Measure theory
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Paul R. Halmos
Measure Theory
Springer-Verlag
NewYork· Heidelberg· Berlin
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Managing Editors
P. R. Halmos
C. C. Moore
Indiana University
Department of Mathematics
Swain Hall East
Bloomington, Indiana 47401
University of California
at Berkeley
Department of Mathematics
Berkeley, California 94720
AMS Subject Classifications (1970)
Primary: 28 02, 28A10, 28Al5, 28A20, 28A25, 28A30, 28A35, 28A40,
28A60, 28A65, 28A70
Secondary: 60A05, 60Bxx
-
Library of Congress Cataloging in Publication
Halmos, Paul Richard,
Measure theory.
Data
1914-
18)
texts in mathematics,
Reprint of the ed. published by Van Nostrand,
New York, in series: The University series
in higher mathematics.
Bibliography: p.
I. Title. II. Series.
1. Measure theory.
[OA312.H26 1974]
515'.42
74-10690
ISBN 0-387-90088-8
(Graduate
All rights reserved.
in
No part of this book may be translated or reproduced
any form without written permission from Springer-Verlag.
by Litton Educational Publishing,
1974 by pringer-Verlag
New York Inc.
@ 1950
Pr°
Inc. and
ed in the United States of America.
ISBN 0-387-90088-8 Springer-Verlag New York Heidelberg Berlin
ISBN 3-540-90088-8 Springer-Verlag Berlin Heidelberg New York
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PREFACE
I
My main purpose in this book is to present a unified treatment
of that part of measure theory which in recent years has shown
itself to be most useful for its applications in modern analysis.
If I have accomplished my purpose, then the book should be
found usable both as a text for students and as a source of reference for the more advanced mathematician.
I have tried to keep to a minimum the amount of new and
unusual terminology and notation.
In the few places where my
nomenclature differs from that in the existing literature of measure theory, I was motivated by an attempt to harmonize with
the usage of other parts of mathematics.
There are, for instance,
sound algebraic reasons for using the terms
and
which
for certain classes of sets-reasons
are more cogent than
the similarities that caused Hausdorff to use
and
The only necessary prerequisite for an intelligent reading of
the first seven chapters of this book is what is known in the
algebra and analysis.
United States as undergraduate
For the
convenience of the reader, § 0 is devoted to a detailed listing of
exactly what knowledge is assumed in the various chapters.
The
beginner should be warned that some of the words and symbols
in the latter part of §0 are defined only later, in the first seven
chapters of the text, and that, accordingly, he should not be discouraged if, on first reading of § 0, he finds that he does not have
the prerequisites for reading the prerequisites.
At the end of almost every section there is a set of exercises
which appear sometimes as questions but more usually as assertions that the reader is invited to prove. These exercises should
be viewed as corollaries to and sidelights on the results more
"lattice"
"ring"
V
"ring"
"field."
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vi
I
i
MMMil---
PREFACE
formally expounded.
They constitute an integral part of the
book; among them appear not only most of the examples and
the theory, but
counter examples necessary for understanding
also definitions of new concepts and, occasionally, entire theories
that not long ago were still subjects of research.
It might appear inconsistent that, in the text, many elementary
notions are treated in great detail, while,in the exercises,some quite
refined and profound matters (topological
spaces, transfinite numbers, Banach spaces, etc.) are assumed to be known. The material is arranged, however, so that when a beginning student comes
to an exercise which uses terms not defined in this book he may
simply omit it without loss of continuity.
The more advanced
reader, on the other hand, might be pleased at the interplay
between measure theory and other parts of mathematics which
it is the purpose of such exercises to exhibit.
The symbol I is used throughout the entire book in place of
such phrases as "Q.E.D." or "This completes the proof of the
theorem" to signal the end of a proof.
At the end of the book there is a short list of references and a
bibliography. I make no claims of completeness for these lists.
Their purpose is sometimes to mention background reading,
rarely (in cases where the history of the subject is not too well
known) to give credit for original discoveries, and most often to
indicate directions for further study.
A symbol such as u.v, where u is an integer and v is an integer
theorem, formula,
or a letter of the alphabet, refers to the (unique)
or exercise in section u which bears the label v.
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ACKNOWLEDGMENTS
11
Most of the work on this book was done in the academic year
1947-1948 while I was a fellow of the JohnSimon Guggenheim
Memorial Foundation, in residence at the Institute for Advanced
Study, on leave from the University of Chicago.
I am very much indebted to D. Blackwell, J. L. Doob, W. H.
Gottschalk, L. Nachbin, B. J. Pettis, and, especially, to J. C.
Oxtoby for their critical reading of the manuscript and their many
valuable suggestions for improvements.
The result of 3.13 was communicated to me by E. Bishop.
The condition in 31.10 was suggested by J. C. Oxtoby. The
example 52.10 was discovered by J. Dieudonné.
P.R.H.
V11
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CONTENTS
I
I
III
PAGE
Preface........................
Acknowledgments
v
vii
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SECTION
0.Prerequisites......................
CHAPTER
1
I:
SETS
AND
1.Setinclusion......................
2.Unionsandintersections
3. Limits, complements, and differences
4.Ringsandalgebras...................
5. Generated rings and o·-rings
6.Monotoneclasses....................
CLASSES
................
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CHAPTER
II:
MEASURES
7.Measureonrings....................
8.Measureonintervals..................
9. Properties of measures
10.Outermeasures
11.Measurablesets....................
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OUTER
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III:
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CHAPTER
17.Measurespaces
18.Measurablefunctions
IV:
MEASURABLE
30
32
37
41
44
MEASURES
12. Properties of induced measures.
13. Extension, completion, and approximation.
14.Innermeasures
15Lebesguemeasure...................
16.Nonmeasurablesets.......
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22
26
MEASURES
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CHAPTER
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62
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FUNCTIONS
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1X
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CONTENTS
X
SECTION
19.
20.
21.
22.
PACE
Cornbinations of measurable functions
Sequences of measurable functions
Pointwise convergence
Convergencein sneasure.
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Integrable simple functions
Sequences of integrable simple functions
Integrable functions
Sequences of integrable functions.
27. Properties of integrals
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90
V: INTEGRATION
CHAPTER
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95
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107
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CHAPTER VI: GENERAL SET FUNCTIONS
28. Signed measures
29. Hahn and Jordandecompositions
30.Absolutecontinuity
31. The Radon-Nikodym theorem
32. Derivatives of signed measures
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PRODUCT
CHAPTER VII:
33. Cartesian products
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34.Sections
35.Productmeasures
36.Fubini'stheorem
37. Finite dimensional product spaces
38. Infinite dimensional product spaces
SPACES
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CHAPTER VIII:
117
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128
132
137
141
143
145
150
154
TRANSFORMATIONS AND FUNCTIONS
39. Measurable transformations
40.Measurerings.....................
41. The isomorphism theorem
42.Functionspaces....................
43. Set functions and point functions.
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CHAPTER IX' PROBABILITY
44. Heuristic introduction
45. Independence
46. Series of independent functions
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184
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196
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xi
CONTENTS
SECTION
PAGE
47. The law of large numbers
48. Conditional probabilities and expectations
49. Measures on product spaces
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cHAPTER
X:
50.Topologicallemmas
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LOCALLY
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SPACES
COMPACT
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51.BorelsetsandBairesets
52.Regularmeasures
53. Generation of Borel measures
54.Regularcontents
55. Classes of continuous functions
56.Linearfunctionals...................
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CHAPTER
XI:
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XII:
MEASURE
61. Topology in terms of measure
62.Weiltopology.....................
AND
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TOPOLOGY
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IN
GROUPS
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63.Quotientgroups....................
64. The regularity
References
Bibliography
of Haar measure
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277
282
291
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293
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List of frequently used symbols
Index
216
219
223
231
237
240
243
MEASURE
HAAR
57.Fullsubgroups
58.Existence.......................
59.Measurablegroups...................
60.Uniqueness......................
CHAPTER
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L
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11
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§0.
PREREQUISITES
the first
The only prerequisite for reading and understanding
seven chapters of this book is a knowledge of elementary algebra
and analysis. Specifically it is assumed that the reader is familiar
with the concepts and results listed in (1)-(7) below.
(1) Mathematical induction, commutativity and associativity
of algebraic operations, linear combinations, equivalence relations
and decompositions into equivalence classes.
(2) Countable sets; the union of countably many countable
sets is countable.
(3) Real numbers, elementary metric and topological properties
of the real line (e.g.the rational numbers are dense, every open
set is a countable union of disjoint open intervals), the HeineBorel theorem.
(4) The general concept of a function and, in particular, of a
sequence (i.e.a function whose domain of definition is the set of
positive integers); sums, products, constant multiples, and absolute values of functions.
(5) Least upper and greatest lower bounds (calledsuprema and
infima) of sets of real numbers and real valued functions; limits,
superior limits, and inferior limits of sequences of real numbers
and real valued functions.
and the following algebraic rela(6) The symbols +o and
-o,
tions among them and real numbers x:
±o
x(±o)
=
(±o)x
0
=
=
o
x/(±o)
=
1
if x > 0,
if x
0,
if x < 0;
0;
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2
[SEc.
PREREQUISITES
The phrase extended real number refers to a real number
0)
or one
of the symbols ±oo.
(7) If x and y are real numbers,
xUy
=
max
xny=min
{x,y }
{x,y}
à(x+ y +
=
g) (x)
=
y
\),
=¼(x+y-lx-y\).
Similarly, if f and g are real valued
fng are the functions defined by
(JU
\x
-
and
f (x) Ug (x)
functions, then fUg
g) (x)
(fn
=
f (x) 0 g (x),
respectively.
The supremum and infimum of a sequence
of real numbers are denoted by
and
U:_1
x,
respectively.
.1
and
{x,}
xx,
In this notation
lim sup, x,
=
and
lim inf, x.
=
0 2-1 U:-- xU:-i A:-- x--
In Chapter VIII the concept of metric space is used, together
with such related concepts as completeness and separability for
of functions on metric
metric spaces, and uniform continuity
spaces. In Chapter VIII use is made also of such slightly more
sophisticated concepts of real analysis as one-sided continuity.
In the last section of Chapter IX, TychonofPs theorem on the
compactness of product spaces is needed (for countably many
factors each of which is an interval).
In general, each chapter makes free use of all preceding chapters; the only major exception to this is that Chapter IX is not
needed for the last three chapters.
In Chapters X, XI, and XII systematic use is made of many
of the concepts and results of point set topology and the elements
of topological group theory.
We append below a list of all the
relevant definitions and theorems.
The purpose of this list is not
to serve as a text on topology, but (a) to tell the expert exactly
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[sEc.
3
PREREQUISITES
0]
which forms of the relevant concepts and results we need, (b) to
tell the beginner with exactly which concepts and results he should
familiarize himself before studying the last three chapters, (c) to
put on record certain, not universally used, terminological conventions, and (d) to serve as an easily available reference for
things which the reader may wish to recall.
Topological Spaces
A topologicalspace is a set X and a class of subsets of X, called
the open sets of X, such that the class contains 0 and X and is
closed under the formation of finite intersections and arbitrary
(i.e. not necessarily finite or countable) unions. A subset E
of X is called a Ga if there exists a sequence
U,} of open sets
{
such that E
U,. The class of all Ga's is closed under the
formation of finite unions and countable intersections. The topological space X is discrete if every subset of X is open, or, equivalently, if every one-point subset of X is open. A set E is closed
if X
E is open. The class of closed sets contains 0 and X and
is closed under the formation of finite unions and arbitrary intersections. The interior, E°, of a subset E of X is the greatest open
set contained in E; the closure, E, of E is the least closed set containing E. Interiors are open sets and closures are closed sets;
if E is open, then E°
E. The
E, and, if E is closed, then Ë
closure of a set E is the set of all points x such that, for every open
X.
set U containing x, EDU ¢ 0. A set E is dense in X if E
A subset Y of a topological space becomes a topological space
(a subspace of X) in the relative topologyif exactly those subsets
of Y are called open which may be obtained by inters
g an
open subset of X with Y. A neighborhood of a point x in X
[orof a subset E of X] is an open set containing x [oran open set
containing E]. A base is a class B of open sets such that, for
every x in X and every neighborhood U of x, there exists a set
B in B such that xeB c- U. The topology of the real line is
determined by the requirement that the class of all open intervals
be a base. A subbase is a class of sets, the class of all finite intersections of which is a base. A space X is separable if it has a
countable base. A subspace of a separable space is separable.
°.1
=
--
=
=
=
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4
[SEC.
PREREQUISITES
Û)
An open covering of a subset E of a topological space X is a
class K of open sets such that EcU K. If X is separable and
K is an open covering of a subset E of X, then there exists a
Of
countable subclass {Ki, K2,
i Which is an open covering
of E. A set E in X is compact if, for every open covering K of E,
K,} of K which is an open
there exists a finite subclass {Ki,
covering of E. A class K of sets has the nnite intersection property if every finite subclass of K has a non empty intersection.
A space X is compact if and only if every class of closed sets with
the finite intersection property has a non empty intersection. A
if there exists a sequence {C,}set E in a space X is «-compact
of compact sets such that E
C.. A space X is locally
U:-1
compact if every point of X has a neighborhood whose closure is
compact. A subset E of a locally compact space is bounded if
there exists a compact set C such that Ec C. The class of all
bounded open sets in a locally compact space is a base. A closed
subset of a bounded set is compact.
A subset E of a locally com«-bounded
if there exists a sequence {C,} of compact
pact space is
sets such that Ec U:-1Cs. To any locally compact but not
compact topological space X there corresponds a compact space
X* containing X and exactly one additional point x*; X* is called
the one-point compactification of X by x*. The open sets of X*
are the open subsets of X and the complements (in X*) of the
closed compact subsets of X.
If {Xs: i e I} is a class of topological spaces, their Cartesian
product is the set X
X {X;: i e I} of all functions x defined
on I and such that, for each i in I, x(i) e X;. For a fixed io in
be an open subset of X;,, and, for i ¢ io, write Ei
Xe;
I, let
of open sets in X is determined by the requirement that
the e
the class of all sets of the form X {Es: i e I} be a subbase. If
the function is on X is defined by (s(x) x(i), then is is continuous. The Cartesian product of any class of compact spaces is
•
•
•
·
·
·,
=
=
=
=
compact.
A topological space is a Hausdorff space if every pair of distinct
points have disjoint neighborhoods.
Two disjoint compact sets
A compact
in a HausdorE space have disjoint neighborhoods.
subset of a HausdorE snace is closed. If a locally compact space
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5
PREREQUISITES
[Sac. 0]
or a separable space, then so is its one-point
A real valued continuous function on a compact
is a Hausdorff space
compactification.
set is bounded.
For any topological space X we denote by 5 (orf (X)) the class
of all real valued continuous functions f such that 0 5 f(x) 5 1
for all x in X. A Hausdorff space is completely regular if, for
every point y in X and every closed set F not containing y, there
is a function f in 5 such that f(y) 0 and, for x in F, f(x) 1.
A locally compact Hausdorff space is completely regular.
A metric space is a set X and a real valued function d (called
0 if and only
distance) on X × X, such that d(x,y) 2 0, d(x,y)
if x y, d(x,y) d(y,x), and d(x,y) 5 d(x,z) + d(z,y). If E and
F are non empty subsets of a metric space X, the distance between
them is defined to be the number d(E,F)
inf {d(x,y):
xe E,
xo} is a one-point set, we write d(E,xo) in place
ye F}. If F
{
of d(E, {xo}). A sphere (withcenter xo and radius ro) is a subset
E of a metric space X such that, for some point xo and some positive number ro, E
{x: d(xo,x) < ro}. The topologyof a metric
space is determined by the requirement that the class of all
spheres be a base. A metric space is completely regular. A closed
set in a metric space is a Ga. A metric space is separable if and only
if it contains a countable dense set. If E is a subset of a metric
d(E,x), then f is a continuous function and
space and f(x)
Ë {x: f(x) 0}. If X is the real line, or the Cartesian product
of a finite number of real lines, then X is a locally compact separax.)
ble Hausdorff space; it is even a metric space if for x
(xi,
y.) the distance d(x,y) is defined to be
and y
(yi,
(Ti..i(xx ys)2 ½. A closed interval in the real line is com*
pact set.
A transformation T from a topological space X into a topological
space Y is continuous if the inverse image of every open set is
open, or, equivalently, if the inverse image of every closed set is
closed. The transformation
T is open if the image of every open
set is open. If B is a subbase in Y, then a necessary and sufficient
condition that T be continuous is that T¯I(B) be open for every
B in B. If a continuous transformation T maps X onto Y, and
if X is compact, then Y is compact. A homeomorphism is a one
=
=
=
=
=
=
=
=
=
=
=
=
=
·
·
·
·
·,
--
-
·,
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6
PREREQUISITES
[SEC.
Û)
to one, continuous transformation of X onto Y whose inverse is
also continuous.
The sum of a uniformly convergent series of real valued, continuous functions is continuous.
If f and g are real valued continuous functions, then fUg
and fng are continuous.
Topological Groups
A group is a non empty set X of elements for which an associative multiplication
is defined so that, for any two elements a and
b are solvable. In every
b and ya
b of X, the equations ax
group X there is a unique identity element e, characterized by
the fact that ex
xe
x for every x in X. Each element x
of X has a unique inverse, x¯¯I, characterized
by the fact that
xx¯
e. A non empty subset Y of X is a subgroup
x x
if x¯ yeY whenever x and y are in Y. If E is any subset of a
x¯I,
E¯I
of
of
elements
where
all
the
the
form
X,
is
set
group
xe E; if E and F are any two subsets of X, EF is the set of all
elements of the form xy, where xeE and ye F. A non empty
subset Y of X is a subgroup if and only if Y¯¯¯IY
c Y. If xe X,
it is customary to write xE and Ex in place of {x}E and E{x}
respectively; the set xE [orEx] is called a left translation [orright
translation] of E. If Y is a subgroup of X, the sets xY and Yx
are called (left and right) cosets of Y. A subgroup Y of X is
invariant if xY
Yx for every x in X. If the product of two
cosets Yi and Y2 of an invariant subgroup Y is defined to be
YiY2, then, with respect to this notion of multiplication, the class
of all cosets is a group Ž, called the quotient group of X modulo
Y antladenoted by X/Y. The identity element 2 of Ñ is Y. If
Y is an invariant subgroup of X, and if for every x in X, x(x)
is the coset of Y which contains x, then the transformation «
is called the projection from X onto 2. A homomorphism is a
transformation
T from a group X into a group Y such that
T(x) T(y) for every two elements x and y of X. The
T(xy)
projection from a group X onto a quotient group Ž is a homo=
=
=
=
=
=
=
=
=
morphism.
A topologicalgroup is a group X which is a Hausdorff space
such that the transformation
(fromXXX onto X) which sends
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lsEc.
7
PREREQUISITES
0]
A class N of open sets containing
e in a topological group is a base at e if (a) for every x different
from e there exists a set U in N such that x e' U, (b) for any two
V,
sets U and V in N there exists a set W in N such that WCUn
(c) for any set U in N there exists a set V in N such that
V-IV c U, (d) for any set U in N and any element x in X, there
exists a set V in N such that Vc xUx¯I,
and (e) for any set U
in N and any element x in U there exists a set V in N such that
Vx c U. The class of all neighborhoods of e is a base at e; conversely if, in any group X, N is a class of sets satisfying the conditions described above, and if the class of all translations of sets
of N is taken for a base, then, with respect to the topology so
defined, X becomes a topological group. A neighborhood V of e
V¯*; the class of all symmetric neighboris symmetric if V
hoods of e is a base at e. If N is a base at e and if F is any closed
set in X, then F
{UF: Ue N}.
The closure of a subgroup [or of an invariant subgroup] of a
topological group X is a subgroup [or an invariant subgroup) of
X. If Y is a closed invariant subgroup of X, and if a subset of
X/Y is called open if and only if its inverse image (under
the projection x) is open in X, then Ž is a topological group and
the transformation « from X onto 1 is open and continuous.
If C is a compact set and U is an open set in a topological group
X, and if CC U, then there exists a neighborhood V of e such
that VCV c U. If C and D are two disjoint compact sets, then
there exists a neighborhood U of e such that UCU and UDU
are disjoint. If C and D are any two compact sets, then C¯*
and CD are also compact.
A subset E of a topological group X is bounded if, for every
x,}
neighborhood U of e, there exists a finite set {xi,
(which,
in case E ¢ 0, may be assumed to be a subset of E) such
that Ec UL1 x;U; if X is locally compact, then this definition
of boundedness agrees with the one applicable in any locally compact space (i.e.the one which requires that the closure of E be
compact).
If a continuous, real valued function / on X is such
that the set N(f)
f is uniformly
{x: f(x) y 0} is bounded, thennumber
continuous in the sense that to every positive
e there
(x,y)into x¯*y is continuous.
=
=
=
·
=
·
·,
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8
PREREQUISITES
[SEc.
.
0]
corresponds a neighborhood U of e such that f(xi) f(x2) < e
whenever xix2¯·1 e U.
A topological group is locally bounded if there exists in it a
bounded neighborhood of e. To every locally bounded topological group X, there corresponds a locally compact topological
determined to
group X*, called the completion of X (uniquely
within an isomorphism), such that X is a dense subgroup of X*.
Every closed subgroup and every quotient group of a locally
|
compact group is a locally compact group.
-
|
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Chapter 1
SETS AND CLASSES
§ 1.
SET
INCLUSION
Throughout this book, whenever the word set is used, it will
be interpreted to mean a subset of a given set, which, unless it is
assigned a different symbol in a special context, will be denoted
by X. The elements of X will be called points; the set X will
be referred to as the space, or the whole or entire space, under
consideration. The purpose of this introductory chapter is to define the basic concepts of the theory of sets, and to state the
principal results which will be used constantly in what follows.
If x is a point of X and E is a subset of X, the notation
xeE
means that x belongs to E (i.e.that one of the points of E is x);
the negation of this assertion, i.e. the statement that x does not
belong to E, will be denoted by
x e' E
.
Thus, for example, for every point x of X, we have
xe X,
and for no point x of X do we have
x e' X.
If E and F are subsets of X, the notation
EcF
E
or F
->
9
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10
SETS AND CLASSES
[SEC.
Ỵ]
means that E is a subset of F, i.e. that every point of E belongs
to F. In particular therefore
ECE
for every set E. Two sets E and F are called equal if and only
if they contain exactly the same points, or, equivalently, if and
only if
EcF
and
FCE.
This seemingly innocuous definition has as a consequence the
important principle that the only way to prove that two sets are
equal is to show, in two steps, that every point of either set belongs also to the other.
It makes for tremendous simplification in language and notation to admit into the class of sets a set containing no points, which
we shall call the empty set and denote by 0. For every set E
we have
OcEcX;
for every point x we have
x e' 0.
In addition to sets of points we shall have frequent occasion to
consider also sets of sets. If, for instance, X is the real line, then
an interval is a set, i.e. a subset of X, but the set of all intervals
is a set of sets. To help keep the notions clear, we shall always
use the word class for a set of sets. The same notations and
terminology will be used for classes as for sets. Thus, for instance,
if E is a set and E is a class of sets, then
EeE
means that the set E belongs to (is a member
the class E; if E and F are classes, then
of, is an element of)
EcF
means that every set of E belongs also to F, i.e. that E is a sub,
class of F.
On very rare occasions we shall also have to consider sets of
classes, for which we shall always use the word collection. If,
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sEc.
11
SETS AND CLASSES
2]
for instance, X is the Euclidean plane and E, is the class of all
intervals on the horizontal line at distance y from the origin, then
each E, is a class and the set of all these classes is a collection.
(1) The
relation C between sets is always reflexive and transitive;
it is sym-
metric if and only if X is empty.
(2) Let X be the class of all subsets of X, including of course the empty set 0
and the whole space X; let x be a point of X, let E be a subset of X (i.e.a member
of X), and let E be a class of subsets of X (i.e.a subclass of X). If u and v vary
independently over the five symbols x, E, X, E, X, then some of the fifty relations of the forms
uev
or ucv
are necessarily true, some are possibly true, some are necessarily false, and some
In particular uev is meaningless unless the right term is a
are meaningless.
subset of a space of which the left term is a point, and ucv
is meaningless
unless u and v are both subsets of the same space.
§2.
UNIONS
AND
INTERSECTIONS
If E is any class of subsets of X, the set of all those points of
X which belong to at least one set of the class E is called the
union of the sets of E; it will be denoted by
UE
or
U {E:
Ee E}.
The last written symbol is an application of an important and
frequently used principle of notation. If we are given any set of
objects denoted by the generic symbol x, and if, for each x, x(x)
is a proposition concerning x, then the symbol
{x:
x(x)
}
denotes the set of those points x for which the proposition x(x)
is true. If {x,(x)} is a sequence of propositions concerning x,
the symbol
{
x: ri
(x),x2(x),
···
}
denotes the set of those points x for which x,(x) is true for every
If, more generally, to every element y of a certain
1, 2,
n
index set T there corresponds a proposition ry(x) concerning x,
then we shall denote the set of all those points x for which the
proposition ry(x) is true for every y in r by
=
·
·
·.
{x:x,(x),
ye T}.
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12
[SEC.
SETS AND CLASSES
2
Thus, for instance,
x eE}
=
E
Ee E}
=
E.
{x:
and
{E:
For more illuminating examples we consider the sets
(t:0&tgl
(=
the closed unit interval),
y2
{(x,y):x2 +
(=
the circumference
=
1}
of the unit circle in the plane), and
···}
1,2,
{n2:n
=
the set of those positive integers which are squares).
In accordance with this notation, the upper and lower bounds (supremum and infimum) of a set E of real numbers are denoted by
(=
sup
E}
{x:xe
inf
and
{x:xe
E}
respectively.
notation will be reserved for the
In general the brace
formation of sets. Thus, for instance, if x and y are points, then
denotes the set whose only elements are x and y. It is
important logically to distinguish between the point x and the
set x} whese only element is x, and similarly to distinguish
between the set E and the class E} whose only element is E.
The empty set 0, for example, contains no points, but the class
contains exactly one set, namely the empty set.
For the union of special classes of sets various special notations
If, for instance,
are used.
{ }
·
·
·
{x,y}
{
{
{0}
E
then
=
{Ei, E2 }
UE=U{E,:i=1,2}
is denoted by
Ei U E2,
if, more generally,
E
=
{Ei,
·
·
·
,
E,
}
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[sEc.
is a finite class
13
SETS AND CLASSES
2]
of sets, then
U{Es:i=1,···,n}
UE=
is denoted by
Ei U
·
U En
·
·
U:-1Ei.
or
If, similarly, {E,} is an infinite sequence of sets, then the union
of the terms of this sequence is denoted by
Ei U E2 U
···
UT.1Es.
or
More generally, if to every element y of a certain index set r
there corresponds a set E,, then the union of the class of sets
er}
{E,:y
is denoted by
U,
If the index set
E,
,,
U, E,.
or
we shall make the convention
r is empty,
U, E,
that
0.
=
The relations of the empty set 0 and the whole space X to
the formation of unions are given by the identities
and
EUO=E
EUX=X.
More generally it is true that
EcF
if and only if
EUF=F.
If E is any class of subsets of X, the set of all those points of
X which belong to every set of the class E is called the intersection
of the sets of E; it will be denoted by
E
or
{E: E eE}.
Symbols similar to those used for unions are used, but with the
symbol U replaced by n, for the intersections of two sets, of a
finite or countably infinite sequence of sets, or of the terms of
any indexed class of sets. If the index set T is empty, we shall
make the somewhat
startling
convention
erE,=X.
that
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14
[SEc.
SETS AND CLASSES
are several heuristic motivations
of them is that if Ti and
r2 are two
which fi c r2, then clearly
There
,
e r2
E,
D
,
2)
for this convention. One
(nonempty) index sets for
e
r,
E,,
and that therefore to the smallest possible r, i.e. the empty one,
we should make correspond the largest possible intersection.
Another motivation is the identity
e
r2 o r2 E,
=
,
e
r,
E,
n
,
e
r2
E,,
valid for all non empty index sets ri and T2. If we insist that this
identity remain valid for arbitrary ri and r2, then we are committed to believing that, for every T,
Û,erE,=
writing E,
=
verE,
yerooE,=
X for every
y
D
,,oE,;
in T, we conclude that
2
0
E,
X.
=
Union and intersection are sometimes called join and meet,
respectively.
As a mnemonic device for distinguishing between
U and n (which,by the way, are usually read as cup and cap,
respectively), it may be remarked that the symbol U is similar
and the symbol n is
to the initial letter of the word
similar to the initial letter of the word
The relations of 0 and X to the formation of intersections are
given by the identities
"union"
"meet."
and
EDO=0
EDX=E.
More generally it is true that
EcF
if and only if
EDF=E.
Two sets E and F are called disjoint if they have no points in
common, i.e. if
EnF=0.
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[sEc.
15
SETS AND CLASSEs
2]
A disjoint class is a class E of sets such that every two distinct
sets of E are disjoint; in this case we shall refer to the union of the
sets of E as a disjoint union.
We conclude this section with the introduction of the useful
concept of characteristic function. If E is any subset of X, the
function XE, defined for all x in X by the relations
1 if xe E,
0 1f x e' E,
is called the characteristic function of the set E. The correspondence between sets and their characteristic functions is one to one,
and all properties of sets and set operations may be expressed
by means of characteristic functions. As one more relevant illustration of the brace notation, we mention
E
(1) The formation of
=
{
X)
XE
x:
=
.
and associative,
unions is commutative
EUF=FUE
Ỵ
i.e.
EU(FUG)=(EUF)UG;
and
the same is true for the formation of intersections.
(2) Each of the two operations, the formation of unions and the formation
of intersections, is distributive with respect to the other, i.e.
EO (FU G)
=
(EO F) U (EO G)
and
EU (FO G)
=
(EU F) O (EU G).
More generally the following extended distributive laws are valid:
and
(3) Does
FDU
(E:EeE}
FU
{E:EeE}
=
U (EO
F:EeE}
{EU F:EeE}.
=
the class of all subsets of X form a group with respect to either of
the operations U and O ?
(4) Xo(x) 0, Xx(x) 1. The relation
-=
-=
is valid for all x in X if and
only
if EC
F.
If EOF
=
A and EUF
=
B,
then
xx
=
XEXF
=
XE
Xy
and
xa
=
XE
and
the identities in (4), expressing
xx
have generalizations to Enite, countably inßnite,
(5) Do
intersections?
xx U x,.
in terms of xx and
xa
unions
and arbitrary
Ÿ Xr
-
xx
=
XP,
and
