Methods of modern mathematical physics volume 1 functional analysis michael reed, barry simon

Later volumes describe various advanced topics in functional analysis and give numerous applications in classical physics, m odem physics, and partial differen- tial equations.. A lthoug

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Library of Congress Cataloging in Publication Data

Reed, Michael.

Methods of modern mathematical physics.

Vol 1 Functional analysis, revised and enlarged edition

Includes bibliographical references.

CONTENTS: v 1 Functional analysis.-v 2 Fourier

analysis, self-adjointness.-v 3 Scattering theory.-v 4

Analysis of operators.

1 Mathematical physics I Simón, Barry.joint

author II Title.

P reface

This book is the first o f a multivolume series devoted to an exposition o f func- tional analysis methods in modem mathematical physics It describes the funda­mental principies o f functional analysis and is essentially self-contained, al- though there are occasional references to later volumes We have included a few applications when we thought that they would provide motivation for the reader Later volumes describe various advanced topics in functional analysis and give numerous applications in classical physics, m odem physics, and partial differen- tial equations

This revised and enlarged edition differs from the first in two m ajor ways First, many coileagues have suggested to us that it would be helpful to include some material on the Fourier transform in Volume I so that this important topic can be conveniently included in a standard functional analysis course using this book Thus, we have included in this edition Sections IX 1, IX 2, and part of

IX 3 from Volume II and some additional material, together with relevant notes and problem s Secondly, we have included a variety o f supplementary material

at the end o f the book Some o f these supplementary sections provide proofs of theorems in Chapters I I - I V which were omitted in the first edition W hile these proofs m ake Chapters I I - I V more self-contained, we still recommend that stu- dents with no previous experience with this material consult more elementary texts O ther supplementary sections provide expository material to aid the in­structor and the student (for example, “ Applications o f Compact O perators” ) Still other sections introduce and develop new material (for exam ple, “ Minimi- zation o f Functionals ” )

It gives us pleasure to thank many individuáis:

The students who took our course in 1970-1971 and especially J E Taylor for constructive comments about the lectures and lecture notes

L Gross, T Kato, and especially D Ruelle for reading parts o f the manu- script and for making numerous suggestions and corrections

F Armstrong, E Epstein, B Farrell, and H Wertz for excellent typing

M Goldberger, E Nelson, M Simón, E Stein, and A W ightman for aid and encouragement

April 1980

M athem atics has its roots in num eroiogy, geom etry, and physics Since the tim e o f N ew ton, the search for m athem atical m odels for physical phenom ena has been a source o f m athem atical problem s In fact, whole branches o f

m athem atics have grown o u t o f attem pts to analyze particu lar physical situations A n exam ple is the developm ení o f harm onic analysis from F o u rier’s

w ork on the heat equation

A lthough m athem atics and physics have grown ap a rt in this century, physics has continued to stim ulate m athem atical research Partially because

o f this, the influence o f physics on m athem atics is well understood However, the co ntributions o f m athem atics to physics are not as well understood Jt is

a com m on fallacy to suppose th a t m athem atics is im postant for physics only because it is a useful tool for m aking com putations Actually, m athem atics plays a m ore subtle role which in the long run is m ore im portant W hen a successful m athem atical m odel is created for a physical phenom enon, th at is,

a model which can be used for accurate com putations and predictions, the

m athem atical structure o f the model itself provides a new way o f thinking about the phenom enon Put slightly differently, when a model is successful

it is natural to think o f the physical quantities in term s o f the m athem atical objects which represent them and to interpret sim ilar o r secondary phenom ena

in term s o f th e sam e m odel Because o f this, an investigation o f the internal

m athem atical structure o f the model can alter and enlarge o u r understanding

o f the physical phenom enon O f course, th e outstanding exam ple o f this is

N ew tonian m echanics which provided such a clear and coherent picture o f celestial m otions th a t it was used to interpret practically all physical phenom ena T he model itself becam e central to an understanding o f the physical world and it was difficult to give it up in the late nineteenth century, even in the face o f contradictory evidence A m ore m odern exam ple o f this influence o f m athem atics on physics is the use o f group theory to classify elem entary particles

vii

T he analysis o f m athem atical m odels for physical phenom ena is p a rt o f the subject m atter o f m athem atical physics By analysis is m eant both the rigorous derivation o f explicit form ulas and investigations o f the internal

m athem atical structure o f the models In both cases the m athem atical prob- lems which arise iead to m ore general m athem atical questions n o t associated with any particular model A lthough these general questions are som etim es problem s in puré m athem atics, they are usually classified as m athem atical physics since they arise from problem s in physics

M athem atical physics has traditionally been concerned w ith th e m athe­

m atics o f classical physics: m echanics, fluid dynam ics, acoustics, potential theory, and optics The m ain m athem atical too! fo r the study o f these branches o f physics is the theory of ordinary and p artiai differential equations and related areas like integral equations and the calculus o f variations This classical m athem atical physics has long been part o f curricula in m athem atics and physics departm ents However, since 1926 the frontiers o f physics have been concentrated increasingly in quantum m echanics and the subjects opened

up by the q uantum theory: atom ic physics, nuclear physics, solid State physics, elem entary particle physics The central m athem atical discipline for the study o f these branches o f physics is functional analysis, though the theories o f group representations and several com plex variables are also

im portant Von N eum ann began the analysis o f the fram ew ork o f q uantum

m echanics in the years foliowing 1926, b u t there were few attem p ts to study the structure o f specific quantum systems (exceptions would be som e o f the work o f Friedrichs and Rellich) This situation changed in the early 1950’s when K ato proved the self-adjointness o f atom ic H am iltonians and G árd in g and W ightm an form ulated the axiom s for quantum field theory These events dem onstrated the usefulness o f functional analysis and pointed out the m any difficult m athem atical questions arising in m odern physics Since then the range and breadth o f b oth the functional analysis techniques used and the subjects discussed in m odern m athem atical physics have increased enorm ously The problem s range from the concrete, for exam ple how to com pute or estím ate the point spectrum o f a p articular o perator, to the general, for example the rep resen taro n theory o f C “"-algebras The techniques used and the general approach to the subject have becom e m ore abstract A lthough

in some areas the physics is so well understood th at the problem s are exercises

in puré m athem atics, there are other areas where neither the physics ñ o r the

m athem atical m odels are well understood These developm ents have had severa! serious effects not the least o f which is the difficulty o f com m unication between m athem aticians and physicists Physicists are often dismayed at the breadth o f background and increasing m athem atical sophistication which are required to understand the models M athem aticians are often frastrated by

their own inability to undersíand the physics and the inability o f physicists

to form úlate the problem s in a way th a t m athem aticians can understand

A few specific rem arks are ap p ro p riate The prerequisite for reading this volume is roughly the m athem atical sophistication acquired in a typical

u ndergraduate m athem atics education in the U nited States C h ap ter I is intended as a review o f b ackground m aterial W e expect th a t the reader will have som e acquaintance w ith pa rts o f the m aterial covered in C hapters II—IV and have occasionally om itted proofs in these chapters w hen they seem uninspiring and u n im p o rtan t for the reader

T he m aterial in this book is sufficient for a two-sem ester course A lthough

we taught m ost o f the m aterial in a special one-sem ester course a t Princeton which m et five days a week, we d o not recom m end a repetition o f that, either for facuity or students In order th a t the m aterial may be easily adapted for lectures, we have w ritten m ost o f the chapters so th at the earlier sections contain the basic topics while the iater sections contain m ore specialized and advanced topics and applications F o r example, one can give students the basic ideas ab o u t unbounded operators in nine or ten lectures from Sections 1-4 o f C hapter VIII O n the o th er hand, by doing th e details o f the proofs and by adding m aterial from the notes and problem s, C h ap ter VIII could easily becom e a one-sem ester course by itself

Each chapter o f this book ends with a long set o f problem s Som e o f the problem s fill gaps in the text (these are m arked with a dagger) O thers develop altérnate proofs to the theorem s in the text or introduce new m aterial We have also included h arder problem s (indicated by a star) in order to challenge the reader W e strongly encourage students to do the problem s It is trite but true th a t m athem atics is learned by doing it, not by w atching other people do it

We hope th a t these volumes will provide physicists w ith an access to

m odern abstract techniques and th a t m athem aticians will benefit by learning the advanced techniques side by side with their applications

II: HILBERT SPACES

xi

III: BANACH SPACES

5 The Baire category theorem and its consequences 79

3 Functions o f rapid decease and the tempered distribuíions 133

Appendix The N-representation fo r S f and 9 ” 141

4 ¡n du ctive lim its: g e n e r a liz e d fu n c tio n s a n d w eak Solutions o f

7 Topologies on locally convex spaces: duality theory and the

Appendix Polars and the M a ckey-A ren s theorem 167

VI: BOUNDED OPERATORS

Vil: THE SPECTRAL THEOREM

VIII: UNBOUNDED OPERATORS

2 Symmetric and self-adjoint operators: the basic criterion

9 The polar decomposition fo r closed operators 297

THE FOURIER TRANSFORM

1 The Fourier transform on 5 f(U n) and 6^'(U n), convolutions 318

2 The range o f the Fourier transform: Classical spaces 326

3 The range o f the Fourier transform: Analyticity 332

SUPPLEMENTARY MATERIAL

IV A The R iesz-M a rko v theorem fo r X = [0 , 1 ] 353

V.5 Proofs o f some theorems in nonlinear functional analysis 363

Contents of Other Volumes

I X T h e F o u rie r T ra n sfo rm

X S e lf-A d jo in tn e s s a n d th e E x iste n c e o f D yn a m ics

Volum e III: Scattering Theory

X I S c a tte rin g T h eo ry

Volum e IV: A n a ly sis of O perators

X I I P e rtu rb a tio n o f P o in t S p e c tra

X I I I S p e c tra l A n a ly sis

I: Preliminaries

The beginner should not be discouraged if he finds that he does not have the prerequisites

1.1 Sets and functions

We assum e th a t the reader is fam iliar w ith sets and functions but it is

a p p ro p ria te to standardize ou r term inology and to introduce here abbrevia- tions th a t will occur th ro u g h o u t the book

I f X is a set, x e X m eans th a t x is an elem ent o f X ; x $ X m eans th at x is

n o t in X T he clause “ for all x in X " is abbreviated (V x e X ) and “ there

exists an x g X such t h a t ” is abbreviated (B x e J Í) T he sym bol {jc|P(x)} stands fo r the set o f x obeying the condition (o r conditions) P(x) I f A is a subset o f X (denoted A cz X ), the sym bol X \A represents the com plem ent of

A in X , th a t is X \A = { x e X \ x $ A ) M ore generally, if A and B are subsets

o f X , then A \B = { x \ x e A , x $ B} W hen we discuss sets w ith a topology, Á will always denote the closure o f the set A Finally, the set o f ordered pairs K*> y ) Ix 6 X , y e 7} is called the C artesian product o f X an d Y and is denoted X x Y.

We will use the w ords “ fu n c tio n ” and “ m a p p in g ” interchangeably In ord er to em phasize th a t certain functions / depend on two variables, we will

som etim es w rite /( • ,• ) • T he sym bol f ( - , y ) denotes the function o f one variable obtained by picking a fixed valué o f y for the second variable A

1

linear function will also be called an operator o r a linear transform ation O ur

functions will always be single valued; so a function from a set X to another

valué in Y for each x e X If A <= X , then f[ A ) = {/ ( x ) | x e A} is a subset of

7 and f ~ x[B] = ( x |/ ( x ) e B} is a subset o f X if B cz Y f [ X ] will usually be called the range o f f and will be denoted R an f X is called the domain o f /

A function f will be called injective (or one-one) if fo r each y e R an f there

is at m ost x e X such th a t f ( x ) ~ y \ f is called surjective (or onto) if

R an / = y If / is both injective and surjective, we will say it is bijective T he

restriction o f / to a subset A o f its dom ain will be denoted by f \ A

If X 3 A we define the characteristic function %A(x) as

There are tw o set theoretic notions which are slightly deeper th an m ere notation, so we will discuss them to some extent A relation i ü o n a set X is a subset R o f X x X ; if <x, y > e R, we say th at x is related (or R -related) to

y and write x R y

D e f in itio n A relation R is called an equivalence relation if it satisfies: (i) (Vx e X ) x R x [reflexive]

(ii) (Vx, y e X ) x R y implies y R x [symmetric]

(iii) (Vx, y, z e X ) x R y and y R z implies x R z [transitive]

The set o f elements in X th a t are related to a given x e X is called the

equivalence class o f x, denoted usually as [x]

It is easy to pro ve:

T h eo rem 1.1 Let R be an equivalence relation on a set X T hen each

x e X belongs to a unique equivalence class.

Thus, under an equivalence relation, a set divides up in a n atu ral way into disjoint subsets

Exa m p le 1 (the integers m od 3) Let X be the integers and w rite x R y

if x — y is a m últiple o f 3 T his equivalence relation divides the integers into

three equivalence classes:

set y, denoted b y / : X -» Y or x J + Y or x i-» /(x ) will have o n e a n d only one

x 6 A

x $ A

[0] — - 6 , - 3 , 0 , 3, 6 , }

[1] = { , - 5 , - 2 , 1 , 4 , 7 , }[2] = - 4 , - 1 , 2 , 5, 8 , }

E x a m p ie 2 (the real projective line) Let R denote the real line and Iet

X be the nonzero vectors in IR 2 ( = IR x IR) We write x R y if there is some

a e IR with x = ay The equivalence classes are lines through the origin (with

<0, 0 ) rem oved)

Next, we discuss Z o rn ’s lemma

D e f in itio n A relation on a set X which is reflexive, transitive, and anti- sym m etric (th a t is, x R y and y R x im plies x = y ) is called a partial ordering

If i? is a p artial ordering, we often w rite x < y instead o f xR y.

E x a m p le 3 Let X be the collection o f all subsets o f a set Y Define

A -< B if A c: B T hen -< is a partial ordering.

We use the w ord “ partial ” in the above definition because two elements

o f X need n o t obey x «< y o r y -< x I f fo r all x and y in X , either x -< or

y -< Jt, X is said to be linearly ordered F o r example, R with its usual order <

is linearly ordered

N ow suppose X is partially ordered by -< and Y c X An elem ent p e X

is called upper bound for 7 i f y < p fo r all y e Y I f m e X and m < x implies

x = m , we say m is a m axim al element o f X

D epending on one’s starting po in t, Z o rn ’s lem m a is either a basic assump- tion o f set theory or else derived from the basic assum ptions (it is equivalent

to the axiom o f choice) W e take Z o rn ’s lem m a and the rest o f set theory as

given

T h e o re m 1.2 (Z orn’s lemma) Let X be a nonem pty partially ordered set

w ith the p ro p erty th at every linearly ordered subset has an upper bound in X

T hen each linearly ordered set has som e upper bound th at is also a maximal

elem ent o f X

Finally, we will use H alm os’ | to indicate the conclusión o f a proof

1.2 Metric and normed linear spaces

T h ro u g h o u t this w ork, we will be dealing with sets o f functions o r operators

or other objects and we will often need a way o f m easuring the distance

between the objects in the sets It is reasonable to define a notion o f distance

th a t has the m ost im p o rtan t properties o f ordinary distance in IR3

D e f in itio n A m etric space is a set M and a real-valued function d (*, •)

on M x M which satisfies:

(i) d (x , y) > 0

(ii) d(x, y ) = 0 if and only if x = y

(iii) d(x, y ) = d(y, x)

(iv) d(x, z) < d (x , y ) + d (y, z) [triangle inequality]

The function d is cafied a m etric on M.

We often cali the elements o f a m etric space points N otice th at a m etric

space is a set M together with a m etric function d\ in general, a given set X

can be m ade into a m etric space in different ways by em ploying different

m etric functions W hen it is n o t clear from the context which m etric we are

talking about, we will denote the m etric space by <M, d ) , so th a t the m etric

is explicitly displayed

E x a m p le 1 Let M — IR" w ith the distance between two points x =

<xl s , x„> and y = <ylt given by

d(x, y ) = J ( x t - y x)2 + • • • + (xH - y„)2

E x a m p le 2 Let M be the unit circle in IR2, th a t is, the set o f all pairs o f

real num bers <a, /?) w ith a 2 + /?2 = 1, and let

E x a m p le 3 Let M = C[0, 1], the continuous real-valued functions on

[0, 1 ] with either o f the m etrics

d i i f , g) = m ax | / O ) - g(x) | d 2( f, g) = P | f ( x ) - g (x) j d x

*€[0,1]

N ow th a t we have a notion o f distance, we can say w hat we m ean by convergence

D efinition A sequence o f elem ents {x:„}“=1 o f a m etric space <M, d ) is

said to converge to an element x e M , if í/(jc, jc„) -» 0 as « -> oo We will often denote this by x„— í_» x or lim ,, ^ x„ = x If x n does not converge to x , we will w rite x n—/ —>x.

In Exam ple 2, d x(p, p') < d 2(p, p') < n d x(p, p') which we will write

d x < d 2 < n d \ T hus p n dt > p if and only if í 2 But in Exam ple 3, the

m etrics induce distinct notions o f convergence Since d2 < d x »/im p lie s/ „ —íl-» /, b u t the converse is false A counterexam ple is given by the functions

g n defined in Figure 1.2, which converge to the zero function in th e m etric d 2

F ig u r e 1.2 The graph o f g„(x).

but which do not converge in the m etric d x This m ay be seen by introducing

the im p o rtan t notion o f Cauchy sequence

D e f in itio n A sequence o f elem ents {*„} o f a m etric space <M, d ) is called a Cauchy sequence if (Ve > 0)(3iV) n, m > N implies d(xn , Jtm) < e.

P ro p o siiio n Any convergent sequence is Cauchy

P ro o f G iven x„^> x and e, find N so n > N implies d (x„ , x) < e/2 Then

n, m > N implies d(x„, x:m) < d(x„ , x ) + d(x, x m) < je + }e |

We now return to the functions in Figure 1.2 It is easy to see th a t if n ^ m, d\{g„, g m) = 1 • T h u s g n is not a C auchy sequence in <C[0, 1], d f ) an d therefore not a convergent sequence Thus, the sequence {g „} converges in <C[0, 1], d2)

D e fin itio n A m etric space in which all C auchy sequences converge is called complete

F o r example, IR is com plete, but O is not I t can be show n (Sections 1.3 and

1.5) th at <C[0, 1], d f ) is com plete b u t <C[0, 1], d 2} is not T he exam ple o f Q and R suggests w hat we need to do to an incom plete space X to m ake it complete W e need to enlarge X by adding “ all possible limits o f C auchy sequences.” T he original space X should be dense in the larger space X

D efiniüon A function / from a m etric space <X , d > to a m etric space

< Y, p> is called continuous a t x if /(* „ ) <r,p>-» f { x ) whenever x„ > x.

We have already had an exam ple o f a sequence o f elem ents in C[0, 1] with

f n J 2— > 0 but f„ dl/ ~» 0 T hus the identity function from <C[0, 1 ] , ^ ) to

<C[0, 1], í/j) is not continuous b u t the identity from <C[0, 1], to

<C[0, 1], d2y is continuous.

D efínition A bijection h from <X , d > to < Y, p> which preserves the

m etric, th at is,

p(h(x), h(y)) = d(x, y )

is called an isometry It is autom atically continuous <X, d ) and <7, p ) are

said to be isom etric if such an isom etry exists

Isom etric spaces are essentially identical as m etric spaces; a theorem con-

cerning only the m etric stru ctu re o f ( X , d ) will hold in all spaces isom etric

family o f equivaíence classes o f C auchy sequences under this equivalence

relation O ne can show th at for any tw o C auchy sequences lim ,,.^ d (xn, y n)

exists and depends only on the equivalence classes o f {x„} and {y„} This lim it

defines a m etric on A? and A? is com plete Finally, m ap M into M by taking x into the constant sequence in w hich each x„ equals x M is dense in M and

the m ap is isom etric |

T o com plete ou r discussion o f m etric spaces, we want to introduce the notions o f open and closed sets T he reader should keep the exam ple o f open and closed sets on the real line in m ind

D efínition Let <X, d > be a m etric space:

(a) T he set { x |x e X , d(x, y ) < r} is called the open ball, B ( y ; r), o f radius

r ab o u t the p o in t y.

(b) A set O cz X is called open if (Vy e 0 )(3 r > 0) B ( y ; r) cz O.

(c) A set N c: X is called a neighborhood o f y e N if B ( y ; r) c N for some

r > 0.

(d) Let E c A l A point x is called a limit point o f E, if (V r > 0)

B (x; r) n (£\{x}) 7^ 0 , th a t is, x is a lim it p o in t o f E if E contains points other than x arbitrarily near x.

(e) A set F <= X is called closed if F contains all its lim it points.

( f ) If G cz X , x e G is called an interior point o f G, if G is a neighborhood

o f x

T he reader can prove for him self the following collection o f elem entary statem ents:

T h eo rem 1.4 Let <X, d ) be a m etric space:

(a) A set, O, is open if and on!y if X \ 0 is closed.

(b) x m ——■ > x if and only if fo r each neighborhood N o f x, there exists an

M so th at m > M implies x m e N.

(c) The set o f interior points o f a set is open

(d) The unión o f a set E w ith its lim it p o in ts is a closed set (den o ted by E and called the closure of E).

(e) A set is open if and only if it is a neighborhood o f each o f its points

One o f the m ain uses o f open sets is to check for convergence usingTheorem I.4.b and in particu lar to check for continuity via the following criteria, the p ro o f o f which we lea ve as an exercise:

Th eo rem 1.5 A function /(•) from a m etric space X to an o th er space Y

is continuous if and only if for all open sets O a Y , / -1 [O] is open.

Finaliy, we w arn the reader th at often in incom plete m etric spaces, closed sets may not appear to be closed at first glance F o r exam ple, [^, 1) is closed in (0, l) (with the usual m etric)

We com plete this section with a discussion o f tw o o f the central concepts of functional analysis: norm ed linear spaces and bounded linear transform ations

D e f m itio n A normed linear space is a vector space, V, over IR (or C) and a function, ||-|j from V to IR which satisfies:

(i) í|u¡| > 0 for all v in V

(ii) \\v\\ — 0 if and only if v — 0

(iii) liai'H = ¡a | ||v || for al! v in V and a in IR (or C)

(iv) ||u + w |¡ < ¡|y|| + IM¡ for all u and w in V

D e f m itio n A bounded-linear transform ation (o r bounded op erato r) from

a norm ed linear space <K,, || ||t > to a norm ed linear space <F2 , || ¡|2> is a

function, T, from V¡ to V2 which satisfies:

(i) T(otv + pw) = a T(v) + fiT(w ) (Vu, w e F)(Va, fi e IR o r C)

(ii) F o r some C > 0, ||T y ||2 < C||t>|li

T he sm allest such C is called the norm o f T, w ritten ||T || or ||T ||1>2 Thus

are norm ed linear spaces O bserve also th a t any norm ed linear space <F, ¡| • |¡>

is a m etric space when given the distance function d(v, w) — ||i? — w\\ There

is thus a notion o f continuity o f functions, and for linear functions this is precisely captured by bounded linear transform ations T he p ro o f o f this fact

is left to the reader

T h e o re m 1.6 Let T be a linear tra n sfo rm a ro n between tw o norm ed

linear spaces T he follow ing are equivalent:

(a) T is continuous a t one point.

(b) T is continuous a t all points.

X can be m ade into a norm ed linear space in exactly one n atural way

All these concepts are well illustrated by the following im p o rta n t theorem and its p r o o f :

T h e o re m 1.7 (the B.L.T theorem ) Suppose T is a bounded linear tran s­

fo rm a ro n from a norm ed linear space <Vj, || • jli > t o a com plete norm ed linear

space <F2 , || * ||2>- T hen T can be uniquely extended to a bounded linear tran sfo rm atio n (with the sam e b o und), T, from the com pletion o f Vl to

<y2 , ih i2>.

P roof Let Vx be the com pletion o f V¡ F o r each x in Vx, there is a sequence

o f elements {x„} in Vx with x n -* x as n -*• oo Since x n converges, it is Cauchy,

so given s, we can find N so that n, m > N implies |jx„ — x m||, < £/||Tj¡ Then

iirx„ - T x m||2 = !!T(X„ - x m)I!2 < \\T\\ \\XH - x j l j < e which proves th at Tx„ is a Cauchy sequence in V2 Since V2 is com plete, T x n - » y for som e y Set

T x = y We m ust first show th at this definition is independent o f th e sequence

x„ —► x chosen If x„ -* x and x ' -*■ x, then the sequence x x, x i, x 2 , x 2 , -*• x

so T x x, T x \ , - » y for some y by the abo ve argum ent T hus lim Tx'„ = y = lim T x „ M oreover, we can show T so defined is bounded because

¡ | T x | | 2 = lim !!Tx„|j2 (see Problem 8)

n-* oo

< ílm C |¡ xn f| i (see A ppendix to 1.2)n-» ao

= Cllxll,

Thus f is bounded T he proofs o f linearity and uniqueness are left to the

reader |

We can use this theorem to give a very elegant definition o f the R iem ann

integral Let PC[a, b] be the family o f bounded piecewise continuous func­ tions on [> a, b], which are continuous from the right, th at is, lim *^ / ( x ) = f ( y ) and for which limxt>, / ( x ) exists at each y and is equal to f ( y ) for all but finitely m any y N orm PC with the norm

!l/l!oo = sup |/ ( x ) |

x e [a, b]

Let x 0 , , x„ be a p artitio n o f the interval [a, b], x 0 = a, x„ = b Let x,-(x)

be the characteristic function o f [x j-i.x ,-) except for xÁ x ) which is the characteristic function o f [X n -i.x J A function on [a, b] o f the form

X?= i with s¡ real is called a step function (to see why, draw its graph).

The set o f all step functions for all possible finite p artitions is a norm ed linear space with the norm

< m a x ¡¿,¡ £ l*< ~ * ¡ - i I

í= i

^ \ \ l s iXi\ U b ~ a) / is a bounded linear tran sfo rm atio n Since the real num bers are com plete, / can be uniquely extended to S , the com pletion o f S (by the B.L.T theorem )

T he extended transform ation / ( / ) , restricted to PC is called the Riem ann

integral and is denoted by

While this m ethod does not appear as the m ost intuitive definition of the

R iem ann integral, it will be seen upon reflection that the p ro o f is really just the “ u s u a l” p ro o f p u t into the language o f com pletion and the B.L.T theorem It ilfustrates a m ain point o f general philosophy in functional analysis: In order to define som ething on a norm ed linear space, it is often convenient to define it on a dense set and extend it by the B.L.T theorem The reader should try his hand at constructing the R iem ann-S tieltjes integral (Problem 11) By using the sam e m ethod, we can define the R iem ann integral

for continuous functions taking valúes in any complete norm ed linear space,

in particular, for com plex-valued functions

A p p en d ix to 1.2 Lim sup and lim inf

Lim sup and lim in f are notions which m ay be unfam iliar to the reader, so

we sum m arize their definition and properties

D e f in itio n Let A cz (R be a nonfinite bounded set Let lim pt(/4) = set of limit points o f A Then the lim it superior o f A is defined by

a

lim sup A ~ ¡Tm A — sup{xr | x e lim pt(/4)}

Similaríy

lim inf A = !im(/4) = inf {x j x 6 lim pt(/4)}

R em arks 1 W hen A is bounded, lim p t(A ) is always nonem pty by the

B olzano-W eierstrass theorem

2 If A is not bounded above, one defines fim A = +00 I f A is bounded above and lim p t(^ ) = 0 one defines fim A = — ce.

3 fim A is actually in lim p t(A) F o r let b = fim A and let e > 0 be given

We can find a g lim pt(/4) so \b — a\ < e¡2 Since a g lim pt(/4), we can find

d e A with |a — d\ < e¡2; so given e, we find d e A w ith \b — d \ < e, th at is,

F or a sequence {a„}, we say b e lim pt{a„} if for all N and all e, there is an

n > N with j b — a„ | < £ We define fim(£7„) = sup{¿?| b e lim pt{úrn}}.

Finally, let us sum m arize the properties o ffim (all for bounded sets; it is a useful exercise to decide which extend to unbounded sets)

Proposition

(a) íím(<7„ + b„) < fim an + fim bn

(b) fim an bn < (fim <3n)(fim b„) if an, bn > 0

(c) fim(ctf„) = c fim an if c > 0

(d) lim(c¿7„) — c fim an if c < 0

i.3 T h e Lebesgue integral

We ha ve ju st seen th a t C[a, b] has tw o quite reasonable m etrics on it In

Section 1.5 we will see that it is a com plete m etric space in the m etric

dx ( f , g ) = sup | / ( x ) - 0 ( x ) j

x e [ a , 6]

In the other m etric we considered, d 2( f, g) = \\f — g\\x with H/Mfi—

¡g |/i(x)| dx, C[a, b] is not complete T o see this for C[0, 1], let f„ be given as

in Figure 1.3 It is not hard to see t h a t /„ is C auchy in j|-|¡,, b u t it does not

converge to any function in C[a, 6]; rather, in an intuitive sense, it “ converges ”

to the characteristic function o f [f, £] (which is, o f course, not in C[0, 1]!)

We can always com plete C[a, b} in |¡ • ¡li realizing elements o f the com pletion

as equivalence classes o f C auchy sequences o f continuous functions; this realization is not notew orthy for its transparency T he exam ple above suggests we m ight also be able to realize elements o f the com pletion as functions If we do realize them as functions, we should be able to define the integral |/ ( x ) | d x (merely as d 2( f, 0)!) for any f in the com pletion.

T he simplest way to realize elem ents o f the com pletion as functions is to

tu rn the above analysis a ro u n d : one introduces an extended notion o f integral

on a bigger space than C[a, b] \ cali it l)[a , ¿] We will prove l) is com plete, so

by general argum ents the closure o f C in L1 is com plete (and it turns out

C = L1)

N ow , how can one extend the notion o f Riem ann integral? The usual

definition o f the Riem ann integral is based on dividing the domain of / into

finer and finer pieces F o r “ nasty ” functions, this m ethod does not w ork and

T h e Le b e s g u e integral

so a different m ethod is needed— the simplest m odification is to divide the range into finer and finer pieces (Figure 1.4) T his m ethod depends m ore on the function and so has the possibility o f w orking for m ore types o f functions

We are thus interested in sets f ' x[a, b] and their size We suppose we have

a size function p on sets which generaiizes p([a, b]) = b — a We will shortiy

retu rn to this size function and see th a t not all sets have a “ size.” W e will then

restrict the types o f / by dem anding th at f ~ x[a, h] have a “ size.” Looking at

Figure 1.4, we define f o r / > 0

T hen £ 2n ( / ) > £ „ ( / ) so th at l i m ^ * £ 2* i f ) = S^P „ ( £ 2« ( / ) ) exists (it may

be co) This limit is defined to be ¡ f dx W e re m a rk th a t for technical purposes

(that is, proving theorem s!) one m akes a different defínition which can be shown to agree w ith this defínition only after a lot o f w ork The defínition

as lim £ 2" i f ) is how ever the best to keep in m ind when thinking intuitively.

Thus, we have transferred the problem to one o f defining an extended notion o f size We m ust first decide w hat sets are to have a size W hy n o t ail sets ? There is a classica! exam ple (see also Problem 13) which shows th at not all sets in R 3 can have a size if we want th at size to be invariant under rotations and translations (and n o t to be trivial, such as assigning zero to all sets):

it is possible to break u p a unit ball into a finite num ber o f wild pieces, move the pieces around by ro tatio n and translation and reassem ble the pieces to get two balls o f radius one (B anach-T arski paradox) Thus, all sets can n o t

have a size, and so som e family 88 o f sets will be the “ m easurable sets.” W hat properties do we want 88 to have? We w ould íike both / - 1 [[0, a)] and

f ~ x[[a, co)] to be m easurable ( / > 0) so we w ould like 88 to have the p roperty:

A e 88 implies R\/4 e 88 Also, when / is continuous, we w ant f ~ x[{a, ¿>)] to

be in 88, so 88 should contain the open sets Finally, we w ant to have

if the A„ are m utually disjoint (to meet our intuitive notion o f size) so we would like (J®=1 A„ e 88 if each A„ is in 88.

Defínition The Borel sets o f IR is the smallest family o f subsets o f R with the following properties:

(i) The fam ily is closed under com plem ents

(i¡) The family is closed u nder countable unions

(iii) The fam ily contains each open interval

T o see th a t such a smallest family exists we note th at if { ^ a}«e A is a collec- tion o f families obeying (i), (ii), and (iii), then so does f ) aeA Thus the intersection o f al! families obeying (i)—(iii) is the smallest such family.

N ow we define the Lebesgue m easures o f sets in á?, the Borel sets in R

D efinition Let J be the family o f all countable unions o f disjoint open

intervaís (which is ju s t the fam ily o f open sets) and let

T his notion o f size has four crucial properties:

T h e o re m 1.8

(a) t i { 0 ) = 0

(b) l f {/4n}®=1 c= á? and the A„ are m utually disjoint (A n n A m — 0 , all

m * n), then ^(U®= i ¿n) = & i

(c) = inf{¿¿(/) | B a /, / is open}

(d) p(B) = su p b u (C )|C c B, C is compact}

The infinite sum in (b) contains only positive term s, so it either converges

to a finite num ber o r diverges to infinity, in which case we set it equal to oo.(c) and (d) say th a t any Borel set can be approxim ated “ from the outside ”

by open sets and from the inside by com pact sets We rem ind the reader that

on the real Une a set is com pact if and only if it is closed and bounded.

We have thus extended the usual notion o f size o f intervaís and we define the family o f functions we will consider in the obvious way:

Definition A fu n c tio n / is called a Borel function if and only if f ~ 1 [(n, b)]

is a Borel set for all a, b.

It is often convenient to allow o u r functions to take the valúes ± o o on small sets in which case we r e q u i r e / - í [{±oo}] to be Borel

P ro p o sitio n / is a Borel if and only if, fo r all B e Ú8,

(see Problem 14)

(which may be infinite) F o r any B e define

ii{B) = in f /i( /)

This last p roposition implies th at the com position o f two Borel functions

is Borel M any books deal with a slightly larger class o f functions th an the

Borel class They first define a set M to be m easurable if one can write

M u A x — B u A 2 where B is Borel and A¡ cz with B¡ Borel and p(B¡) = 0

(thus they add and subtract “ u n im p o rtan t ” sets from Borel sets) A m easur­

able function is then defined as a function, f for which f ~ l [{a, ¿)] is always

m easurable It is no longer true th a t f ° g is m easurable if / and g are, and

m any technical problem s arise In any event, we deal only with Borel sets and functions and use the words Borel and measurable interchangeably.

Borel functions are closed under m any operations:

Proposition (a) If f g are Borel, then so are / + g, fg , m ax { f g] and

m in { / g) If / i s Borel and X e i?, / / i s Borel.

(b) I f each f n is Borel, « = 1 , 2 , , and f„ ( p ) - * f( p ) fo r all p, then /

Since | / 1 = m a \ { f —/} , | / | is m easurable i f / is.

As we sketched above, given/ > 0, one can define \ f d x (which m ay be oo )

If j ¡ / ¡ d x < o o , we w rite f e i f 1 and define j f d x = J / + d x — J / _ dx where /+ — m a x { / 0}; / _ = m ax{— f 0) I £ x{a, tí) is the set o f functions on (a, tí)

which are in i f 1 if we extend them to the whoíe real line by defining them to

be zero outside o f (a , tí) If f e ¿?l (a, tí), we w rite \ f d x = / dx W e then

have :

T h eo rem 1.9 Let / and g be m easurable functions T hen

(a) I f f g e i f !(a, b), so a r e / + g and X f for all X e R.

(b) I f |^ | < / a n d / e i f 1, then ^ e i f 1

(c) 1 ( / + 9) dx — ¡ f d* + í 9 d x i f / and g are in i f /

(d) ¡ j / ¿ x | < J | / ¡ dx i f / i s in i f 1

(e) If / < g, then \ f d x < \ g dx, if / and g are in i f 1.

(f) I f / i s bounded and m easurable on —oo < a < b < o o , then / e i f 1 and

This theorem shows th at J has all the nice properties o f the R iem ann integra! even though it is defined for a larger class o f functions

The properties that m ake the space L1 (which we will shortly define) com plete are the following absolutely essential convergence theorem s:

is Borel

T h e o re m 1.10 (m onotone convergence theorem ) Let /„ > 0 be m easur-

able Suppose f„(p) -* f ( p ) for each p and th at / B+Í(p) > f„ (p ) all p and n (in which case we write f n/ f ) - If J f„(p) dp < C for all n, then f e and

J l f ( p ) ~ f n ( p ) \d p - + 0 as n - * oo.

T h e o re m 1.11 (dom inated convergence theorem ) Let f„(p) - > /( p ) for each p an d suppose |/„ ( p ) | < G{p) fo r all n and som e G e Ja?1 T h e n / e J2?1

and j | f ( p ) - f „ ( p ) i dp -* 0 as n -► oo.

In the latter case, we say G dom inates the pointw ise convergence T h at a

dom inating function exists is crucial F o r example, le t/„(x ) = (1 /«)fy-„,„](*)•

T hen f„(x) -*■ 0 fo r each x, b u t J | /„ j d x = 2 so j j f„(x) j d x does not go to zero In this case, it is not h ard to see th a t sup„ |/„ (x )| = G(x) is not in i ? !.

We are alm ost ready to define as a m etric space by letting p ( f , g) =

í \ f — 9\ dx W e can n o t quite do this because j \ f — g\ d x = 0 does not

im ply / s g (for example, / and g m ight differ at a single point) T hus, we

first define th e notion of alm ost everyw here (a.e.):

D efinitio n We say a condition C(x) holds alm ost everyw here (a.e.) if

{xj C(x) is false} is a subset o f a set o f m easure zero

D efinitio n We say two functions f g e i ? 1 are equivalent i f f ( x ) = g(x) a.e (this is the sam e as saying f \ f — g\ d x = 0).

D efinition T he set o f equivalence classes in i ? 1 is denoted by as Ü

1} w ith th e norm ¡|/j|j = J | / | d x is a norm ed linear space.

T hus an elem ent o f L1 is an equivalence class o f functions equal a.e In

partic u la r w hen f e Ü , the symbol f ( x ) for a p a rticu lar x does not m ake sense

Nevertheless we continué to w rite “ / ( x ) ” b u t only in situations w here state-

m ents are independent o f a choice from the equivalence class T hus, for

exam ple, f n(x) - * f ( x ) f o r almost all x is independent o f the representatives

chosen for / and f „ By this replacem ent o f pointw ise convergence w ith p o in t­

wise convergence alm ost everywhere, the tw o convergence theorem s carry over from i ? 1 to L1

H aving cautioned the reader th a t f ( x ) is “ technically m eaningless” for

f e L1, we rem ark th at in certain special cases it is m eaningful Suppose f e l }

has a representative / (that is, / is a function; / an equivalence ciass o f func­tions) which is continuous T hen no other representative o f / is continuous,

so it is natural to w rite f { x ) for J{x).

T he critical fact a b o u t L1 is :

T h eo rem 1.12 (R iesz-Fisher) L1 is com plete

P ro o f Let f„ be C auchy in L 1 It is enough to prove som e subsequence

converges (see Problem 3) so pass to a subsequence (also labeled /„) w ith

l l / „ - / „ + ilit < 2 - Let

m

ffm(x) = £ | f n(x) ~ f a+l(x) |

n = 1

Let g w be the infinite sum (which m ay be oo) T hen g m/ g w and

í 19m i ^ X »=i ll/ñ —fn +1II — I , so by the m o notone convergence theorem ,

g ^ e Ü T hus < oo a.e As a result

/« (* ) = / l W - X ifn(x) - f n + l(x))

n= 1

converges pointw ise a.e to a function f ( x ) M oreover, ¡ / m( x )S <

l/i(x ) | + #oq(*) e L1 so /„-* ■ / in L 1 by the dom inated convergence theorem |

T his p ro o f has a corollary (see Problem 17):

C o ro liary If /„ - » / i n L1, then some subsequence f ttl converges pointwise

a.e to /

As a final result which brings us full circle to ou r original m o tiv a tio n :

P ro p o sitio n C[ay b] is dense (in ||-Hj) in l ) [ a ,b ], i.e L1 is the com ­

p arts are in I) [a, b] W hen no confusión arises, we will denote this space, with

the norm

l l / l l i - f ‘ l / 1 *

also by i ) [a, b] T he integral o f a com plex-valued function is defined by

¡ f d x = J R e ( /) d x + i jIm ( / ) d x

1.4 Abstract measure theory

One o f the m ost im p o rtan t tools which one com bines with ab stract func- tional analysis in the study o f various concrete m odels is “ general ” m easure theory, th at is, the theory o f the last section extended to a m ore abstract setting

The sim plest way to generalize the Lebesgue integral is to w ork with functions on the real line and w ith Borel sets but to generalize the underlying

m easure; we consider this special case o f ab stract m easure theory first Recall

th at the Lebesgue integral was constructed as foilows W e started w ith a

notion o f size for intervals, p([a, b]) = b — a, and extended this in a unique

way to a notion o f size for a rb itra ry Borel sets A rm ed with this notion o f size fo r Borel sets, the integral o f Borel functions was obtained by m easuring sets o f the form 6]) We found the vector space ^ ( [ 0 , 1], dx) con­structed in the last section is ju st the com pletion o f C[0, 1] w ith the m etric

d2(f, g) = j¿ \ f { x ) — g(x) | dx, where we needed only the R iem ann integral

to define d 2 ■

N ow suppose an arb itrary m on o to n e function a(x) is given (th at is, x > y

implies a(x) >: a(y)) It is not h a rd to see th a t the lim it from the right, lime_ 0 a(x + je j) and the limit from the left,lim e_ 0 a(x — |e |) exist; we w rite

them as a(x + 0) and a(x — 0) respectively Since (a, b) does n o t include the points a and b, it is n atu ral to define pa((a, b)) = a (b - 0) — a (a + 0) From

this n otion o f size for intervals, one can construct a m easure on Borel sets

o f R, th a t is, a m ap pa: & -*• [0, oo] w i t h = Yj°= i if B¡ o Bj — 0 and t¿a( 0 ) = 0 By construction, this m easure has the regularity p roperty

pa(B) = sup{p(C) \C cz B, C compact}

= in f{ /¿(0 )|B c- O , O open}

Also, p(C) < ce for any com pact set C A m easure w ith these two regularity

properties is called a Borel measure In p articu lar, ¿1) = a (¿ + 0) -

a (a — 0) One can then construct an in te g ra l/ - * J / dpa (we will also write

J / da) which has properties (a)-(e) o f T heorem 1.9; it is called a Lebesgue- Stieltjes integral Ü([a, b], da) and Ü (U , da) can be form ed as before These spaces o f equivalence classes o f functions are com plete in the m etric p ( /, g) —

j \ f ~ 9\ da, anc* analogues o f the m onotone and dom inated convergence theorem s hold The continuous functions C[a, b] form a dense subspace o f b], da)\ p u t differently, Ü([a, b], da) is the com pletion o f C[a, b] with

the metric pa(/, g) — | f — d\ da where we need only use the R iem an n Stieltjes integral to define pa (see Problem 11).

-Let us consider three examples which iilustrate the variety o f Lebesgue- Stieltjes measures

Exam p le 1 Suppose a is continuously differentiable T hen p a(a, b) = (dajdx) dx w here d x is Lebesgue m easure, so it is to be expected (and is

indeed true!) th at

Thus, these m easures can essentially be described in term s o f Lebesgue

m easure

E xam p le 2 Suppose th at a(x) is the characteristic function o f [0, oo)

T hen p a(a, b) = 1 if 0 e (a, tí) and is 0 if 0 ^ {a, tí) The m easure one gets out is very easy to describe: p j f i ) = 1 if 0 e B , and p j j f ) = 0 if 0 £ B T he

reader is invited to construct explicitly the integral and convince him self th at

This m easure da is know n as the D irac m easure (since it is ju st like a

5 function) Let us consider Ü (U , da) in this case In i f 1 we have p (/, g) =

1 /(0) — g(0) | so p ( /, g) — 0 if and only if /( 0 ) = p(0) As a result, we see that

the equivalence classes in L1 are completely described by the v a lu é /(0 ) so th at

L}(U, da) is just a one-dim ensional vector space! N otice how different this is from the case o f Ll(U, dx) w here the valué o f a “ fu n c tio n ” at a single point

is not defined (since elements o f L1 are equivalence classes)

Exa m p le 3 O ur last example m akes use o f a fairíy pathological function,

a(x), which we first construct Let S be the subset o f [0, 1]

j y < f e = / ( o ) www.TheSolutionManual.com

H< > < - )—( H -1 -) -*■■) -( - )—w —I

F i g u r e 1.5 The Cantor set.

th a t is, rem ove the m iddle third o f w hat is not in S at each stage and add it

to S, see Figure 1.5 The Lebesgue m easure o f S is + 2(£) + 4(^7-) + ' ' • = 1 Let C = [0, 1]\5 It has Lebesgue m easure 0 C, which is know n as the C an to r

set, is easy to describe if we w rite each x e [0, 1] in its base three decimal expansión T hen x e C if and only if this base 3 expansión has no l ’s Thus C

is an u ncountable set o f m easure 0 T o see this, m ap C in a one-one way onío

[0, 1] by changing 2’s into l ’s and viewing the end result as a base 2 num ber

Now co n stru ct a(x) as follows: set a ( x ) — -j on (y, y ); a(X) = y on (y, J-);

a(x) = | on (•§-, £), etc.; see Figure 1.6 Extend a to [0, 1] by m akíng it

con-F i g u r e 1.6 The Cantor function.

tinuous T hen a is a n onconstant continuous function with the strange property th at a'(x ) exists a.e (with respect to Lebesgue m easure) and is zero a.e N ow , we can form the m easure Since a is continuous, fia({p}) ~ 0

for any set {p} with only one point Nevertheless, p a is co ncentrated on the set

C in the sense th a t pa([0, 1]\C) = p a(S) — 0 O n the other hand, the Lebesgue

m easure o f C is zero T hus and Lebesgue m easure “ liv e ” on com pletely difierent sets

In a sense we now m ake precise, these three examples are m odels o f the most general Lebesgue-Stieltjes measures Suppose is a Borel m easure on

First, let P = {x | g({x}) # 0}, th a t is, P is the set o f puré points o f g Since g is

Borel [/¿(C) < oo for any com pact set], P is a countable set Define

T hen g pp is a m easure and g cont = g - g pp is positive g cont has the p ro p erty

Mcont((pj) — 0 f ° r P> th a t is, it has no puré points and g pp has only puré points in the sense th a t g pp(X ) = £ „ * PPP(M )-

Definition A Borel m easure g on ¡R is called continuous if it has np p u ré points g is called a puré point measure if g (X ) = ^ X E Í/i(x) for any Borel

set X

Thus, we have se e n :

T h eo rem 1.13 A ny Borel m easure can be decom posed uniquelyinto a

sum pe = ¡xpp + g cont w here ¿ícont is continuous and g pp is a p u ré p o in t m easure.

We have thus generalized Exam ple 2 by allowing sums o f D irac m easures

Is there any generalization o f Exam ples 1 and 3 ?

D efinition We say th a t ¡i is absolutely continuous with respect to (w.r.t.)

Lebesgue measure if there is a fu n c tio n ,/, locally L1 (th at is, j* j / ( x ) | d x < oo

for any finite interval {a, b)) so th a t

for any Borel function g in L ^R , dg) We then w rite d g = f dx.

T his definition generalizes Exam ple 1; we will eventually m ake a different (but equivalent!) definition o f absolute continuity

D efinition We say g is singular relative to Lebesgue measure if and only

if g(S) = 0 for som e set S where R\S has Lebesque m easure 0.

T he fundam ental result is:

T h eo rem 1.14 (Lebesgue decom position theorem ) Let g be a Borel measure T hen g = ¿tac + ¡using in a unique way w ith g ac absolutely continuous w.r.t Lebesgue m easure a n d w ith g iint singular relative to Lebesgue m easure.

IÍ „ ( * ) = X K M ) = /<(J» n X )

x e P n X

T hus T heorem s 1.13 and 1.14 tell us th a t any m easure g on IR has a canonical decom position g — g pp + g ac + g sing w here g pp is puré p oint, g ac absolutely continuous w ith respect to Lebesgue m easure, and g iing is continuous and singular relative to Lebesgue m easure T his decom position will recur in a

quantum -m echanical context where any State will be a sum o f bound States, scattering States, and States w ith no physical in terp retatio n (one o f our hardest jo b s will be to show th a t this last type o f State does n o t occur; th a t

is, th a t certain m easures have g sing = 0 ; (see C h ap ter X III).)

This com pletes o u r study o f m easures on IR T he next level o f generalization involves m easures on sets w ith som e underlying topological s tru c tu re ; we will retu rn to study this case o f interm edíate generality in Section IV.4 T he m ost general setting lets us deal w ith an a rb itra ry set We first need an abstraction

The definition o f m easure is o b v io u s(!):

Definition A measure on a set M w ith <7-ring 0% is a m ap g: 0t -* [0, oo]

with the p ro p e rtie s:

(a) p { 0 ) - 0

(b) ^ ( U ^ ) = if A t n A j = 0 for all i # /

W e shall often speak o f the m easure space <M, g } w ithout explicitly men- tioning 0%, b u t the tr-ring is a crucial elem ent o f the definition O ccasionally, we will w rite <M, 0%, g ) F o r certain pathologically “ b ig ” spaces, one wants to

use the n o tio n o f c-ring rath e r th an cr-field, b u t to keep things simple, we will consider m easures on cr-fields and will suppose the whole space isn’t too big in the sense:

Definition A m easure g on a cr-field & is called <r-finite if and only if

M ~ ! A¡ with each g(A¡) < oo.

We will suppose all o u r underlying m easures are cr-finite

Defm ition Let M , N be sets with a-fields £% and 3F A m ap T: M -* N

is calied measurable (w.r.t 01 and 3F) if and only if 'iA e 0F, T ~ l [A] e 0t A

m ap / : M -* IR is called m easurable if it is m easurable w.r.t an d the Borel sets o f IR

Given a m easure g on a m easure space Ai, we can define \ f dg for any positive real-valued m easurable function on M and we can form d¡i), the set o f integrable functions and Ü (M , dg), the equivalence classes o f func­ tions in if71 equal a.e.[/j] As in the case <M, d g } — <1R, d x ) y the following

crucial theorem s hold:

0 < f ( x ) < f 2(x) < • • • and f ( x ) — lim„_ „,/„(*), then f e S£ 1 if and only if

l i m ^ x |!/n|!¡ < oo and in th a t case l i m ^ ^ W f - f J U = 0 and l i m , ^ II^IU =

il/lli-T h eo rem 1.16 (dom inated convergence theorem ) If / „ e L?(M, dg), lim „_oc/ n(x) = f ( x ) a.e.[¿í], and if there is a G e 1) with |/„(x )¡ < G(x) a.e.|ju],

for all n, then / e L1 and lim „_00 \\f — /„üj = 0

T h eo rem 1.17 (F a to u ’s lemma) I f f„ E S £ x, each f n(x) > 0 and if

i í m l l / J , < oo, t h e n /( x ) = !im /n W is in i f 1 and l l / ^ < H m ll/J ,

N ote In F a to u ’s lem m a nothing is said ab o u t lim,,.,,*, ||/ —/ „ |lt

T h eo rem 1.18 (R iesz-Fisher theorem ) l} ( M ,d g ) is com plete.

One also has the idea o f m utually singular:

Defm ition Let g, v be two m easures on a space M w ith a-field 01 We say that g and v are m utually singular if there is a set A e 0t with fx(A) = 0, v(M \A ) = 0.

It is useful to take a vJeaker looking defm ition o f absolute continuity which

is essentially the opposite o f singular:

Defm ition We say v is absolutely continuous w.r.t g if and only if g(A) = 0 implies v(/4) = 0.

T h at this definition is the same as the previous one is a consequence of: