STUDIES IN LOGIC AND
THE FOUNDATIONS OF MATHEMATICS VOLUME 119
Editors
J. BARWISE, Stanford D. KAPLAN, LosAngeles H. ...
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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
NORTHHOLLAND 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
NORTHHOLLAND 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 Cataloginginqublicstionh 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 0444879277
519.2
PRINTED IN THE NETHERLANDS
8528540
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 twosemester 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 nexttolast draft. We thank the staff of NorthHolland 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 7879 academic year in order to learn about Abraham Robinson's Theory of Infinitesimals (socalled "NonStandard 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 Sintegrability and studied product measures. Sintegrability 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 Fubinitype theorem due to H. Jerome Keisler [1977] as well as a mixed Fubinitype 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. HooverPerkins and Lindstrom. Chapter 5 is devoted to 'path properties' of processes. Our treatment of paths with only jump discontinuities is a little different from LindstromHooverPerkins'. 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 measuretheoretically in time on a hyperfinite scheme. follow Keisler [1984] again, but extend his results to include "previsible 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 'Loebspace' 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
settheoretic
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 nonarchimedean.
"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 dfold
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
Sdense
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
ddimensional 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
Scontinuous
s
the
f 2 0
where
SCauchy 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
Scompleteness
infinite
limit
is a n almost
n.
of
n
such
g,(t).
Scontinuous
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
Scontinuity 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
nearstandard
A
It follows that if
and
let
carrier prouided
that
for
a
i s an
Stite measure i f
a[Ud]
has nearstandard 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 nearstandard 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 nearstandard 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 nonlocally
compact spaces a distinction must be made between "limited" and "nearstandard'' 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
Stite 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
Stite
measure
the
countabLy
is
the
(weakstar)
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 nearstandard carrier have to do with it?
Clearly,
a
0
st'
is always a Borel measurebut
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
Scontinuous at each (nearstandard 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). Slim 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 (stl(
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. Slim 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
Slimit.
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,
Slim
P[T~
is
The
p(At),
independent of
= 0
is approximately
J(lAt)
_ {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)
Scontinuous 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
(4.3.67)
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
SCauchy
there exists an
E[lYJYkl]
to an
is an
P[lXP1
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 Atstopping 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
Pmeasurable 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 Sintegrable.
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
Gtstopping 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
ralmost
Section
333
6.6 Predictable Processes
(6.6.12) DEFINITION:
A
basic
almostprevisible
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[(stl(r,)
r[( ( tj. tj+ll v stl(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)
Pcontinuity 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 almostpreuisible
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) = {Hl(b)
Every basic function
:
H E V
&
H takes at most the values 0
almostprevisible
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 almostprevisible 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 almostprevisible
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 almostprevisible sets to form an algebra we must also include sets of the form
(r.m)
x A,
for
A
in the
Pcompletion 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
Slim(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,
6tlifting 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 )
ddimensional
[1972] Theorem 1 . 1 .
DavisGundy'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
Sintegrability 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
Slim[6M.6M](t) t tr
= inf{st[6M,6M](t)},
Slim[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
Atdecent path sample for some infinitesimal A t actually has a
[6M,6M]
as the process M.
in
T6,
but
6tdecent path sample for the same
The proof that
has a
[6M,bM]
6t
6tdecent
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
ddimensional
6tstopping time whose standard
is a u(u).
6tlocal martingale and
satisfies
Slim M(t) t lo
= st[M(u)]
E{st[6N,6NI(l)}
 s > E{
=
The
two extremes of
uSintegrable. [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 Sintegrable and (7.2.5) completes the proof.
Our next result says nearly the same sums.
uequivalent
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] = (1p)
PCT1
>
t]
k
or
T
s
means
t at
"success"
limited multiple of
happens
8
71
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 G6M. t
N(t)
where
G
=
is deterministic and bounded,
G(l+k6t)
= (1)
k
.
This makes
aN(t6t)
= p1
(l)kl(l)k(lp)
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(nsr

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 *xpl
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 xpl * vx xpl
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
xx (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 Scontinuous 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 Scontinuous
and
is Scontinuous 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 Scontinuous relative to
D
aR+
,
only i f for every standard positive
there exists a standard positive for all x
,
y
in D
Ixyl < e
be
0
in
OR+
in
E
,
such
, implies
If(x)f(y)l
y
and