Measure Theory and Functional Analysis

205 Pages·2013·1.156 MB·English

by  Nik Weaver

MEASURE THEORY AND
FUNCTIONAL ANALYSIS
8813hc_9789814508568_tp.indd 1 8/4/13 9:55 AM
May29,2013 22:38 WorldScientificBook-9inx6in fm
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
MEASURE THEORY AND
FUNCTIONAL ANALYSIS
Nik Weaver
Washington University in St. Louis, USA
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
8813hc_9789814508568_tp.indd 2 8/4/13 9:55 AM
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
MEASURE THEORY AND FUNCTIONAL ANALYSIS
Copyright © 2013 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and
retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
ISBN 978-981-4508-56-8
Printed in Singapore
May29,2013 22:38 WorldScientificBook-9inx6in fm
Preface
ThisbookisbasedonasetofnotesIdevelopedoverseveralyearsofteaching
agraduatecourseonmeasuretheoryandfunctionalanalysis. Itsfocalpoint
is the stunning interplay between topology, measure, and Hilbert space
exhibitedinthespectraltheoremanditsgeneralizations. Theprerequisites
are minimal: readers need to be familiar with little beyond metric spaces
and abstract real and complex vector spaces.
Ihavestrivento eliminate unnecessarygenerality. Thus,wheneverpos-
sibleIassumetopologicalspacesaremetrizable,measurespacesareσ-finite,
Banach spaces are either separable or have separable preduals, and so on,
if there is any advantage in doing so. My rationale is that the objects of
central importance in the subject all seem to be, in various senses, essen-
tially countable, whereas the essentially uncountable setting houses a raft
of pathology of no obvious interest. There are other benefits, as well: the
machineryofgeneralizedconvergence(i.e.,netsandfilters)becomeslargely
superfluous, and appeals to the axiomof choice can generallybe weakened
to countable choice or even dropped altogether. I wonder how many ana-
lystsrealizethattheHahn-Banachtheorem,famousforitsnonconstructive
nature,requiresnochoiceprincipleatallinthesettingofseparableBanach
spaces.
Expert readers will notice numerous minor innovations throughout the
book. PerhapsthemostfruitfuloriginalideaismyincorporationofHilbert
bundles into the spectraltheorem,a device Iintroducedin my bookMath-
ematical Quantization (CRC Press,2001). When I was a graduate student
afriendadvisedme thatthe multiplicationoperatorversionofthe spectral
theoremistheformyouunderstand,butthespectralmeasureversionisthe
form you use. This is a pithy way of pointing out that although multipli-
cation operators are more intuitive than spectral measures, they appear in
v
May29,2013 22:38 WorldScientificBook-9inx6in fm
vi Measure Theory and Functional Analysis
spectral theory in a noncanonical and therefore somewhat inelegant man-
ner. TheHilbertbundleapproachneatlyresolvesthisdilemma. Usingonly
the elementary notions of Hilbert space direct sums and tensor products,
one is able to formulate a more canonical multiplication operator version
of the spectral theorem which, moreover, transparently exhibits both the
underlyingspectralmeasureanditsmultiplicity. Evenmorebenefitsaccrue
whenwegeneralizespectraltheorytofamiliesofcommutingoperators: the
standard structure theorems for concrete abelian C*- and von Neumann
algebras are augmented with spatial information which not only tells us
that such algebrasare abstractly isomorphic to C (X) and L (X) spaces,
0 ∞
but also cleanly exhibits the way these abstract spaces are situated within
(H).
B
I wish to express my gratitude to all of my students who took this
course over the past several years. Those were some very talented classes,
and teaching them was a real pleasure.
This work was partially supported by NSF grant DMS-1067726.
Nik Weaver
WhenI’mworkingonaproblem,Ineverthinkaboutbeauty,I
thinkonlyhowtosolvetheproblem. ButwhenIhavefinished,
if the solution is not beautiful, I knowit is wrong.
— Buckminster Fuller
May29,2013 22:38 WorldScientificBook-9inx6in fm
Contents
Preface v
1. TopologicalSpaces 1
1.1 Countability. . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . 9
1.4 Metrizability and separability . . . . . . . . . . . . . . . . 13
1.5 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6 Separation principles . . . . . . . . . . . . . . . . . . . . . 21
1.7 Local compactness . . . . . . . . . . . . . . . . . . . . . . 24
1.8 Sequential convergence . . . . . . . . . . . . . . . . . . . . 27
1.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2. Measure and Integration 35
2.1 Measurable spaces and functions . . . . . . . . . . . . . . 35
2.2 Positive measures . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 Premeasures. . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4 Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . 47
2.5 Lebesgue integration . . . . . . . . . . . . . . . . . . . . . 52
2.6 Product measures. . . . . . . . . . . . . . . . . . . . . . . 59
2.7 Scalar-valued measures . . . . . . . . . . . . . . . . . . . . 63
2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3. Banach Spaces 75
3.1 Normed vector spaces . . . . . . . . . . . . . . . . . . . . 75
3.2 Basic constructions . . . . . . . . . . . . . . . . . . . . . . 83
vii
May29,2013 22:38 WorldScientificBook-9inx6in fm
viii Measure Theory and Functional Analysis
3.3 The Hahn-Banach theorem . . . . . . . . . . . . . . . . . 89
3.4 The Banach isomorphism theorem . . . . . . . . . . . . . 95
3.5 C(X) and C (X) spaces . . . . . . . . . . . . . . . . . . . 100
0
3.6 Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.7 Ideals and homomorphisms . . . . . . . . . . . . . . . . . 109
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4. Dual Banach Spaces 119
4.1 Weak* topologies . . . . . . . . . . . . . . . . . . . . . . . 119
4.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.3 Separation theorems . . . . . . . . . . . . . . . . . . . . . 127
4.4 The Krein-Milman theorem . . . . . . . . . . . . . . . . . 131
4.5 The Riesz-Markovtheorem . . . . . . . . . . . . . . . . . 134
4.6 L1 and L spaces . . . . . . . . . . . . . . . . . . . . . . 141
∞
4.7 Lp spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5. Spectral Theory 157
5.1 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.2 Hilbert bundles . . . . . . . . . . . . . . . . . . . . . . . . 164
5.3 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.4 The continuous functional calculus . . . . . . . . . . . . . 176
5.5 The spectral theorem. . . . . . . . . . . . . . . . . . . . . 182
5.6 Abelian operator algebras . . . . . . . . . . . . . . . . . . 187
5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Notation Index 197
Subject Index 199
May29,2013 22:38 WorldScientificBook-9inx6in fm
Chapter 1
Topological Spaces
1.1 Countability
We adopt the convention that 0 is not a natural number; thus N =
1,2,3,... .
{ }
Definition 1.1. A set is countably infinite if there is a bijection between
it and N. It is countable if it is either finite or countably infinite. It is
uncountable if it is not countable.
Countability conditions of various types will be assumed liberally
throughoutthis book. Assuming that asetis countable canbe veryconve-
nient because this means that its elements can be indexed as (a ), with n
n
rangingeitherfrom1toN forsomeN orfrom1to ,ineithercasegiving
∞
us the ability to deal with them sequentially. Actually, the hypotheses we
impose usually will not assert that the main set of interest is itself count-
able,butratherthatinsomewayitsstructureisdeterminedbyacountable
amountofinformation. This informalcommentmightmakemoresenseaf-
ter we discuss separability and second countability in Section 1.4.
ClearlyNiscountablyinfinite,sinceitistriviallyinbijectionwithitself.
The set of even natural numbers is also countably infinite via the bijection
n 2n, and as the set of odd natural numbers is obviously in bijection
↔
with the set of even natural numbers, it is countably infinite too.
This shows that a countably infinite set (the natural numbers) can be
split up into two countably infinite subsets (the even numbers and the
odd numbers). Conversely, with a moment’s thought it also shows that
the union of two disjoint countably infinite sets will again be countably
infinite: we can put one set in bijection with the even numbers and the
otherinbijectionwiththeoddnumbers,andthencombinethetwomapsto
1
May29,2013 22:38 WorldScientificBook-9inx6in fm
2 Measure Theory and Functional Analysis
establishabijectionbetweentheunionofthetwosetsandN. Forinstance,
wecanuse this ideatoshowthatthe setofintegersZiscountablyinfinite.
Define f :Z N by
→
2n if n>0
f(n)=
( 2n+1 if n 0;
− ≤
this is a bijection that matches the positive integers with the even natural
numbersandthe negativeintegersandzerowith the oddnaturalnumbers.
Next we observe that subsets and images of countable sets are always
countable.
Proposition 1.2. Let A be a countable set.
(a) Any subset of A is countable.
(b) Any surjective image of A is countable.
Proof. (a) We take it as known that any subset of a finite set is finite,
so assume A is countably infinite. Let f : N A be a bijection and let
→
B be any subset of A. If B is finite we are done, so assume B is infinite.
Then f 1(B) mustbe an infinite subsetof N, so it has a smallestelement,
−
a second smallest element, etc. Let n be the smallest element of f 1(B),
1 −
n the next smallest, and so on; then the map k f(n ) is a bijection
2 k
7→
between N and B. So B is countably infinite.
(b) Suppose f : A B is a surjection. Create a map g : B A by,
→ →
for each b B, letting g(b) be an arbitrary element of f 1(b). Then g is a
−
∈
bijection between B and a subset of A, and it follows frompart (a) that B
must be countable. (cid:3)
This proposition illustrates why it is helpful to have a special term
(“countable”)for setswhich areeither finite orcountably infinite. Itis not
true that any subset of a countably infinite set is countably infinite, nor is
it true that any surjective image is countably infinite.
Having said that, in analysis the unqualified word “sequence” usually
means“infinitesequence”,i.e.,asequenceindexedbyN,andwewillfollow
this convention. If we want to consider finite sequences we will explicitly
use the qualifier “finite”.
Earlier we used the fact that the natural numbers can be partitioned
intoevennumbersandoddnumbersinordertoshowthattheunionoftwo
disjoint countably infinite sets is alwayscountably infinite. This resultcan
be strengthened to say that a union of countably many sets, each of which
is countable, will always be countable. We can show this by partitioning