Measure theory and integration (second edition)
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MEASURE THEORY AND INTEGRATION
"Talking of education, people have now a-days" (said he) "got a strange
opinion that every thing should be taught by lectures. Now, I cannot see
that lectures can do so much good as reading the books from which the
lectures are taken. I know nothing that can be best taught by lectures,
except where experiments are to be shewn. You may teach chymestry by
lectures — You might teach making of shoes by lectures!"
James Bosweil: Life of Samuel Johnson, 1766
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ABOUT OUR AUTHOR
Gearoid de Barra was born in the city of Galway, West Ireland and moved
as a young boy to Dublin where he spent his schooldays. He then studied
mathematics at University College Dublin, National University of Ireland
where he gained his BSc. Moving to England, he graduated from the
University of London with a PhD for research on the convergence of
random variables, an area of application of some of the material covered in
this book. He then transferred to Hull University in Yorkshire for a teaching
appointment; and afterwards spent two summers in 1975 and 1988 in
Australia, teaching and researching at the University of New South Wales.
More recently, he became Senior Lecturer at the Royal Holloway College,
University of London, to continue teaching and research related to aspects
of operator theory and measure theory involving ideas from the material in
this book.
He has enjoyed teaching university mathematics at all undergraduate and
postgraduate levels, including many courses on measure theory and its
applications to functional analysis, from which source this book has
developed.
The first edition was the standard text in the departments of
mathematics at both Cardiff University and Royal Holloway College, and
has attracted attention in Canada and Scandinavia. It was also translated into
Italian as Teoria del/a Misura è deli integrazione.
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Measure Theory and Integration
C. de Barra, PhD
Department of Mathematics
Royal Holloway
University of London
wP
Oxford
Cambridge
Philadelphia
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New Delhi
Published by Woodhead Publishing limited.
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First published. 1981
Publishing Limited. 2003
Updated edition published h'
Reprinted by Woodhead Publishing limited. 2011
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(I de Rarra. 2003
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Contents
Preface
.
Notation
9
11
Chapter 1 Preliminaries
Set Theory
1.1
1.2 Topological Ideas
1.3 Sequences and 1Mm
1.4 Functions and Mappings
1.5 Cardinal Numbers and Countability
1.6 Further Properties of Open Sets
1.7 Cantor-like Sets
15
17
18
21
22
23
23
Chapter 2 Measure on the Real Line
2.1
2.2
2.3
2.4
2.5
2.6
Lebesgue Outer Measure
Measurable Sets
Regularity
Measurable Functions
Borel and Lebesgue Measurability
Hausdorff Measures on the Real Line
Chapter 3 Integration of Functions of a Real Variable
Integration of Non-negative Functions
3.1
3.2 The General Integral
3.3 Integration of Series
3.4 Riemann and Lebesgue Integrals
Chapter 4 Differentiation
The Four Derivates
4.1
27
30
37
42
45
54
60
68
71
77
S
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Contents
6
4.2
4.3
4.4
4.5
4.6
Continuous Non-differentiable Functions
Functions of Bounded Variation
Lebesgue's Differentiation Theorem
Differentiation and integration
The I.ebesgue Set
Chapter 5 Abstract Measure Spaces
5.1
Measures and Outer Measures
5.2 Extension of a Measure
5.3 Uniqueness of the Extension
5.4 Completion of a Measure
5.5 Measure Spaces
5.6 integration with respect to a Measure
.79
81
84
87
90
93
95
99
100
102
105
Chapter 6 Inequalities and the L" Spaces
6.1
Thel/Spaces
109
6.2
6.3
6.4
6.5
Convex Functions
Jensen's Inequality
The Inequalities of Holder and Minkowski
111
Completeness of L'ip)
113
115
118
Chapter 7 Convergence
7.1
7.2
7.3
7.4
Convergence in Measure
Almost Uniform Convergence
Convergence Diagrams
Counterexamples
Chapter 8 Signed Measures and their Derivatives
8.1
Signed Measures and the Hahn Decomposition
8.2 The Jordan Decomposition
8.3 The Radon-Nikodym Theorem
8.4 Some Applications of the Radon-Nikodym Theorem
8.5 Bounded Linear Functionals on
121
125
128
131
133
137
139
142
147
Chapter 9 Lebesgue-Stieltjes Integration
9.1
9.2
9.3
9.4
9.5
9.6
Lebesgue-Stieltjes Measure
Applications to Hausdorff Measures
Absolutely Continuous Functions
Integration by Parts
Change of Variable
Riesz Representation Theorem for C(J)
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153
157
160
163
167
172
Contents
Chapter 10 Measure and Integration in a Product Space
10.1 Measurability in a Product Space
10.2 The Product Measure and Fubini's Theorem
10.3 Lebesgue Measure in Euclidean Space
10.4 Laplace and Fourier Transforms
Hints and Answers to Exercises
Chapter 1
Chapter 2
Chapter 3
Chapter 4
ChapterS
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
7
176
179
185
189
197
198
204
209
211
215
220
223
227
230
References
236
Index
237
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Preface to First Edition
This book has a dual purpose, being designed for a University level course on
measure and integration, and also for use as a reference by those more interested
in the manipulation of sums and integrals than in the proof of the mathematics
involved. Because it is a textbook there are few references to the origins of the
subject, which lie in analysis, geometry and probability. The only prerequisite is
a first course in analysis and what little topology is required has been developed
within the text. Apart from the central importance of the material in pure
mathematics, there are many uses in different branches of applied mathematics
and probability.
In this book I have chosen to approach integration via measure, rather than
the other way round, because in teaching the subject I have found that in this
way the ideas are easier for the student to grasp and appear more concrete.
Indeed, the theory is set out in some detail in Chapters 2 and 3 for the case of the
real line in a manner which generalizes easily. Then, in Chapter 5, the results for
general measure spaces are obtained, often without any new proof. The essential
L" results are obtained in Chapter 6; this material can be taken immediately after
Chapters 2 and 3 if the space involved is assumed to be the real line, and the
measure Lebesgue measure.
In keeping with the role of the book as a first text on the subject, the proofs
are set out in considerable detail. This may make some of the proofs longer than
they might be; but in fact very few of the proofs present any real difficulty.
Nevertheless the essentials of the subject are a knowledge of the basic results
and an ability to apply them. So at a first reading proofs may, perhaps, be
skipped. After reading the statements of the results of the theorems and the
numerous worked examples the reader should be able to try the exercises. Over
300 of these are provided and they are an integral part of the book. Fairly
detailed solutions are provided at the end of the book, to be looked at as a last
resort.
Different combinations of the chapters can be read depending on the
student's interests and needs. Chapter 1 is introductory and parts of it can be
read in detail according as the definitions, etc., are used later. Then Chapters 2
and 3 provide a basic course in Lebesgue measure and integration. Then Chapter
4 gives essential results on differentiation and functions of bounded variation, all
for functions on the real line. Chapters 1, 2, 3, 5, 6 take the reader as far as
general measure spaces and the L" results. Altematively, Chapters 1, 2, 3, 5, 7
9
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10
introduce the
Preface
reader to convergence in measure and almost uniform
convergence. To get to the Radon-Nikodym results and related material the
reader needs Chapters 1, 2, 3, 5, 6, 8. For a course with the emphasis on
differentiation and Lebesgue-Stieltjes integrals one reads Chapters 1, 2, 3, 4, 5,
8, 9. Finally, to get to measure and integration on product spaces the appropriate
route is Chapters 1, 2, 3, 5, 6, 10. Some sections can be omitted at a first
reading, for example: Section 2.6 on Hausdorff measures; Section 4.6 on the
Lebesgue set; Sections 8.5 and 9.6 on Riesz Representation Theorems and
Section 9.2 on Hausdorff measures.
Much of the material in the book has been used in courses on measure theory
at Royal Holloway College (University of London). This book has developed
out of its predecessor introduction to Measure Theory by the same author
(1974), and has now been rewritten in a considerably extended, revised and
updated form. There are numerous proofs and a reorganization of structure. The
important new material now added includes Hausdorif measures in Chapters 2
and 9 and the Riesz Representation Theorems in Chapters 8 and 9.
Ode Barra
Preface to Second Edition
The material in this book covers several aspects of classical analysis including
measure, integration with respect to a measure and differentiation. These topics
lead on to many branches of modem mathematics. So some notes have been
added which indicate some of the directions in which the material leads. These
are less formal than the main text and contain some references for further
reading.
Ode Barra
Royal Holloway, University of London
January 2003
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Notation
Notation is listed in the order in which it appears in the text.
0: end of proof.
iff: if, and only if.
3: there exists.
v: given any.
x CA: x is a member of the set A.
A c B, (A 2 B): set A is included in (includes) the set B.
A C B: set A is a proper subset of the set B.
Ex: P(x)I. the set of those x with property P.
CA: the complement of A.
0: the empty set.
U, fl: union, intersection (of sets).
A —B. the set of elements of A not in B.
A A B = (A — B) U (B — A): the symmetric difference of the sets A, B.
Z: integers (positive or negative).
N: positive integers.
Q: rationals.
R: real numbers.
?(A): the power set of A, i.e. the set of subsets of A.
A X B: the Cartesian product of the sets, A, B.
[xl: the equivalence class containing x (Chapter 1), or, in Chapter 2, etc., the
closed interval consisting of the real number x.
Ix, the metric space consisting of the space X with metric p.
A: the closure of the set A.
G5 set: one which is a countable intersection of open sets.
set: one which is a countable union of closed sets.
11
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Notation
12
inf A, sup A: infirnum and supremum of the set A.
Urn sup x,,, liin
upper and lower limits of the sequence (x,,).
x(a—),
x(a+): left-hand, right-hand limits of x at a. So x(a+) is the function
whose value at a is lim
x,, = o(n"): ,f1' x,, 0.
x,, = O(n"):
x,, is bounded.
-* a,
> a. Similarly flx+),ftx—), etc.
characteristic function of the set A (= I on A, = 0 on CA).
Card A: the cardinality of A.
the cardinality of N.
(2(a): (in Theorem 9, Chapter 1) the equivalence class containing a.
Cantor-like sets.
etc.: the 'removed intervals';
'1,1
the 'residual intervals', for the Cantor-like sets.
N(x,e): the set [z': It—xi
m*: Lebesgue outer measure.
m: Lebesgue measure.
A+x= [y+x:yEA1.
1(1): the length of the interval I.
a-algebra (usually 8): a class closed under countable unions and complements
and containing the whole space.
Intervals: of the form [a, b) unless stated otherwise.
the a-algebra of Lebesgue measurable sets.
a.e.. almost everywhere; i.e. except on a set of zero measure.
a-algebra of Borel sets.
= —min(f, 0).
max(f,
r=
esssupf= thf[a:fCaa.e.J .essinff=
a.e.j.
lim A,, lim sup A,, lim infA1: the limit, upper limit, lower limit of the sequence
of sets IA,).
r'(A)= (x:/(x)EEA].
r= [x—y:x,yETJ (Chapter2).
d(A,B)' inf Fix —yI:x EA,y EBJ.
h: a Hausdorff measure function.
H;,6: the 'approximating measure' to l-lausdorff measure.
H;: Hausdorff outer measure when h(t) = tP.
modulus of continuity of the functionf.
H(A): Hausdorff measure corresponding to the Hausdorff measure function h.
ffdx: integral (over the whole line) off with respect to Lebesgue measure.
ffdx: integral off over the set F.
0, p4.':
usually simple functions, taking only a finite number of non-negative values.
Rffdx: Riemann integral off.
ifi: absolute value of the real (or in Chapter 10, complex) functionf.
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Notation
13
logx. the natural logarithm of x.
5D' SD: upper and lower Riemann sums given by the dissection D.
L(a, b): functions integrable on (a, b).
fh(X)=f(X + h).
upper and lower right derivates.
IX, 11: upper and lower left derivates.
Pf[a, b], N1[a, b], T1[a, b]: positive, negative and total variations of f over
[a,b].
p (or p4, n (or n/I, t (or t1) the corresponding sums for a partition.
BV[a, hi: set of functions of bounded (total) variation on [a, bJ.
1(w): length of the polynomial ir.
81(x): the lump' offatx.
ftc, 6) = (f(d) —f(c))/(d —c), wheref is a function of a single variable.
f'(x) = df/dx.
F: Conventionally, the indefInite integral off.
'1? : ring of sets (closed under unions and differences).
S (R): a-ring generated by R.
2KQR): hereditary a-ring generated by 'It.
a-finite measure: one for which the space is a countable union of sets of finite
measure.
pt: any outer measure, or the outer measure defined by p.
8 *: class of p*measurable sets.
p: measure obtained by restricting /1* to S *, also the completion of the measure
p.
g: a-ring obtained on completing measure p on S.
(1, 8 j: measurable space.
11,8 ,pI: measure space.
f= limf,,: pointwise liniit,f(x) =
eachx.
ffdp: integral off with respect to p.
ffdp: integral off over the set E.
L(X, p): class of functions integrable with respect to p.
or L"(p): the class of measurable functions with f f1P dp Coo functions equal a.e. being identified.
(f IfI" dp)1'T', the 9-norm off.
'4i of: composite function, (ji of)(x) =
/',, -÷ f a.u.: f, -+ almost uniformly (uniform convergence with an exceptional
set of arbitrarily small measure).
pip: measures v, p mutually singular, i.e. i4A) = p(C,4) = 0 for some measurable
A.
v = — ii: Jordan decomposition of the signed measure v.
total variation of the signed measure v.
p '
if I,, =
f
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Notation
14
dv/dp: Radon-Nikodym derivative of v with respect to p.
[ji]: (Chapter 8) indicates that an identity holds except on a set of zero p-measure,
(or zero Ipi-measure for a signed measure p).
Ox II: norm of the vector x.
= sup (If(x)l) = supremum norm off.
P,: Lebesgue-Stieltjes measure, with g a monotone increasing left-continuous
function and Pg([a, b)) =g(b) —g(a).
$'g: the Pg-measurable sets.
I I dPg or f f dg: integral of I with respect to the Lebesgue-Stieltjes measure
derived fromg. Also ffdg wheng EBV[a, bJ (Definition 4, Chapter 9).
pr1: the measure such that pf' (E) =
(E)).
C(T): the set of functions continuous on the interval I with supremum norm
&: elementary sets, i.e. union of a finite number of disjoint measurable rectangles.
it,
Ix X
monotone classes (Chapter 10).
product of measurable spaces.
Y, S X
U': (x,y)EE].
E":y-sectionofE= [x:(x,y)EE].
fl: class of sets, depending on context (Chapter 10).
p X v: the product measure. (So (p X PXA X B) = p(A) v(B)).
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CHAPTER 1
Preliminaries
In the chapter we collect for reference the various mathematical tools needed in
later chapters. As the reader is presumed to be familiar with the content of a
first course on real analysis, we are concerned not with setting up the theory
from stated
but with giving definitions and stating results and theorems
about sets, sequences and functions which serve to fIx the notation and to make
it clear in what form elementary results will be used. Proofs are provided for the
less familiar results. in section 1.7 we describe in some detail the special sets of
Cantor. These sets and the functions associated with them will be referred to
frequently in later chapters.
The standard abbreviations: iff 'if and only if, 3 'there exists', v 'given any',
'implies', will be used as required. The end of a proof is indicated by the
symbol 0.
1.1 SET ThEORY
Whenever we use set theoretic operations we assume that there is a universal set,
X say, which contains all the sets being dealt with, and which should be clear
from the context. The empty set is denoted by 0; x E A means that the element
x belongs to the set A. By A
B we mean that x E A x E B; A C B is strict
inclusion, that is,A
B and there existsx withx EBandxnotmA
We denote by [x:F(x)] the set of points or elements x of X with the property
P. The
CA of A is the set of points x of X not belonging to A; CA
obviously depends on the sets X implied by the content — in fact X is usually
the set of real numbers, except in Chapter 10. We will denote the union of two
sets by A U B or of a collection of sets by U
where I denotes some index
set, or by
:F(a)J — the union of all sets
such that a has the property
P. Sinularly for intersections A fl B, etc. Then unions and intersections are
The
linked by the De Morgan laws:
=
C(flAa) =
enceA —B =Á ñ CB;A
—B)U(B—A)isthesymmetricdifference
of A and B, some properties of which are listed in Example 1. The Cartesian
product XX Y is the set of ordered pairs [(x,y): x EX,y E YJ. We wifi denote
15
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[Ot. I
Preliminaries
16
the real numbers by R, the integers by N, the set of all integers by Z = [0, ± 1, ±
2, . . .], the set of rationals by 0, and n-dimensional Eucidean space by
so
that
is the set of n-tuples (x1,.. .
considered as a vector space with the
usual inner product. Notations for intervals are [a,b] = [x: a
[a,b)
[x:
(i)
EF=
E = F,
(v) For any sets E, F, G we have E F c (E
11
(vi)U
1=1
G) U (G
F),
fl
11
1=1
Solution: (i) is obvious from the symmetry of the definition.
To obtain (ii) use the identity C(E
F) =
(CE
By symmetry this must equal the right hand-side.
(lii) (E F) (G H) = ((F E) G) H by
fl CF) U (E fl F), to get
(i) and (u)
=
=
=
(iv) is obvious.
(v)We haveE—Fc (E—G)U(G—F)andF—Ec (F—G)U(G—E),so
taking the union gives the result.
(vi) This follows from the more obvious inclusion
U E1 — U F1 = Li (E1 — F1).
Example 2: LetE1
E2
...
.... Show thatL) (E1 —E1)=E1
Solution: This is just an application of De Morgan's laws with E1 as the whole
space.
Principle of Finite Induction. Let P(n) be the proposition that the positive integer
n has the property P. If P(1) holds and the truth of P(n) implies that ofF(n+1),
then P(n) holds for all n E N. It is to this property of positive integers that we
are appealing in our frequent 'proofs by induction' or in inductive constructions.
DefinItion 1: An equivalence relation R on a set E is a subset of E X E with the
following properties:
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Topological Ideas
SeC. 1.21
17
(i)(x,x)ER foreachxEE,
(ii)(x,y)ER
(iii) if (x,y)ER and (y,z)ER then (x,z)ER.
We write x " y if(xy) ER. Then R partitions E into disjoint equivalence classes
such that x and y are in the same class if, and only if, x " y. For, writing [xJ =
by (i), so
x EEl = E. Also by (ii) and (iii),
Lz: z " xl, we have x E
for any x,y E, either [xJ = [y] or [xl
[y] = O,so the sets [xJ are the
required equivalence classes.
In Chapter 2 we will need the Axiom of Choice which states that if [Es: a E AJ
is a non-empty collection of non-empty disjoint subsets of a set X, then there
exists a set V ỗ X containing just one element from each set
1.2 TOPOLOGICAL IDEAS
A quite broad class of spaces in which we can consider the ideas of convergence
and open sets is provided by metric spaces.
Definition 2: A nonnegative function p on the ordered pairs [(x,y) = x E X,
y E I] is a
on X if it satisfies
(i) p(xy)= 0 if, and only if,x
(ii) p(x,y) = p(y,x),
(ill) p(x,z)
IX,pI forms a
space. If we relax the condition that the distance between
distinct points be strictly positive so that (i) reads: p(x,y) = 0 if x = y, we
tam
a pseudo-metric. We will be especially concerned with the space R and,
briefly in Chapter 10, with A". But the idea of convergence in a metric space is
implied in many of the definitions of Chapters 6 and 7.
A set A in a
space
is open if given x EA there exists e >0 such
that [y: p(y,x) <€1 A. That is: A contains an 'e-neighbourhood' of x, denoted
N(x,e). So X and 0 are open and it also follows that any union of open sets is
open and that the intersection of two open sets is again open. The class of open
sets of X forms a topology on X. We now define various other ideas which can
be derived from that of the metric on X. The properties that follow inunediately
are assumed known. In the case of the real line we list various properties which
will be needed, in Theorem 1 and in the later sections.
A set A is closed if CA is open. The closure A of a set A is the intersection of
all the closed sets containing A, and is closed. A point x is a limit point of A if
given e >0, there exists y E A, y * x, with p(y,x)
non-empty open set, so that a nowhere dense set in A is one whose closure
contains no open interval. A is said to be a perfect set if the set [x:x a limit
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_
[Cli. 1
Preliminaries
18
point of A] is A itself. If the set A may be written as A =
G1,
where the sets
1=1
G,
are open, A is said to be a G6 -set; if A U F1 where the sets F1 are closed,
then A is an F0-set. Clearly the complement of a G6 -set is an Fa-set.
In Chapter 10 we will refer to the relative topology on a subset
A of R",
namely the class of sets G of the form H flA where H is an open set in fl". This
class of sets forms a topology on A.
We will assuim the notion of supremum (or least upper bound) of a set A of
real numbers, denoted usually by sup[x: x E Al or by sup[x: P(x)1 , where P is
the property satisfied by x. In the cases where we use this notation the set in
x1 for the suprequestion will be non-empty. For a finite set we will write
mum of the relevant set. Similarly we will write mf[x: x EAI for the infimum
(or greatest lower bound) or mm Xj in the finite case.
1
We will need the following important property of the real numbers.
Theorem I (Heme-Borel Theorem): If A is a closed bounded set in R and A c
U Ga, where the sets Ga are open and I is some index set, then there exists a
crEJ
finite subcollection of the sets, say [G1, f =
1,
. .
.,
n] whose union contains A.
Exercises
1. (i) Let p be a pseudo-metric of a space E and write x " y if p(x,y) = 0.
Show that this defines an equivalence relation on E and describe the equivalence classes to which it gives rise.
(ii) In the notation of p.' 17,1 let
be defined by p*([xl, [yJ) = p(x,y);
show that
is a metric on the set of equivalence classes.
2. Show that A is nowhere dense iff cA is dense.
3. Show by examples that G6 and F0-sets may be open, closed or neither
open nor closed.
& Show that In a metric space each point is a closed G5 -set.
S. Show by examples that Theorem 1 can break down if either of the conditions
A closed or A bounded is omitted.
1.3 SEQUENCES AND LIMITS
A numerical sequence
is a function from N to R. We define the upper
=inf[ sup xm: n EN]. If there is no ambiguity
J
m>n
possible we will write this as lim sup x,,. Similarly: lim
N
is the lower limit of
}.
If lim sup x,
hin
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inf Xm: fl E
m>n
we write their common
Sequences
Sec. 1.31
value as lAm x,,, the limit of
—un inf (—xe).
and Limits
19
}. From the definitions we get easily lim sup x,, =
If we consider a function from A to A we get the analogous definition: the
upperlimit of at a0 is lim sup= inf[ sup
xa: h >0] . Similarly we define
O
tional notation we
In the more usual func-
write lim sup x(a) and urn inf x(a), where x(a) is a
valued function. A property of upper and lower limits is given by the following
example.
Example 3: Prove that if Jim Ya exists, then urn sup (xa + ya) = tim sup xa +
a-÷a0
Iijii
lim inf (xa +
= lim inf xa + lim
-'a0
where all the limits are
supposed finite.
Solution: We prove the first equation. Write
urn sup (xa + ya), 12 = urn sup xa, 13 = urn
Given
0<
0, there exists & > 0 such that xa <12 +
and
<13 +
when
I <&. SO Xa +
<12 +13 + 2€ in this range, and as is arbitrary
<12 + 13. Conversely: there exists &' > 0 such that xa +
+ and
>13 — when 0 < a — a0 I <&', so in this range xa (xa + Ya) — Ya <
+2€andsol2
givingtheresult.
A similar result holds for sequences
We will be particularly concerned with 'one-sided' upper and lower limits,
and express these in functional notation:
— a0
limsupx(t)=inf[ sup x(a—u):h>O],
0
liminfx(t) =sup[ inf x(a—u):h>O].
0
If these quantities are equal, we say that km x(t) exists and we write this
limit as x(cr—). Note that x(a—) need not be defined, although lim sup x(t) and
tim ml x(t) always are. Replacing a — u by a + u in these definitions we get
a—
Jim supx(t),
lim ml x(t) and, if it exists, x(a+).
is monotone increasing, and we write x,,t, if for each
The sequence
x,, for each n; so if a sequence Is both monotone increasing and
N,
is a
monotone decreasing it is constant. We will assume the result that if
monotone sequence and is bounded, then it has a limit, and we write x,, t x or
ii
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[Ot I
Preliminaries
20
x as appropriate; if tx,,) is monotone but not bounded, then x,, -+
x,,
x,,
oo or
the case may be.
A sequence
is a Cauchy sequence if for any positive e there exists N such
that lx,, —
< e for n,m > N. We will assume the result that a sequence converges if, and only if, it is a Cauchy sequence. We define a Cauchy sequence of
elements of a
space similarly, requiring that P(X,,, Xm) <e for n,m > N.
Then the space is a complete metric space if every Cauchy sequence converges,
as
I
so that the result assumed above is that R forms a complete metric space with
p(x,y)=
We
use the o- and 0-notations; so that if fx,, is a sequence of real numbers,
then x,, = o(n") means that
n
x,,
x,, =
means that
x,,
x,, = 0(1)
means that
is bounded. For functions from R to R: f(x) = o(,g(x)) as
x -÷ a means that given e> 0, there exists > 0 such that fr(x)t <
for
0 < — a I <5 ; flx) = o(g(x)) as x -÷ means that given e > 0, there exists
K > 0 such that lf(x)I <
for x > K. Similarly J1:x) = O(g(x)) means that
there exists M> 0 such that lf(x)l Mk(x)I as x -÷ a or x -÷ as the case may
be.
We also need some properties of double sequences {x,,,,, } which are functions on N X N, and we recall that lim Xnm = x means that given e > 0
is, m
thereexistsNsuchthatifn,m>N, Ix,,,,, —xI
given M>
0 there exists N such that x,,1,, > M for all n,m > N.
If one index, m or n, is kept fIxed,
J is an ordinary 'single' sequence and
we have the usual notation of iterated limits lim lim x11,, and thu tim
Theorem 2: If ix,,,,, } is increasing with respect to n and to m, then lim
n/n-,
x,,,,,
exists and we have
tim
x,,/n = tim lim
= lirn lim
(1.1)
and if any one is infinite, all three are.
Proof: Write Ym = lim
this is clearly defined for each m. Since
for each n, we have Ym
Ym+i. So 11 = limYm exists (it may =
oo).
Write 1 = sup(x,,,m: n, m E N], where we may have 1 =
that lim
Then it is obvious
x,,1,, =1. In the case 1 <°° we wish to show that 1=11. Since x,,,,,,
1 for each n and m, it is obvious that
1. Also, given e >0 let N be such that
IXn,m —lt
and
AsyN
+ e,
and as e is arbitrary, 1
, so 1 =
Similarly we may show that the third limit
in (1.1) exists and equals 1. The case = is considered similarly. 0
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Functions and Mappings
Sec. 1 .4J
In the case of a series
sums,
a1,
=
21
a, we may consider the sequence
} of its partial
and in the case of a double series
a1
So
to each property of sequences there corresponds one of
1=1 i=1
series. For instance, to Theorem 2 corresponds the following result.
Theorem 3: If a1,,> 0 for all i,/ E N, then
=
a11
1,1=1
a1, exists and
j=1 1=1
a1,1,
i=1 1=1
all three sums being infinite if any one is.
We will also need the following elementary properties of series.
Theorem 4: If aj is absolutely convergent, that is,
1=1
a1 is convergent to a fInite sum.
1a11 < oo, then
the series
a convergent series, with sums, say, and
Theorem 5: If a1 4 0, then
for eachn,
Exercise
6.
Let 4) be a monotone function defined on [a,bl Show that
and ỗb(b)
exist.
1.4 FUNCI1ONS AND MAPPINGS
Functions considered will be real-valued (or, briefly, in Chapter 10, complexvalued) functions on some space X. In many cases the space X will be R. if the
function f is defined on X and takes its values in Y we will frequently use the
Y and f: Y -÷ Z, then the composite function
notation f: X -÷ Y. If g: X
=f(g(x)).
fog: X-÷Z is defined by
The domain of f is the set [x: 1(x) is definedj. The range of f is the set
Y we writef1 (A) Ix: x E X,
[y: y = f(x) for some xJ. If f: X Y and A
X we write flB) = [y: y = fix), x E B] . The function f is
fix) E A], and if B
a one4o-one mapping of X onto Y if the domain off is X, the range off is Y,
and /1:x1) = f(x2) only if x1 x2. hi this case f1 is a well-defined function on
Y. The identity mapping is denoted by 1 so I x = x. If a
function
is a one-to-one
mapping, then on the domain of f,f1 of= I and on the range off,fof' =
1.
The function f extends the function g or is an extension of g if the domain of
f contains that of g, and on the domain of g, f = g. Frequently, in applications,
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Preliminaries
22
[Cli. 1
this definition for set functions, that is, functions whose domains are
classes of sets and which take values in R.
If [fe: a EA] is a set of functions mapping X A, we will write
for
we will use
the functionf defined on Xby
= sup
a EAJ. The notations sup I,
I
and
max(fg) have the obvious meanings, and we denote similarly inlima and
minima.
Definition 3:
n 1, 2,. . ., andfbe
functions on the space I.
uniformly if given >0, there exists n0 E N such that
Then 1,,
f
x El] <e forn >n0.
sup
Elementary results concerning continuity and differentiation will be used as
required, as will the definitions and more familiar properties of standard functions. A standard result on continuous functions which we will assume known is
the following.
Theorem 6: Let
be a sequence of continuous functions,
uniformly;
then
f is a continuous function.
1,,
X -÷ A and let
Statements about sets can be turned into ones about functions using the
following notation.
Definition 4: Let the set A be contained in the space X; then the characteristic
function of A, written
is the function on X defmed by:
= 1 for
A step function on R is one of the form
a1 br,, where
= 1,..
n denote
disjoint intervals. An example of such a function which will be used Is sgn x
which is defined as: sgn x = 1 for x> 0, sgn 0 = 0, sgn x = —1 for x <0.
1.5 CARDINAL NUMBERS AND CARDINALITY
Two sets A and B are said to be equipotent, and we write A B, if there exl3ts
a
impping with domain A and range B. A standard result of set theory
(cf. [11], p. 99) is that with every set A we may associate a well-deflned object,
Card A, such that A "-' B if, and only if, Card A = Card B. We say that Card A>
Card B if for some A' C A we have B "-' A' but there is no set B' C B such that
A ".' B'. We assume the result: for any set A, Card A
we write Card P (A) = 2a.
If A is fInite, we have Card A = n, the number of elements in A. If A N
we write Card A =
and describe A as an infInite countable set. If A R we
write Card A = c. It is easy to show that if Jig any interval, open, closed, or halfopen, and if I contains more than one point, then Card I = c. Another result is
that if [Aj: I E
NJ
is a collection of countable sets, then Card L) A, =
1=1
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Also
Further Properties of Open Sets
Sec. 1 .6J
Card 0 =
23
In Chapter 2, Exercise 45, we need the following result: Card
[f: f: A -+ Rj = 2C; for an explicit proof see
,
p. 50.
1.6 FURThER PROPERTIES OF OPEN SETS
Theorem 7 (LindelOf's Theorem): If
= [Ia: a E AJ is a collection of open
intervals, then there exists a subcollection, say [Ii: i = 1, 2, . . . ], at most
countable in number, such that U
11
I,= U
aEA
Proof. Each x E 'a is contained in an open interval J, with rational end-points,
such that J1
Ills for each a; since the rationals are countable the collection
[J,] is at most countable. Also, it is clear that U 'a = U J,. For each i choose
aEA
1=1
anintervall1 of
such thatl1
Jj. Then U 4 = U J,
aEA
(=1
U
so we get the
1=1
identity and the result. 0
If the subcollection obtained is finite, we make the obvious changes of
notation.
Theorem 8 (LindelOf's Theorem in Rn): If
is a collection of
= [Ga: a E
open sets in R", then there exists a subcollection of these, say [G,: i = 1,2,.. .1'
at most countable in number, such that U G1 = U Ga.
aEA
1=1
Since [t: It — xI <rJ C Ga for some r> 0, there exists an open 'cube'
with the sides of length
such that x E Ta C Ga, and a 'rectangle' J,
with rational coordinates for its vertices and containing x may be chosen within
Ta. The proof then proceeds as in R. (We have written lx — y I for the usual
distance between points x ,y of R".) 0
1'heorezn 9: Each non-empty open set G in R
vals, at most countable in number.
is
the union of disjoint open inter-
Proof: Following Definition 1, p. 16, write a "-'b if the closed interval [a,b], or
[b,aJ if b
classes. Let C(a) be the equivalence class containing a. Then C(a) is clearly an
interval. Also C(a) is open, for if k E
then (k e, k + e) ỗ G for suffiso G is the union of disjoint
ciently small e. But then (k e, k + e) c
intervals. These are at most countable in number by Theorem 7.0
1.7 CANTOR-LIKE SETS
We now describe the Cantor-like sets. These, and the functions defined on them,
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[Oi. I
Preliminaries
24
are particularly useful for the construction of counter-examples. A special case —
the Cantor ternary set or Cantor set — is sufficient for many purposes and will
be described separately.
The construction is inductive. From [0,11 remove an open Interval
with
centre at 1/2 and of length < 1. This leaves two 'residual intervals', J11, J1,2,
each of length < 1/2. Suppose that the nth step has been completed, leaving
closed intervals
.,J,,2n, each of length < 1/2". We carry out the (n + l)st
step by removing from each
k and
formed
an
of length < 1/2". Let 1',,
open interval
2"
=U
k1
and
,k with the same centre as
let P= fl
n1
P,1.
Any
set P
in this way is a Cantor-like set.
In particular P contains the end-points of each J,,
2fl-1
UU
n1 k1
Since [0,11 — P
=
k, an open set, P is closed. Since P contains no interval, indeed
contains no interval of length > T", it follows that Pi8 nowhere dense.
The set P is perfect since if x E F, then for each n, x
for some k,. So if
for any positive we choose n such that 1/2" <€, then the
of
lie in (x — €, x + e). But these end-points belong to P, so x is a limit point of P.
A particular case which will be useful is when 1(J11) = l(J1,2) = < 1/2,
=
l(J2,1) = . =
etc., where 1(1) denotes the length of the interval I.
each
So at each stage residual interval is divided in the same proportions as the original
interval [0,11. Denote the resulting Cantor-like set, in this case, by
Slightly
more generally, let 1(J11) = l(J1,2) =
1/2), l(J2,1) = . =
=
etc., so that at each stage the residual intervals are equal but the proportions are
allowed to change from stage to stage. Denote the resulting set in this case by Pt,
where = fbi,
for each n. Use is made of Pt
. . .). Note that
and
in the next chapter.
We may vary the construction by choosing the removed open intervals
centre', with centres a fIxed combination y, 1 — of the
of the
where 0
The Cantor Set P
From the interval [0,1] first remove (1/3, 2/3), then (1/9, 2/9) and (7/9,8/9),
etc., removing at each stage the open intervals constituting the 'middle thirds'
of the closed intervals left at the previous stage. This gives a special case of the
previous constructions, with the residual closed intervals at the nth stage, J,,1,...,
each of length 1/3", the open intervals Ifl, also being of length 1/3". if
2"
= U J,,
k=1
more
k
thenP= fl F,, is the Cantor ternary set or
n1
briefly the Cantor set. That P is uncountable follows from Example 4
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