

infinite

n satisfying the probability above. P to pick an infinite n E n[m n 1. This T~ P P' P 1 At-stopping time with b = a+ n'

Use saturation is the claimed

-

( 6 . 4 . 6 ) EXERCISE:

Show that i f

p

is only

an internal (nonstopping)

T

P-measurable we may still select so

that the remaining conclusions

in (6.3.14)are true, in particular.

(6.4.7)

st Y ( T ) = X(st(s)).

EXAMPLE:

This example shows one kind of difference between and

st X ( T )

It is based on a remark in Hoover 81 Perkins

%(st(T)).

[1983] and an example in Lindstrom [1980].

On the

*finite

jump function is a.s. (4.1.5)

j :

set

W + *IR

finite but

not

W

(where

R = W lr )

choose an internal

with a symmetric distribution which S-integrable.

to obtain a distribution:

For

example, apply

304

Chapter

Notice that

x

P[lJl

=

m]

= 0, but

6: Hvperfinite Evolution

E[lJl]

=

m.

Define a process

by

0

where

at

is a positive infinitesimal.

L

The infinitesimal jump at 1 t = - + Gt. The functions

are

1 t = 2

announces the finite jump

2

Gt-stopping times for each

N

m

and

1 and 0 for r < -, we see 2 2 that X(st(T,)) = X(l) with the same distribution as j while 1 =: 0 and 1 P[st X(T~) = O] + 1 as m + m , since X(z) T~ = 5,

Since

X(r)

= j(ul12)

N

when

Ijl

N




H(r.o) = a.

0. Then

s -1 [K(r.w)

(6.6.10)

and

#

H

H.

r = 0.

and

is almost previsible and

H(r.o)l

the

approximations to obtain a previsible equal to

if

= 0.

usual G

simple

function

which is

r-almost

Section

333

6.6 Predictable Processes

(6.6.12) DEFINITION:

A

basic

almost-previsible

process

is

bounded

a

process of the form

m




to find an

P -

1 s*

t k = max[t

is empty, any

To see

Let E U :

t k Z sk Z a

is a larger inner approximation -1 tk Z a since qk st (a) x R.

This means that

r"s,.t,l

Extend

the sequence

{sk,tk}

x

n1 >

P

1 - E.

to an internal sequence and use

Section

335

6.6 Predictable Processes

Robinson's

Sequential

Lemma

and

the

Internal

Definition

Principle to select an infinite n s o that sn Z tn 1 n[[sk,tk] x R] > p - i; f o r all k < n. Let s = min[sk t = max[tk

and

:

k

n].

Now, having chosen sj z rj,

so

gl

imally

gj's

is determined at if

k




J

n

tl

so = 0 tl z sl.

but

sl+t,

Also increase r[(st-'(O)

with

s.

infinites-

[tl.l])

x

R] = 0.

The latter is possible by the remarks of the previous paragraph. Choose each t

>

t

s +L.

j

tJ

5 j tm+l = 1.

j = l.**-.m

for

sJ

and

n

w[(st-l(r,)

r[( ( tj. tj+ll v st-l(rj.rj+l

(0) if

G

(tj.tj+l ]

j = 0).

x

[t,,l])

R] = 0.

Let

j = O.l.--*.m.

Then for each

(replacing

in this manner, that is,

by

[O,tl]

1)

x

n] = 0

and replacing

This means that for these

j

I s

(rj*rj+ll and

gj

by

with

as above,

r[st

by the

G(t,w)

P-continuity of

f

H(st(t).w)]

= 0

7.

The next result is also helpful in stochastic integration.

336

ChaDter 6:

HvDerfinite Evolution

(6.6.14) LEMMA:

V

S u p p o s e that preuisible

processes

If

space.

processes

V

with

ualues

of

a

in

bounded almost-

separable

normed

c o n t a i n s the basic almost-preuisible

and

convergence,

is a u e c t o r s p a c e

is

then

closed

V

under

contains

bounded all

pointwise

bounded

almost-

p r e u i s i b l e processes w i t h u a l u e s i n that space.

PROOF : We will show that for any almost previsible set vector

b

in the range space, the function

1y

bI#(r,w)

and any

E 1.

This

proves the lemma because all bounded measurable functions are bounded pointwise limits of sums of these "simple" functions. Let

b

be an arbitrary but fixed range vector and consider

the collection of sets

Z(b) = {H-l(b)

Every basic function

:

H E V

&

H takes at most the values 0

almost-previsible

bIA(w)I(q,s,(r)

set

is

belongs

in

to

Z(b)

V.

V

is a vector

because if

space.

Hi1(b)

Finally,

Z(b)

because

Finite

unions of basic almost-previsible sets belong to

& b)

Z(b)

disjoint because

is a monotone class lim Hm

is either increasing or decreasing,

takes only the values

0

and

b

and belongs to

Monotone Class Lemma (3.3.4) shows that sigma algebra of almost-previsible

the

Z(b)

V.

The

is the whole

sets for each vector

This proves the lemma as remarked above, since all bounded

b.

Section

6.6

functions

337

Predictable Processes

are

limits

of

"simple"

functions

(partition

the

to extend

the

range).

(6.6.15) REMARKS ON EXTENSION TO Only minor

technical changes are required C0.m) x R

results of this section to (5.5.4).

C0.m):

in the framework of

One change which we mention explicitly is this.

In

order for finite disjoint unions of basic almost-previsible sets to form an algebra we must also include sets of the form

(r.m)

x A,

for

A

in the

P-completion of

Also, a b a s i c a l m o s t - p r e u i s t b l e p r o c e s s o n

C0.m)

O(r).

is one of the

form

-

m- 1

where

0 = rl




6t,

1 1 ~ x 1a.s. t

var %(q)

so

S

X(s.0)

z %(q,u),

Chapter 7: Stochastic Integration

354

Next we find one infinitesimal time sample satisfying the

V(t)

opposite inequality. Let know

S-lim(X.V) = ( 2 , var 2)

number 0




t

IAXII

P[max(IV(t)-V(s)-l

var

a.s.. so for every finite natural

there exists

m,

6t-lifting of

be a

:

s.t

E

At]


m

IHI < H




0

d

E

such

and every

N,

there

that f o r 6t

and

and for every internal convex increasing function

*[o.-)

*~ 0 . m )

satisfying

Section

7.2

373

Quadratic Variation of Martinaales

q(0) = 0

9(2x)

and

k*(x)

x

f o r all

E *[O,m)

the following inequalities hold:

PROOF : This

result

finite case

of

follows by the

taking

extension o f )

d-dimensional

[1972] Theorem 1 . 1 .

Davis-Gundy's

* transform

the

of

(the

Burkholder-

While this is a cornerstone

of our theory, we shall not give a proof since i t is a "wellknown standard result."

(7.2.7) PATHWISE PROJECTION OF For

process

any

internal

[6M,6M](t)

of squares.

process

each

t

The (local)

is finite when

r E [O,m)

M,

the

quadratic variation

is increasing for all

o

S-integrability of

(7.2.5) means that except for whenever

[6M.6M]:

o

[6M,6M](t,o) o

e A.

since i t is a sum

[6M.6M]

proved in

in a single null set is also finite.

A,

Hence for

the left and right limits along

S-lim[6M.6M](t) t tr

= inf{st[6M,6M](t)},

S-lim[6M.6M](t) t lr

= sup{st[6M,6M](t)}.

t E Us.

tZr

and

both exist in

IR.

t E

Us

tZr

It follows, (5.3.25).

that

[6M,6M]

has a

Chapter 7: Stochastic Integration

374

At-decent path sample for some infinitesimal A t actually has a

[6M,6M]

as the process M.

in

T6,

but

6t-decent path sample for the same

The proof that

has a

[6M,bM]

6t

6t-decent

path sample, Theorem (7.4.9). uses the machinery we develop for stochastic integration.

We believe that there should be simple

direct proofs of this basic fact, but do not know any. We abuse notation and define a pathwise projected process using the extended standard part:

By the preceding remarks a process with paths

[P.%](r,w)

is indistinguishable from

Dt0.m).

The abuse of notation is

in

justified by (7.2.11) The following is a key technical lemma that tells, us some information about paths of quadratic variation processes.

(7.2.8) LEHMA:

Let

M

suppose that part,

;(a) =

be a u

d-dimensional

6t-stopping time whose standard

is a u(u).

6t-local martingale and

satisfies

S-lim M(t) t lo

= st[M(u)]




E{st[6N,6NI(l)}

- s > E{

=

The

two extremes of

u-S-integrable. [6N.&iN]

st G2d(A,)}

[ st G 2du

by (7.1.5).

these inequalities agree because

Hence

st E{[6N,6N](l)}

G2

= E{st[6N,6N](l)},

is so

is S-integrable and (7.2.5) completes the proof.

Our next result says nearly the same sums.

u-equivalent

Again,

M

summands pathwise give

is a s above.

(7.3.6) PROPOSITION: Suppose

G1

U{(t,w)

*

and

:

G2

I

2 6M -summable and

st Gl(t.w) # st G2(t.w)}

Then the marttngale tnftnttely close to N2(t)

E{

are

= 0.

t

Nl(t) = B G1(s)6M(s) = 2 t G2(s)6M(s), in f a c t ,

max Et[Gl(s)-G2(s)]6M(s) 6t

k*6t] = (1-p)

PCT-1

>

t]

k

or

T

s

means

t at

"success"

limited multiple of

happens

8

7--1

a

Now we define a




1.

limited, then

Let




M(t,w)

T ( W ) T(O)

.(&I).

the first success,

until the first success,

l+t =

T .

so

However,

7: Stochastic Integration

Chapter

416

we have seen that multiple of

1

Z

T

a.s., since

T

is a.s. a finite

a.Hence

*martingale

Next, we define a

1 G-6M. t

N(t)

where

G

=

is deterministic and bounded,

G(l+k6t)

= (-1)

k

.

This makes

aN(t-6t)

= p-1

(-l)k-l(-l)k(l-p)

We know that

p =

6+

noninfinitesimal limited

o(6t)

r

an

N(T) does not have a

first builds up to infinitely near

r

t = 1.

if

t = l+k6t

if

t =

T ( W T

T(O)


m

is

either

for every

m E

standard,

ON

.

PROOF:

We

offer a

Principle; below, Take any limited

proof as an

direct

(APP.1.2)(4),

n

*N

,

Leibniz'

we suggest a different

i.e., a *natural number n

for some (standard) m E N'

n < m

application of

.

proof.

such that

Apply Leibniz' Principle to

the bounded sentence N, ( x < m

vx and 1,

conclude

...

,

m

imply

that

n

x

or

= 0

...

or

x = m )

equals one of the standard

numbers

0,

.

(APP.1.2) REMARK:

*

N

"looks

times copies of

like" N

followed by a dense

Z

,

n

belong to

and

0

ordinal

0

is uncountable.

Specifica11y : (1) If

m = n

m

,

-m

and

- n(

Im

< 1

,

(apply Leibniz' Principle to the 'same' statement in (2) If n € *N

n

*N

are also in

is unlimited and

*

N

m € N'

and unlimited.

,

then

n + m

Therefore, around

then N ). and each

Appendix

454

N

n E

unlimited

(m+n)/2

.

N

m,n E *N

(3) If

Z embedded into the

there is a whole copy of

unlimited part of

are unlimited, then the *integer part of

is also unlimited. Hence, between two disjoint copies of

(4) Once we have shown (l), (APP.l.l)

you can give another proof

using the standard part map: r = st n ;

limited, call

In

- [r]l

and

therefore, n

=

-

In

5

,

nr

,

[nr]

1

,

-

and

n

is

because it is the sum of an smaller

than

1

r , s E ‘(0,l) are standard

,

nr

at an infinite distance appart

ns = nr(ns-r

-

1)

.

ns

of

are each

Hence, the hypernatural

Z

[ nS] lie on two different copies of

.

This

proves that there are uncountably such copies.

(APP.1.3) PROPOSITION: (a) The

set

of

standard *natural

numbers

‘N

is

\ UN

is

external.

(b)

*N

The set of unlimited *natural numbers

external. (c) and

‘X

‘JX c *X (d)

For any set =

*X

,

or

,

X E

X

‘X

is external

either

X

is a finite set

and

the

inclusion

,

of

unlimited

is strict. The

;

. then the hyperreal numbers

(why?), and

other, because

N

of

+ Ir - [ r l l ,

is unlimited and

r < s

real numbers,

rl

standard real number

[ r ] E ‘N

n E *N

(5) If

unlimited

a

n E

if

*

then

which is strictly smaller than infinitesimal

.

Z

Z there is also another (disjoint) copy of

numbers

1

sets of limited scalars

scalars, of infinitesimals

0

,

0

the map

st

,

and

the

1

Appendix

455

,

=

relation

are all external entities.

PROOF : claim that for a subset of an internal set

First, we the

property

of

P(V) :

assume

e

for

*X P sentence

T,V

being internal is equivalent to

,

T

v 6 X Vt6Xp, P' to obtain

hence

*

xP' T

if

internal

apply

teP(v)

xp,

V t e

are

p ;

some

t/

tlvE

V

*

t6

Leibniz'

iff

P(V)

Principle

to then

to

the

( V x ~ t , x e v ) ,

( V x e t, x e v ) ,

iff

T E

is internal, then

belonging T C V ;

and

,

V

*

P(V)

(the

converse

is

obvious). Proofs it

is

of 'externality' are best handled by

contradiction:

usually convenient to show that a set does not have

some

property which is known to hold (by an appropiate application

of

Leibniz' Principle) for internal sets. N :

(a) Consider the following true statement in

E P(N),

V T

*

Its is

*

*

( T is bounded in N)

transform says that if

T

implies

(T has a maximum)

is an internal subset of

*

bounded, i.e., bounded by some member of

maximum,

i.e.,

a

N

*

.

N and

, then it has a

maximum (writte down the whole sentences

in

detail if you do not feel sure about the last assertions). Now regard then

it

would

bounded.

But

member of N '

U

N

have a maximum then

,

N ;

as a subset of

m+l

m

,

if it were

because

it

is

would also be limited, and

internal, certainly hence

a

by (APP.l.l).

(b) The reader can work out a proof similar t o the last one,

by considering the statement W

T & P(N), T # 0

implies

T has a mimimum.

Alternately, sentences, always

one

finite

t r a n s f o r m of a p p r o p i a t e obvious

then

to

X

e

is a f i n i t e set.

X

X = {xl,..,xn)

Now,

E

X

be

X

saturation

infinite,

A c

i n c l u d e d as

property

[A : A

c U x&

i n t e r s e c t i o n , then

x

that

xn ) * ‘X is

is also internal, s o a

of

part

Leibniz’

the s e t

i s a member

x

if

must belong t o ‘X \ {XI

is external,

‘X

*

\ A is finite]

‘X

and t h i s i s absurd:

n o t empty;

‘X

x =

or

and assume

P r i n c i p l e ( s e e ( 0 . 2 . 3 ) ( b ) and ( 0 . 4 . 2 ) ) ,

n

of

given

(APP.l.l)

...

( x = *xl or

then every f i n i t e subset

Hence

external,

f o r some p , t h e n p 2 1 and E X P ’ 9 and by L e i b n i z ’ P r i n c i p l e ,

iff

X E * X

9

let

internal; the

*x _c *xp-l

so

I

be

sets

The f o l l o w i n g proof

i s a g e n e r a l i z a t i o n o f t h e proof of

‘X

x 5 xp-l * vx xp-l

is

has

*N\‘N

internal

would be i n t e r n a l .

‘N

above. I f

by

*

by

boolean o p e r a t i o n s w i t h

internal s e t s ;

( c ) Assume =

can show,

that

give

otherwise

*X

1

Appendix

456

of

this

.

and t h e r e f o r e t h e i n c l u s i o n

U

X c *X

(See a

of an e x t e r n a l s e t i n t o an i n t e r n a l s e t h a s t o be s t r i c t .

h i n t f o r a more d i r e c t proof o f t h i s f a c t i n E x e r c i s e (0.4.5).) (d)

If

internal; -1

x-x (0 \

0

were i n t e r n a l ,

then

u

a similar argument works f o r

N =

*

N

R\ 0

n 0 would a l s o be

.

i s i n t e r n a l ( i n d e e d , even s t a n d a r d ) , i f

{0}

would be i n t e r n a l and) t h e n

*

R \ 0

Since t h e o

map

were i n t e r n a l

would be i n t e r n a l as

well.

i s n o t d i f f i c u l t t o s e e t h a t f o r a map t o be i n t e r n a l i t

It is

necessary

(apply

0 --->

the R

that

both i t s domain and i t s

Internal Definition Principle),

cannot be i n t e r n a l .

range

s o the

be

internal

map

Again by t h e same p r i n c i p l e ,

st if

were i n t e r n a l , t h e s e t o f i n f i n i t e s i m a l s would be i n t e r n a l t o o :

: FJ

1

Appendix

In

451

the

elementary the

rest

of this Appendix,

*

properties of

R

we are to

introduce

that appear very often

some

throughout

book and have in common that they are translations to

this

setting of analogous standard topological properties, but keeping the standard tolerances, whence the

prefix.

IS-!

(AF'P.1.4) DEFINITION:

A hyperreal function

-a

*R

in

if

x,a f dom(f)

-a c

We

is said to be S-continuous implies

f

x E D

if

f

(D C - dom(f)

x = a

is defined for such x

a

relative to

D

f(a)

at

(and

is S-continuous

and

is S-continuous relative

continuous at

=

f(x)

in particular). We say

(and f(x)

say

a

s

5 D c- *R

relative f(a)

x

f

imply

f(x)

in particular).

9 g if f

is

S-

each

in

D

for

a

in particular).

for hyperreal

Similar definitions apply defined

on

subsets of arbitrary metric

case,

x

a

a

means

that

the

spaces

*distance

functions (in this

d(x,a)

is

infinitesimal).

(APP.1.5) PROPOSITION:

Let

f

an internal set. if and

D -C dorn(f)

be an internal function and Then

f

is S-continuous relative to

D

aR+

,

only i f for every standard positive

there exists a standard positive for all x

,

y

in D

Ix-yl < e

be

0

in

OR+

in

E

,

such

, implies

If(x)-f(y)l


y

and