Foundations of quantum mechanics

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FOUNDATIONS OF
QUANTUM MECHANICS
JOSEF M. JAUCH
University of Geneva, Switzerland

A
'V

ADDISON-WESLEY PUBLISHING COMPANY
Reading, Ma.csac/:useu.c

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1968 BY ADDISON-WESLEY PUI3LISIIING COMPANY, INC.

CATAIXE CARl) No. 67-2397K.
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PREFACE

This book is an advanced text on elementary quantum mechanics.
By "elementary" I designate here the subject matter of nonrelativistic

quantum mechanics for the simplest physical systems. With the word
"advanced" I refer to the use of modern mathematical tools and the careful
study of difficult questions concerning the physical interpretation of quantum
mechanics.

These questions of interpretation have been a source of difficulties from
the beginning of the theory in the late twenties to the present day. They
have been the subject of numerous controversies and they continue to worry
contemporary thoughtful students of the subject.

In spite of these difficulties, quantum mechanics is indispensable for

most modern research in physics. For this reason every physicist worth his
salt must know how to use at least the language of quantum mechanics. For
many forms of communication, knowledge of the approved usage of the
language may be quite sufficient. A deeper understanding of the meaning is
then not absolutely indispensable.

The pragmatic tendency of modern research has often obscured the
difference between knowing the usage of a language and understanding the
tneaning of its concepts. There are many students everywhere who pass
their examinations in quantum mechanics with top grades without really
understanding what it all means. Often it is even worse than that. Instead
of learning quantum mechanics in parrot-like fashion, they may learn in
this fashion only particular approximation techniques (such as perturbation
theory, Feynman diagrams or dispersion relations), which then lead them
to believe that these useful techniques are identical with the conceptual basis
of the theory. This tendency appears in scores of textbooks and is encouraged
by some prominent physicists.
This text, on the contrary, is not concerned with applications or approxlinations, but with the conceptual foundations of quantum mechanics. It
is restricted to the general aspects of the nonrelativistic theory. Other fundamental topics such as scattering theory, quantum statistics and relativistic
(Itlailtuni mechanics will be reserved for subsequent publications.
v
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vi

PREFACE

When I wrote this book I had three categories ut readers ninitl: the
student of physics who has already acquired a first knowledge ot qiutiltitni

mechanics, the experienced physicist who is in search ot a (Iceper understanding, and the mathematician who is interested in the iiiatliciiiaticul
problems of quantum mechanics.
The book consists of three parts. Part I, called Matheniatical Foundations, contains in four chapters a sundry collection of mathematical results,
not usually found in the arsenal of a physicist but indispensable for understanding the rest of the book. I have taken special care to explain, motivate,
and define the basic concepts and to state the impoitant theorems. The
theorems are rarely proved, however. Most of the concepts are from functional analysis and algebra. For a physicist this part may be useful as a
short introduction to certain mathematical results which are applicable in
many other domains of physics. The mathematician will find nothing new
here, and after a glance at the notation can proceed to Chapter 5.
In Part 2, called Physical Foundations, I present in an axiomatic form the

basic notions of general quantum mechanics, together with a detailed
analysis of the deep epistemological problems connected with them.

The central theme here is the lattice of propositions, an empirically
determined algebraic structure which characterizes the intrinsic physical
properties of a quantum system.
Part 3 is devoted to the quantum mechanics of elementary particles.
The important new notion which is introduced here is localizability, together
with homogeneity and isotropy of the physical space. In this part the reader

will finally find the link with the conventional presentation of quantum
mechanics. And it is here also that he encounters Planck's constant, which
fixes the scale of the quantum features.

Of previous publications those of von Neumann have most strongly
influenced the work presented here. There is also considerable overlap with
the book by G. Ludwig and with lecture notes by G. Mackey. In addition
to material available in these and other references, it also contains the results
of recent research on the foundations of quantum mechanics carried out in

Geneva over the past seven years.
The presentation uses a more modern mathematical language than is
customary in textbooks of quantum mechanics. There are essentially three
reasons for this:
First of all, I believe that mathematics itself can profit by maintaining
its relations with the development of physical ideas. In the past, mathematics
has always renewed itself in contact with nature, and without such contacts
it is doomed to become pure syniholisni of ever-increasing abstraction.

Second, physical ideas can he expressed much more forcefully and
clearly if they are presented in the appropriate language. The use ot such a
language will enable us to distinguish more easily the difficulties which we
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PREFACE

vii

might call syntactical from those of interpretation. Contrary to a widespread belief, mathematical rigor, appropriately applied, does not necessarily
introduce complications. In physics it means that we replace a traditional
and often antiquated language by a precise but necessarily abstract mathematical language, with the result that many physically important notions
formerly shrouded in a fog of words become crystal clear and of surprising
simplicity.

Third, in all properly formulated physical ideas there is an economy of
thought which is beautiful to contemplate. I have always been convinced
that this esthetic aspect of a well-expressed physical theory is just as indispensable as its agreement with experience. Only beauty can lead to that

"passionate sympathetic contemplation" of the marvels of the physical

world which the ancient Greeks expressed with the orphic word "theory."
About 350 problems appear throughout the book. Most of them are
short exercises designed to reinforce the notions introduced in the text.
Others are more or less obvious supplements of the text. There are also
some deeper problems indicated by an asterisk. The latter kind are all
supplied with a reference or a hint.

There remains the pleasant task of remembering here the many colleagues, collaborators, and students who in one way or another have helped
to shape the content and the form of this book.

My first attempts to rethink quantum mechanics were very much
stimulated by Prof. D. Finkelstein of Yeshiva University and Prof. D.
Speiser of the University of Louvain. It was during the year 1958, when all
three of us were spending a very stimulating year at CERN, that we began
examining the question of possible generalizations of quantum mechanics.
Many of the ideas conceived during this time were subsequently elaborated
in publications of my students at the University of Geneva. I should mention
here especially the work of G. Emch, M. Guenin, J. P. Marchand, B. Misra,
and C. Piron.
In the early stages I profited much from various discussions and c9rre-

sponcknce with Professor G. Mackey. Many colleagues have read and
criticized different portions of the manuscript. I mention here especially
l)r. R. Hagedorn of CERN, whose severe criticism of the pedagogical
aspects of the first four chapters has been most valuable. With Dr. J. Bell
ot CERN, I debated especially the sections on hidden variables and the
measuring process. The chapter on the measuring process has also been
iiitluenced by correspondence with Prof. E. Wigner of Princeton and with
Prot. L. Rosenfeld of Copenhagen. Several other sections were improved
hy criticism from Prof. R. Ascoli of Palermo, Prof. F. Rohrlich of Syracuse,

New York,
dell'Antonio of Naples, and Dr. (1. Baron of Rye, New
'iurk. Prof. C. F. v. Weizsückcr of IIaniburg read and commented on the
section of Chapter 5. Professor (, Mackey read critically
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viii

PREFACE

the entire manuscript and suggested many improvements. I)r. C. Piroti
and Mr. A. Salah read the proofs, and I am much indebted to them For their
conscientious and patient collaboration. To all of them, and ninny others
too numerous to mention, I wish to express here my thanks.
The major portion of the book was written while I had the privilege of
holding an invited professorship at the University of California in Los
Angeles, during the winter semester of 1964. May Professors D. Saxon and
R. Finkelstein find here my deeply felt gratitude for making this sojourn,
and thereby this book, possible. The University of Geneva contributed its
share by granting a leave of absence which liberated me from teaching duties
for three months.
I have also enjoyed the active support of CERN, which, by according
me the status of visiting scientist, has greatly facilitated my access to its
excellent research facilities and contact with numerous other physicists
interested in the matters treated in this book.
It is unavoidable that my interpretation of controversial questions is
not shared by all of my correspondents. Of course, I alone am responsible
for the answers to such questions which appear in this book.
Mrs. Dorothy Pederson of Los Angeles and Mlle. Frances Prost of

Geneva gave generously of their competent services in typing a difficult
manuscript. May they, too, as well as their collaborators, find here my
expression of gratitude.

J. M. J.

Geneva, Switzerland
August 1966

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CONTENTS

PART 1
Chapter

1

1—1

1—2
1—3

1—4
1—5

Mathematical Foundations

Measure and Integral

Some notions and notations from set theory
The measure space
Measurable and integrable functions
The theorem of Radon-Nikodym
Function spaces

3

7
9

14
16

Chapter 2 The Axioms of Hilbert Space
The axioms of Hilbert space
2—2 Comments on the axioms
Realizations of Hilbert space
2—3
2—4
Linear manifolds and subspaces
The lattice of subspaces
2—5

18
19
23

2—1

Chapter

3

3—1

3—2
3—3

3—4
3—5
3—6

24
26

Linear Functionals and Linear Operators

Bounded linear functionals
Sesquilinear functionals and quadratic forms
Bounded linear operators
Projections
Unbounded operators
Examples of operators

t

30
32

33
37

40
42

Chapter 4 Spectral Theorem and Spectral Representation
Self-adjoint operators in finite-dimensional spaces
4—2 The resolvent and the spectrum
The spectral theorem
4—3
4—4 The functional calculus
4—5
Spectral densities and generating vectors
4—6 The spectral representation
4 7 Elgenfunction expansions
4—i

ix
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58

60
62

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CONTENTS

PART 2 Physical Foundations
Chapter

5

5—1

5—2
5—3

5—4
5—5
5—6

5—7
5—8

The Propositional Calculus

Historic-philosophic prelude
Yes-no experiments
The propositional calculus
Classical systems and Boolean lattices
Compatible and incompatible propositions
Modularity

The lattice of subspaces
Proposition systems

68
72
74
78
80
83

84
86

Chapter 6 States and Observables
6—1

6—2
6—3

6—4
6—5
6—6

6—7
6—8

6—9

The notion of state
The measurement of the state

Description of states
The notion of observables
Properties of observables
Compatible observables
The functional calculus for observables
The superposition principle
Superselection rules

90
93

94
97
99
101
101

105
109

Chapter 7 Hidden Variables
7—1

7—2
7—3

7—4

Chapter

8

A thought experiment
Dispersion-free states
Hidden variables
Alternative ways of introducing hidden variables

112
114
116
119

Proposition Systems and Projective Geometries

8—1

Projective geometries

121

8—2

Reduction theory.
The structure of irreducible proposition systems
Orthocomplementation and the metric of the vector space
Quantum mechanics in Hilbert space

124
127
129

8—3

8—4
8—5

131

Chapter 9 Symmetries and Groups
9—1

9—3

9—4
9—5

9 6

The meaning of symmetry
Abstract groups
Topological groups
The automorphisms of a proposition system
Transformation of states
Projective representation of groups
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135
137
139
142

145
146


CONTENTS

Chapter 10
10—1

10—2
10—3

10—4

Chapter 11

Xi

The Dynamical Structure

The time evolution of a system
The dynamical group
Different descriptions of the time evolution
Nonconservative systems

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151
153
155

157

The Measuring Process

Uncertainty relations
160
11—2 General description of the measuring process
163
11—3 Description of the measuring process for quantum-mechanical
systems
164
11—4 Properties of the measuring device
168
11—5
Equivalent states
170
11—6 Events and data

173
11—7
Mathematical interlude: The tensor product
175
11—8 The union and separation of systems
179
11—9 A model of the measuring process
183
11—10 Three paradoxes
185
11—1

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PART 3
Chapter 12

13—2
13—3

13—4
13—5
13—6
13—-7

13 8

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Elementary Particles
The Elementary Particle in One Dimension

Localizability
12—2 Homogeneity
12—3
The canonical commutation rules
12—4 The elementary particle
12—5
Velocity and Galilei invariance
12—6
The harmonic oscillator
12—7 A Hilbert space of analytical functions
12—8
Localizability and modularity

1 3—1

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12—1

Chapter 13

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The Elementary Particle without Spin

Localizability
Homogeneity and isotropy
Rotations as kinematical symmetries
Velocity and Galilei invariance
Gauge transformations and gauge invariance
Density and current of an observable
Space inversion
Time reversal
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Xii

CONTENTS

Chapter 14 Particles with Spin

Spin, a nonclassical degree of freedom
14—2 The description of a particle with spin
14-3 Spin and rotations
14—4 Spin and orbital angular momentum
14—5 Spin under space reflection and time inversion
14—6 Spin in an external force field
14—7 Elementary particle with arbitrary spin

14—1

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Chapter 15 Identical Particles
15—1

15—2
15—3

15—4
15—5

15—6

Assembly of several particles
Mathematical digression: The multiple tensor product
The notion of identity in quantum mechanics
Systems of several identical particles
The Bose gas
The Fermi gas
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Author Index
Subject Index

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293


PART 1
Mathematical Foundations

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CHAPTER 1

MEASURE AND INTEGRAL
Therefore there is no perfect measure of continuous quantity except by means
of indivisible continuous quantity, for example by means of a point, and no
quantity can be perfectly measured unless it is known how many individual
points it contains. And since these are infinite, therefore their number cannot
be known by a creature but by God alone, who disposes everything in number,
weight, and measure.
ROBERT GROSSETESTE,

13th century A.D.

The purpose of this chapter is to acquaint the reader with the modern theory
of integration. Section 1-1 contains some basic notions of set theory together
with a list of terms and formulas. In Section 1-2 we present the notion of
measure space and some properties of measures. We define measures on

c-rings of a class of measurable sets, but we pay no attention to the maximal
extensions of such measures. The following section (1-3) introduces the

measurable and integrable functions and defines the notion of integral.
Section 1-4 introduces the theorem of Radon-Nikodym by way of a trivial
example. The last section (1-5) on function spaces forms the bridge to the
general theory of Hilbert space to be presented in Chapter 2.
1-1. SOME NOTIONS AND NOTATIONS FROM SET THEORY

A collection of objects taken as a whole is called a set. The objects which
make up a set are called the elements of the set. We denote sets by capital
letters, for instance A, B,. , S; and the elements by small letters, for instance
a, b, .. , x. If the element x is contained in the set S we write x e 5; if it
is not contained in the set S we write x S. If every element of a set A is
contained in B we write A a B or B A, and we say A is a subset of B.
If A c B and B a A, then we say the two sets are equal and we write
. .

.

A=B.

The set A u B denotes the set of elements which are either in A or in B
or in both. It will be called the union of A and B.

I
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4

I-i

MEASURE AND INTEGRAL

C
C

(a)

(b)

Hg. 1—1 Relations between point sets: (a) A a B (A subset of B);
(b) A n B = 0 (disjoint sets).

The set A n B denotes the set of elements which are in A as well as in B.
It is called the intersection of A and B.
If A is a subset of a set 5, we define by A' (with respect to 5) the set of
all elements which are in S but not in A. The set A n B' = A — B is called
the difference of A and B, or the relative complement of B in A.

A subset of a general set S can be defined by a certain property m(x).
The set A of all elements which have property m(x) is written

A = {x

:

It means: A is the set of all elements which satisfy property m(x). Thus for
instance the operation A u B can be defined by

AuBr={x:xeAand/orxeB}.
Similarly we write

A nB = {x : xeA and xe B}.
If there exists no element which has property m(x), then the

set

0 = {x : m(x)} defines the empty set. We have always, for any set A c 5:

ØcA,

OuA=A,

ØnA—O,

1/f_—S.

If two sets A and B are such that A n B = 0, they are called disjoint.
All these notions can be easily illustrated and remembered by using
point sets in a plane (see Figs. 1—1 and 1—2).

We shall now go a step further and consider collections of subsets of a
set. We speak then of a class of subsets. Of particular interest are classes
which are closed with respect to certain operations defined above.

(a)

(b)

(c)

Fig. 1—2 Intersection-, union-, and difference-sets: (a) A n B;
B
A n B' (for S the entire plane).
(b) u B; (c) A
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SOME NOTIONS AND NOTATIONS FROM SET THEORY

1-1

5

most useful notion is that of a ring. A nonempty class of subsets is
called a ring B? if, forAeB? and BeG?, it follows that A uBePA and
A — B e B?. Examples of rings are easily constructed. One of the simplest
possible is the ring consisting of an arbitrary subset A a S, together with
the sets 0, A' and S.
Since A — A = 0, every ring contains the empty set. One proves by
is a finite collection of submathematical induction that if' A1, A2, .
The

. . ,

sets in B?, then

and

1:1
Here we have introduced the easily understandable notation

(JA1

A

a

A1 u A2 u

u

ring with the additional property that, for every countable

sequence A1 (i =

1,

2,

. .

.) of sets contained in B?, we have
1=1

A ring is called an algebra (or Boolean algebra) if it contains 5, or equivalently if A e B? implies A' e B?.

The primary purpose for introducing the notions of ring, c-ring, and
algebra of sets is to obtain sufficiently large classes of sets to be useful for a

theory of integration. On the other hand, for the construction of a measure,

the class of sets must be restricted so that an explicit construction of a
measure is possible. This class must contain certain simple sets and for this
reason we want to construct c-rings of sets generated by a certain class of
subsets. How this is done is now to be explained.
11 e is any class of sets, we may define a unique ring called the ring
generated by 1, denoted by G?(t). It is defined as follows: Denote by B?,
(1 e I = some index set) the family of all the rings which contain the class g.
The intersection of all
is again a ring (Problem 9) and it defines the ring
generated by 1:
B?(t) = fl
It is clearly the smallest ring which contains the class S of sets. Furthermore

it is unique. The same procedure can be used for c-rings.

We are now prepared for the most important notion of this section,
the Borel set.

Let Sbe the reallineS= {x
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cxc

+cc}.


6

MEASURE AND INTEGRAL

For

e

we

1-1

choose the set of all bounded semiclosed intervals of the form

[a, b) e {x

a

x 'C b}.

The Borel sets on the real line are the sets contained in the c-ring G?(4')
generated by this class e.
The choice of semiclosed intervals as starting sets might be somewhat
surprising, but there is a technical reason for this. The finite unions of semiopen intervals are a ring (Problem 7), while this is not so for closed or open
intervals. It is, however, a posteriori possible to show that the c-ring generated by the open or the closed interval is also the class of Borel sets. These
properties are not difficult to prove, but they require certain technical devices
which transcend the purpose of this book ([1], §15). We shall therefore
state them without proof:

1) The class of all Borel sets is the c-ring generated by all open or all
closed sets.
2) The entire set S is a Borel set. The c-ring of Borel sets is thus an algebra.

3) Every countable set is a Borel set.

The first of these properties permits an extension of the notion of Borel sets
to certain topological spaces. For instance, in a locally compact Hausdorif
space one defines the Borel sets as the c-ring generated by all closed subsets
[1, Chapter 10].
With this general notion one can define Borel sets, for instance, on an
n-dimensional Euclidean space, on a finite-dimensional manifold, such as a

circle, a torus, or a sphere, and on many other much more complicated
spaces. In many applications which we shall use we have to deal with the
Borel sets for an arbitrary closed subset of the real line.
PROBLEMS

1. (A')' =

A for all sets A. [Note: In order to prove equality of two sets A and B,
one must prove separately A C B and B a A.]

2. AC BimpliesB' CA'.
3. A and B are disjoint if and only if A a B', or equivalently B a A'.

4.(AnB)'=A'uB'.
5.(A—B)'=A'uB.
6. The class of all subsets of a set S is a ring.
< x < -(-x}; then the class of all subsetsofS of the form
7. LetS = {x :

is

a ring.
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THE MEASURE SPACE

1-2

7

8. If a ring B? of subsets of S contains 5, then A e B? implies A' e B?, and vice
versa.

9. The intersection of any family of rings (a-rings) B?, is a ring (a-ring).

1-2. THE MEASURE SPACE

A measure space is a set of elements S, together with a c-ring M of subsets
of S and a nonnegative function 4u(A), defined on all subsets of the class M,
which satisfies certain properties to be enumerated below.
The subsets of the c-ring M are called the measurable sets. We denote
the measure space by (5, M, ji). Sometimes the explicit reference to the
measurable sets and the measure p is suppressed, and we then simply refer
to S as a measure space. Because M is a ring, 0 e M and so the null set is
always measurable. It is always possible to arrange that S e M, too, so
that M is an algebra.
The conditions to be satisfied by the set-function 4u(A) for A e M are
as follows:

1) 0 4u(A) c cc;

2) p(O) = 0;

3) For any disjoint sequence of sets A, e M (i =
A1)

1,

2,

. .

=Ep(AJ.

Property 1 may be relaxed to include infinite measures. Then we can only

require 0 p(A) . cc.

(Most applications, however, will be for finite

measures.) A set function which satisfies property (3) is called c-additive.

The sets A e M with p(A) = 0 are called the sets of measure zero. A
property which is true for all x e S except on a set of measure zero is said
to be true "almost everywhere" (abbreviated a.e.).
Naturally the question arises whether any measures exist and what their
properties are. Instead of entering into these rather difficult questions, we
shall explicitly exhibit two types of measures which we shall use constantly:
the Lebesgue measure and the Lebesgue-Stieltjes measure.
For the Lebesgue measure, the c-ring of measurable sets consists of the
Borel sets on the real line. On the half-open intervals [a, b), with a b, we

define

p{[a, b)} =

b — a.

In the theory of measure, one proves that this set function on the half-open
intervals has a unique extension to the Borel sets such that it satisfies conditions (I), (2), and (3). This extension will be called the Lebesgue measure
on the real line. (Actually the measure can be further extended to the class
of Lebesgue measurable sets, but we shall not need this extension explicitly.)
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8

1-2

MEASURE AND INTEGRAL

The Lebesgue-Stieltjes measure is a generalization of the Lebesgue
measure obtained in the following way: Let p(2) be a real valued, nondecreasing function, defined for — cc A +
and such that

p(A+O)= lim p(A +c)= p(A).
+0

For any semiopen interval [a, b), (a b), we define

p{[a,b)} =p(b)—p(a).
The unique extension of this set function to the Borel sets on the real line

is called the Lebesgue-Stieltjes measure on the real line.
For p(A) = A this measure reduces to the Lebesgue measure. The greater
generality of the Lebesgue-Stieltjes measure is especially convenient for discrete measures.

We obtain a discrete measure by letting p(A) be constant except for a
countable number of discontinuities 2k' where
P(Ak)

= PR

—

0)

+

The measure is then said to be concentrated at the points 2k with the
weights

We say two measures
and P2 are comparable if they are defined on
the same c-ring of measurable sets. Thus all measures on the Borel sets
are comparable.
In the following we shall examine the relation between different comparable measures. The main point is the observation that comparable
measures can be partially ordered. In the following we shall assume all
measures to be comparable without repeating it.
A measure Ri is said to be inferior to a measure P2 if all sets of p2-measure
zero are also of p1-measure zero. Pi is also called absolutely continuous
with respect to
We use the notation Ri -<

Thus Ri -< P2 if and only
if p2(A) = 0 implies p1(A) = 0. Two measures Pi' P2 are said to be equivalent
if both
and
P2 -<

then the same null sets. We write for this relation
Pi
P2 and we note that it is an equivalence relation (Problem 2).
Two measures Pi and P2 are said to be mutually singular if there exist
two disjoint sets A and B such that A u B = S and such that, for every
measurable set X c 5,
The two measures have

p1(AnX)=p2(BnX)=O.
Examples of mutually singular measures are easily constructed (Problem 8).

If a measure p is absolutely continuous with respect to Lehesgue measure,
it is simply called absolutely continuous.
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MEASURABLE AND INTEGRABLE FUNCTIONS

1-3

9

PROBLEMS
1. Let ji. be a discrete Lebesgue-Stie!tjes measure. For every Bore! set A,

Pk1 where the sum extends over a!! Ak, e A.

=

2. The re!ation —' is an equiva!ence re!ation; this means it is reflexive, symmetrica!, and transitive:
imp!ies

(b)
(c)

r-'

,a2 and

r-'

imp!ies

—'

on the rea! !ine are equiva!ent if and on!y if
Two discrete measures
and
they are concentrated in the same points and their respective weights are non-

3.

zero.

4. Every Lebesgue-Stie!tjes measure on the rea! !ine can be decomposed unique!y

into a discrete part and a continuous part, corresponding to the decomposition
into a discrete and a conof the nondecreasing function p(A) = pa(A) +
is constant except on a finite or
tinuous function. The discrete part
countab!y infinite set of points where pa(A) is discontinuous.
Every continuous nondecreasing function
can be decomposed into an
abso!ute!y continuous and a singu!ar function
= pa(A) + p8(A). The
function p8(A) is singu!ar in the sense that its derivative p8'(A) exists a!most
everywhere and is equa! to zero, yet p5(A) is continuous and nondecreasing
([2], Section 25).
*6. Theorem (Lebesgue). A finite nondecreasing function p(A) (or, more generally,
a function of bounded variation) possesses a finite derivative a.e. ([2], Section 4).
Theorem (Lebesgue). The necessary and sufficient condition that a finite, continuous and nondecreasing function is equal to the integral of' its derivative is
that it is absolutely continuous ([2], Section 25).
8. Let
another
be a discrete measure concentrated at the points AS" and
discrete measure concentrated at the points
Then the two measures are
singu!ar with respect to one another if and on!y if
for a!! pairs of
indices i and k.
1-3.

MEASURABLE AND INTEGRABLE FUNCTIONS

'[he theory of measure spaces permits a definition of the integral of functions

which is much more general than the so-called Riemann integral usually
introduced in elementary calculus. This more general type of integral, to
he defined now, is absolutely indispensable for the definition of Hilbert space

and other function spaces used constantly in quantum mechanics.

We start with the definition of a function. A function f is a correspondence between the elements of a set D1, called the domain of f, and a
set A,-, called the range off such that to every x e D1 there corresponds
exactly one element 1(v) e A1. The elements of 0,- are called the argument
of

the

function.
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1-3

MEASURE AND INTEGRAL

We remark especially that we define here what is sometimes called a
single-valued function. It is possible (and, for a systematic exposition, advisable) to treat multivalued functions by reducing them to single-valued
ones. In analytic function theory this procedure leads in a natural manner
to the theory of Riemann surfaces.
We emphasize, too, that a function has three determining elements:
A domain, a range, and a rule of correspondence x —, f(x). It will sometimes

be necessary to distinguish two functions which have different domains,
although in a common part of these domains the value of the two functions
may agree.
f(x)
B

x

of a function f(x).

Fig. 1—3 The inverse

The sets D1 and A1 may be quite general sets—for instance, subsets of
real or complex numbers. In that case we obtain real or complex functions.
But more often they will consist of points in a topological space, functions
in a function space, or even subsets of sets. In the latter case we speak also
of set functions.

An example of a set function of great importance is the following:

Let B be a subset of the range A1; we define the inverse image

(B)

(see Fig. 1—3) by setting

f1(B) =

{x

:f(x)eB}.

If the correspondencef between D1 and A1 is one-to-one, we can define the
inverse function, also denoted byf 1 but with arguments y e A1. The inverse
function
1(y) has domain D1-1 = A1, range A1-1 = D1, and satisfies

p 1(f(x)) = x

for all x e

and

= y

for all y e A1.

t Note that the inverse image is not the inverse function. As a function it is
defined on the subsets of

and its values are subsets of 1),.
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1-3

MEASURABLE AND INTEGRABLE FUNCTIONS

11

Although, strictly speaking, these two identity functions should be distinguished (since in one case the domain is D1, in the other A1), they are
usually considered identical, an assumption which is entirely correct only
if D1= A1.
For the rest of this section we shall consider real-valued functions over
a measure space. A1 is then a subset of the real line B?.

Let (5, M, p) be a measure space and f a real-valued function with
domain D1 = S. We call the function f measurable on S if for every Borel
set B on the real line, the setf1(B) is measurable.
The simplest examples of measurable functions are obtained from the
set functions XAX) defined by
(1

XA(X)

forxeA,

for x A.
=
A set function is measurable if and only if the set A is measurable. Indeed
we find immediately that
—1

XA

(A ifleB,
(B)=10

ifl*B,

so that XAX) is measurable if A e M.
There is a resemblance between measurable functions in a measure space
and continuous functions in a topological space S. A topological space is
defined by the class of all open subsets of S. A function f(x) from S onto
A1 e B? is then sajd to be continuous if the inverse image f - 1(B) of any open
set is an open set in S. One obtains the more general class of measurable

functions by replacing the word "open" by "measurable," in the above
definition of continuous function. The class is more general because (at
least in all the measure spaces which we consider) the open sets are measurable sets. One of the most important problems in measure theory is to
identify the class of measurable functions over a measure space. An efficient
way of doing this is to construct the measurable functions from certain simple
functions by the operations of sums, products, and the passage to the limit.
In what follows we shall describe this process, without giving proofs.

First, one observes that if two functions f and g are measurable, then
the functions

f+g

and

fg,

defined by

(f + g)(x) = f(x) + g(x),
arc

fg(x) = f(x)g(x),

also measurable. From this it follows that the so-called simple fun ctions

defined by

f(x)
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12

1-3

MEASURE AND INTEGRAL

real constants and A, e M, are also measurable.
;
integral of a simple function as

with

J

We define the

fdp

It is a finite number since the ji(A1) are all finite (we admit only finite
measures).

(n = 1, 2, . .). We define

convergence in the measure of such a sequence to a limit function f if, for
every e> 0,
Next we consider sequences of functions

lim p({x :

.

4) =

—

0.

n -.

The

notion of measurability is more general than that of integrability, the

latter being restricted by the condition that there exist a finite-valued integral.
Since simple functions are not only measurable but also integrable, we can

define the integrable functions as follows: A finite-valued function f on a
measure space (S, M, p) is integrable if there exists a sequence of simple
functions f,, such that f,, tends to f in the measure. It follows then that the
numbers
= J d4u tend to a limit which defines the integral of f:

[py = lim

[fndlt.

n—+coj

J

Various theorems then permit the usual operations with integrals.
instance, if f1 andf2 are integrable, so are (jj + f2) and

f

For

and

integral defined here corresponds to what in the elementary integration theory is called the definite integral. There is also a concept which
generalizes the usual notion of the indefinite integral. In the usual definition,
The

the indefinite integral depends on the lower and upper limit of the integration
variable. It is thus an interval function.

More generally we may define a set function v(A) for all A e M by
setting
v(A)

= J
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J