Order-Preserving Maps and Integration Processes. (AM-31), Volume 31 9781400882304

Citation preview

Annals of Mathematics Studies Number 3 1

ANNALS OF MATHEMATICS STUDIES Edited by Emil Artin and Marston Morse 1. Algebraic Theory of Numbers, by

H erm an n W ey l

3. Consistency of the Continuum Hypothesis, by 6.

The Calculi of Lambda-Conversion, by

7. Finite Dimensional Vector Spaces, by 10. Topics

in

Topology, by

K urt G o d e l

A lo n zo C h u r c h

Paul

R.

H a lm o s

S o lo m o n L e fsc h e t z

11.

Introduction to Nonlinear Mechanics, by N.

14.

Lectures on Differential Equations, by

and N.

K rylo ff

S o lo m o n

B o g o l iu b o f f

L efsc h etz

15. Topological Methods in the Theory of Functions of a Complex Variable, by M a r s t o n M o r s e 16.

Transcendental Numbers, by

17.

Probleme General de

19.

Fourier Transforms, by S.

20.

la

C arl

L u d w ig

S ie g e l

Stabilite du Mouvement, by M. B och n er

and

K;

A . L ia p o u n o f f

C h a n d r a sek h a r a n

Contributions to the Theory of Nonlinear Oscillations, Vol. I, edited by S. L e f s c h e t z

21.

Functional Operators, Vol. I, by

22.

Functional Operators, Vol. II, by

23.

Existence Theorems in Partial Differential Equations, by

J o h n von N e u m a n n J ohn

von

N eum ann D orothy

L.

B e r n s t e in 24.

Contributions to the Theory of Games, Vol. I, edited by H. W. A. W. T u c k e r

25.

Contributions to Fourier Analysis, by A. A. P. C a l d e r o n , and S. B o c h n e r

26.

A Theory of Cross-Spaces, by

27.

Isoperimetric Inequalities in Mathematical Physics, by G. G. S z e g o

28.

Contributions to the Theory of Games, Vol. II, edited by H. A. W. T u c k e r

29.

Contributions to the Theory of Nonlinear Oscillations, Vol. II, edited by S. L e f s c h e t z

30.

Contributions to the Theory of Riemann Surfaces, edited by et al.

Zygm und,

W.

and

T r a n su e , M . M o r se ,

R obert Sch atten

31. Order-Preserving Maps and Integration Processes, by 32.

K uhn

Curvature and Betti Numbers, by

K. Yano

and S.

E dw ard

B och n er

P o lya

and

Kuhn

and

L . Ah lfo rs

J.

M cShane

ORDER-PRESERVING MAPS AND IN T EG R A T IO N PROCESSES By EDWARD J. McSHANE

Princeton, New Jersey Princeton University Press 1953

Copyright, London:

1953,

by Princeton University Press

Geoffrey Curriberlege, Oxford University Press L. C. Card 53-6583

Printed in the United States of America

CONTENTS

Introduction

3

Chapter I.

7

Partially Ordered Sets and Systems 1 . 2.

Partial Orderings and Lattices Completeness

7 9

3*

Types of Convergence

1^

h.

Closure

21

5-

Continuity

23

6.

Partially Ordered Groups

26

7- Linear Systems, Multiplicative Systems and Algebras within Partially Ordered Sets Chapter II.

Definition of the Mapping

32 36

8.

The Postulates

36

9*

U-elements and L-elements

39

1 0 . Lattice Properties of the Classes of U- and

b1

L-elements 1 1 . Definition of the Mapping or Integral

Chapter III.

Lattice Properties, Convergence and Measurability

^5 ^8

1 2 . Further Lattice Properties of the Classes of

131 ^.

U-elements and L-elements

^8

Lattice Properties of the Class of Summable Elements

53

Monotone Sequences of Summable Elements

15- Dominated Convergence 1 6 . Structure of the Set of Summable Elements 17- Measurable Elements

v

55 59

6b 65

vi

CONTENTS

Chapter IV.

Algebraic Operations

18.

Interchange of Order of Operations

19•

Addition, Scalar Multiplication and Binary

69 69

Multiplication

71

20.

Summability of Products

76

21.

Operations on Measurable Elements

77

Chapter V.

Real-valued Functions

81

22.

Integrals of Real-valued Functions

81

23.

Measurable Functions and Lebesgue Ladders Fubini1s Theorem

87

2k . Chapter VI.

Applications

93 95

25.

Measure in Locally Compact Spaces

26.

Functions on Locally Compact Spaces

ioo

27-

A Non-absolutely Convergent Integral

1 03

28.

Remarks on Operators in Hilbert Spaces

1 08

29.

Spectral Resolution of a Bounded Hermitian Operator

30.

125

Spectral Resolution of Complex Partially Ordered ♦-algebras with Unit

Bibliography

118

Spectral Resolution of Real Partially Ordered Algebras with Unit

32.

113

Bochner’s Generalization of the Bernstein-Widder Theorem

31•

95

131 155

ORDER-PRESERVING MAPS AND INTEGRATION PROCESSES

INTRODUCTION The various definitions of the Lebesgue integral and its generali­ zations may be classified, if we wish, into two subsets, one making essen­ tial use of some kind of norm or modulus, and another, less numerous, in which the central role is played by some order relation. For example, bocnner’s mLegrau (..bocnner ij 01 a iuncLion wnose values lie on a uanacn space involves formation of successive approximations which converge in the sense of the norm in the Banach space. Again, in Stone's treatment [Stone 1] of the integral, order properties are first used to define a norm in a func­ tion space, and then the integral is defined by means of a convergence according to this norm. On the other hand, the definition of the Riemann integral by use of the Darboux upper and lower integrals rests on order properties, and so does Daniell's definition [Daniell 1] of integral, gen­ eralizing the Lebesgue and Radon (or Lebesgue - Stieltjes) integrals. The Perron integral, too, is defined as the function which is simultaneously the lower bound of a set of overestimates and the upper bound of a set of underestimates, and thus is one of those making essential use of order. Daniell develops the integral by assuming that a subset E of the lattice F of all real-valued functions on a set T is mapped into the set of real numbers by an order-preserving mapping I satisfying certain requirements. He then shows how this mapping can be extended, giving a mapping I whose domain contains E and whose range is contained in the reals; this mapping is order-preserving, and also has the closure and continuity properties expressed in the well-known convergence theorems of the Lebesgue theory. Thus Daniell's theory can be regarded as a study of a special case of the following problem. We are given two partially ordered sets F and G, and an order-preserving mapping I of a subset E of F into G. We seek conditions on E, I , F, and G that will enable us to extend the domain of definition of IQ, producing say a mapping I of FQ into G, in such a way that the enlarged domain FQ shall have some useful closure properties and the extended mapping I shall have some useful kind of continuity property on its domain. It is to the study of this problem that the following pages are devoted. Because it is

^References are to the brief bibliography at the end of the study. 3

k

INTRODUCTION

an exploitation of the order properties almost exclusively, it is to be expected that the territory which it more or less naturally takes in is in part different from that covered by the integration theories based largely on norms.

If it gracefully furnishes interesting applications, its exist­

ence is justified.

We believe that the examples exhibited in the last

chapter will show this to be the case. For the sake of simplicity we restrict the problem by assuming that

F

is a lattice.

tially ordered set

This is not a great restriction, since every par­

F

can be embedded in a lattice, and if the process of

extension yields a domain containing points of that lattice not in the original

F

we have the privilege of ignoring them.

one of the applications in the last chapter

F

In fact, in all but

is isomorphic with the

lattice of extended-real-valued functions on some domain. with regard to

G

is different.

tion 29 the values of on a Hilbert space.

I

The situation

For example, in the application in Sec­

lie in the set

G

of bounded hermitian operators

These may be regarded as defining quadratic functions

on Hilbert space, and thus embedded in the lattice of real functions on the Hilbert space.

But the chief importance of the extended mapping

that its values also are bounded Hermitian operators.

I

is

If we knew only that

the values corresponded to some real function on the Hilbert space, the result would have been without interest. that the image space

G

Therefore we refrain from assuming

is a lattice.

This in fact constitutes one of the chief differences between the present treatment of integration and others that have preceded it.

Thus,

for example, in the fundamental paper of H. Freudenthal [Freudenthal 1 ] there is developed a type of Riemann-Stieltjes integral with values in a "partially ordered module," which is not merely a partially ordered linear system as we here define the term, but is a lattice.

Likewise, M. H. Stone

[Stone 3] and HidegorS Nakano [Nakano 1 , 2 ] define integrals whose values lie in a lattice. is not trivial.

The difference, insofar as it concerns our present aims, For example, a (suitably closed) commutative algebra of

hermitian operators is actually a lattice, but this is by no means super­ ficially evident; while on the other hand it Is obvious that this algebra is partially ordered. To the best of my knowledge, there are two important publications on integration processes involving partially ordered spaces which are not lattices. These are [Bochner 1 ] and [Bochner and Ky Fan 1 ] . In both of these a Riemann-Stieltjes type of integral is used, in the former to extend the Bernstein-Widder theorem, and in the latter to obtain a representation of the general distributive order-preserving mapping of functions continuous on an interval (or circumference) into a partially ordered space.

The

principal result of [Bochner 1] is very close to the application here in Section 3 0 .

INTRODUCTION

5

One result of working with partially ordered sets instead of lattices is that we must first develop an adequate theory of closure, completeness, convergence and continuity.

This we do in the first chapter,

a trifle more extensively than is absolutely essential for later use.

It

is our strong suspicion that some interesting portions of lattice theory can be usefully generalized to less restricted partially ordered sets. Our methods are of course related to those of Daniell.

Some

rather drastic changes necessarily result from replacement of the lattice of real-valued functions on a set

T

by the general lattice

F*

and the

replacement of the real number system by the partially ordered set

G.

But

in addition to these there are a couple of changes that would affect the discussion even of the case considered by Daniell. the use of auxiliary partial orderings. by the relation

The set

valued functions on an interval )>.

where by

f »

F

is partially ordered

in terms of which it is a lattice; but there may also

be other partial orderings useful as auxiliaries. relation

One of these concerns

J

For example, the real­

are partially ordered by the usual

But we can also partially order them by the relation g

we mean that to each point

a neighborhood on which the supremum of

g

x*

of

J

»,

there corresponds

does not exceed the infimum of

f . When we develop the Lebesgue-Stielt jes integral by extending the ele­ mentary integral on step-functions, we use systems of step-functions directed by

»

or by its reverse

«,

and thereby avoid the sometimes

tedious discussion of the phenomena at the boundaries of the intervals of constancy of the step-functions.

This same device is also of service in

the study of the Riemann-Stielt jes integral, as will be shown in a joint paper with T. A. Botts [McShane and Botts l]. for

»,

With other special meanings

the auxiliary partial ordering also proves useful in the other

examples in this paper. The other departure from Daniell*s pattern is a relaxation of the requirement that the set integral

I

E

of elementary functions on which the elementary

is defined should be a lattice.

This relaxation can be

pictured with the help of the family of polynomials on an interval Since the (real) functions continuous on such functions

g-],g2,gj

the function

point is the middle one of the values of But if

e^,e2,

a polynomial.

and

e^

J.

form a lattice, for any three

mid(g 1 ,g2 ,g^)

whose value at each

g 1 ,g2,g-*> is also continuous.

are polynomials their "middle" is not necessarily

However, in this case, if

f - mid ( e ^ e ^ e ^ )

J

f

is any function such that

has a positive lower bound, by Weierstrass* theorem we

can find a polynomial e on J such that f - e and e - mid(e 1 ,e2 ,e^) both have positive lower bounds on J. Our substitute for the requirement that

E

element

be a lattice in e

F

is just the analogue of demanding that such an

exist (and the dual requirement, where

a positive lower bound).

mid ( e ^ e ^ e ^ )

Even this we ask only when

I^e^

and

- f

has

I0 ^e2 ^

INTRODUCTION

6 have a common upper bound.

This relaxation is useful, for example, in

studying the spectral resolution of bounded hermitian operators on a Hilbert space, as discussed in Section 2 9 ; for here the natural starting point is the family of polynomials in the operator.

It is also useful in the appli­

cations studied in Sections 3 0 , 31 and 3 2 , for similar reasons. the requirement that there exist an when

IQ (e1)

and

I0 (e2)

e

Because

as just described is imposed only

have a common bound, the integral obtained is

not necessarily absolutely convergent.

In fact, as one application of the

general theory we obtain (in Section 2 7 ) an integral which closely resembles the Perron integral, and may be identical with it. Our principal type of convergence based on order properties shows a close affinity with "pointwise" rather than "uniform" convergence. instance, in Section 3 we show that when

F

For

is the set of extended-real-

valued formations on some domain, order-convergence is the same as pointwise convergence.

When

F

is the set of bounded hermitian operators on

Hilbert space, order-convergence coincides with (eventually) bounded strong convergence.

Accordingly, the spectral resolution of an operator

we obtain serves to put the Borel-measurable functions [-||B||,||B||] f(B),

f

B

which

on the interval

into correspondence with operators, designated as usual by

forming a commutative algebra; the integral composes each

f(B)

out

of limits of linear combiiiations of certain projection-operators forming the resolution of the identity for

B.

When we consider algebras of hermitian

operators (as in Section 31), we map the elements of the algebra on Borelmeasurable functions on a product of certain intervals; this is in contrast with the Gelfand-Neumark representation by means of simpler (i.e., continu­ ous) functions on a more complicated domain. bounded hermitian operator and

For instance, if

E^, - « < A. < »

identity, the least algebra containing

B

B

is a

is its resolution of the

and all the

E^

is by the

present theory represented by the Borel-measurable functions on the interval [-||B||,||B||]. would require

The representation by continuous functions on a domain D

D

to be of much greater complexity.

These considerations may lend some interest to the theorems in Sections 31 and 32 concerning partially ordered algebras, since we again obtain theorems which, when interpreted in the specialization to operators on Hilbert space, give the "strong" rather than the "uniform" theory. The examples which we have considered do not exhaust the possi­ bilities, nor have the individual examples been studied as exhaustively as possible. It is hardly to be expected that the present approach would yield startlingly new results in such well-worked fields as LebesgueStieltjes integration or the spectral resolution of a bounded hermitian operator.

In familiar settings we are content to obtain familiar theorems;

the novelty, if any, lies In the unification, whereby such apparently diverse processes as Perron-type integration and spectral resolution appear as instances subsumed under a single theory.

C H A P T E R

I

PARTIALLY ORDERED SETS AND SYSTEMS

$ 1 . PARTIAL ORDERINGS AND LATTICES

A binary relation partial ordering; of (1 .1 )

(a)

If

(b)

If

F

>

between pairs of elements of a set

F

is a

if it satisfies the conditions

a > b

and

a > b

b > c

and

then

b > a

a > c.

then

a = b.

(Sometimes (1 .1 a) alone is used as the definition of a partial ordering; then relations satisfying (l.la,b) are called proper partial orderings. But we shall adhere to the terminology above.) ing of that in

F,

>

F),

and

a > b

is defined to mean that

If

>

is a partial order­

a > b

ora = b,

is another partial ordering.

It is reflexive

and is the same as

was itself reflexive.

>

if

>

partial ordering whose symbol includes the mark ive, but the absence of

=

=

we

see

(a> a for all

(as in

a

Thus any ))

is reflex­

in the symbol does not imply the absence of

reflexivity. Let b > a.

be partially ordered by

We verify at once that

element all

F

b

x


b

unique.

Lower bounds and the infimum of

It S

when it exists, will be designated

has just two members, say

usually write a V b for V s and a A b for As . If a set F is partially ordered by ^>,

S = (a,b),

we

sois eachsubset

FQ .

But the situation here is not as simple as the analogous one in topology. For if S is a subset of F Q , it may have a supremum when regarded as a subset of

Fq

it may have a

but not when regarded as a .subset of supremum in F Q

F, or vice versa;

when regarded as a subsetof F Q 7

and a

or

I.

8 different (1 ) F 2 < x

PARTIALLY ORDERED SETS AND SYSTEMS

supremum in F

when regarded as a subset of F.

the rationals, F 2

] such that the elements of

D),

and for two such elements

^

F

are the

and ^ 2,

the

relation ^ y 2 moans that ^ (6))^2 (5) f*0I> all 6 in D. It is obvious that > is a partial ordering of F. Since we shall not have any use for cartesian products except when the sets are partially ordered, we shall usually abbreviate the product symbol to X.F., partial ordering being understood as above. For tion” of

on

"projection" of tions on

F^

(2 .6 )

Ffl S

shall mean on

F^

F D,

of

Let F.

S

F, the "projec­ of F,

the

S.

[F,X

partially ordered sets a subset

for a subset

the meaning of the in

shall mean the set consisting of the projec­

of the elements of THEOREM.

(6);

0 0

be the cartesian product of

[Ffl

6 in

In order that S

D.

Let

it isnecessary and sufficient that for each the projection of

S

on

F^

S

be

have a supremum in

6

in

shall have a supremum

(with respect to >$); in. this case, the supremum of S is the element of F whose projection on F^ is the supremum of the projection of is also true.

S

on

F ^ • The dual

12

I.

PARTIALLY ORDERED SETS AND SYSTEMS

If S has a supremum

4>

for each

an upper bound for the projection of let

b*

S

6

in

be an upper bound for the projection of

4(6) for

6

D , ^ 6',

in

while

6'

S

in

F ^ ,.

4'(6') = b*. Then 4' ^ ^,4(6'). That is,

D,

in

Let

is an

is

and

4'(6) =

upper

4(6*) is the supremum of the projection of S on F ^ . Conversely, if 4(6) = V[projection of S on F^] for each 6 , 4 is an upper bound for S. Given any other upper bound 4' of S, we have for each 6 in D that 4'(6) y^4(6) for all \p In S, so 4'(6) is an upper bound for the projection of S on F ^ , and 4(6) • Hence 4 is the supremum bound for

of

S.

S,

4' ^

4(6)

the projection

F ^ . Fix a

so

and

The dual is established similarly. COROLLARY.

(2.7)

If for each

6

D

in

the set

F ft

is

Dedekind-complete under the partial ordering

the o 13 also Dedekind complete.

cartesian product If a lattice

F

does not contain a greatest element, we can

adjoin a new ideal element (usually denoted by all

x

in

adjoin one.

F.

Likewise if F

If

F

enlarged lattice.

More: 0,

F

I > x

each non-empty set

S C F

0

for we can

ct)

n(ce) = V(f(oel):at

exists, and also

we define The lower

be a net of

F.

o-lim sup f(oe) « in A

dually.

A(n(oe):ct

in

If for each in A)

A

a

and

exists,

to be A(n(a):oe in A). o-lim inf f (a) is defined ° ln A

To simplify the notation we shall usually abbreviate the symbol o-lim sup f(a)

to

o-lim sup^fta)

th? ln ^ lower limit.

or to

o-lim sup f,

and likewise for

16

I.

(3-3)

PARTIALLY ORDERED SETS AND SYSTEMS in

A)

is a net of ele­

ments of a partially ordered set

COROLLARY.

If

F,

and

o-lim sup^fCa)

(f(ce):ot

and

o-lim inf^fta)

exist, then

( i)

o-lim supa f(a) > o-lim inf^f (a);

(ii)

o-lim^fCa)

and exists if and only if

o-lim supa f(o«)

and

o-lim in^f(ce)

equal, and in that case

are

o-limtff(ce)

is

equal to their common value. Conclusion (i) is evident.

Consider (ii).

lower limits are equal, the set

(n(a):ct

dually defined set

A)

(3*1)•

(m(a):a

Conversely, if

in

exists, let

For each

n

we have

n(oe')< n,

and therefore

o-lim sup^f (a)

N,

If the upper and

defined in (3 *2 ) and the

A)

have all the properties specified in

o-lim^ffa)

in (3 .1 ).

in

in

f(ce)

M and

N

is eventually

he as stated

< n,

o-lim supa f(a) < n.

< A N = o-lim^f (a) .

a*

so for some

Hence

Similarly o-lim infa f (a) >o-lim^f (a)

which with (i) completes the proof. In particular, if

F

is a complete lattice,

limits of every net of elements of

F

theupper and lower

are both defined.

There is some interest in finding the relationship between con­ vergence of a net in a product-space and convergence of its projections into the factor spaces. (3 .L)

THEOREM. spaces, [F,>]

Fg

F-, 6

Let

0

in

D

be a collection of

being partially ordered by

>^ ;

be the cartesian product

(f(a):oe

in

A)

and let

? =

Let

be a net of elements of

F.

In order

-M-

that

f

be o-convergent to an element

f

of

F, it

is necessary and sufficient that ( i)

there exist elements that eventually,

(ii)

If Suppose We choose for

h*

is satisfied.

If

m',n'

in

of

F

D,

such and

the projection

f^ of

on

F, shall be o-convergent to the ® * * projection f^ of f on F ^ . D is a finite set, (i) may be omitted. first that

o-lim f(ot) = f *.

any element of

M

and for

Let h"

are the projections of

they are directed by If

6

for each f

h',h"

h* < f (ct) < h” ,

be as in (3 .1 ).

M,N

F^

are

m',n'

m,n

N;

(i)

respectively on

respectively, and by (2 .6 ) V

are inrespectively, there exist

tively whose projections on

M,N

any element of

in M,N

Fg

= f^ respec­

respectively, so eventually

3m1


and to have

is eventually true f

m^

>^,

For each finite subset

D - D .

of finite to be

directed by

m^

all elements for

M^,N^

and for each

true that

so

17

^ n ‘* Hence (ii) holds. Conversely, suppose (i) and (ii) satisfied.

there A

TYPES OF CONVERGENCE

f*

that

f(oe) > m. * — is o-convergent to f .

is finite we can take

DQ

and hM ] . This is easily seen to for supremum; and if The set

N

m

is in M,

it

can be defined analogously,

Two special cases occur often enough in later pages to make it worth while to mention them as examples: (3-5)

COROLLARY.

If

H

Is the set of all extended-

real-valued functions on a domain ordering that in

Dg,

h 1 < h2

if

then a directed system

ments of

H

and only

if

Bg.

H

If

a domain

D^,

with the partial

h ^ x ) < h2 (x)

for all

(ha :ce in

is convergent to an element h^(x)

converges to

hQ (x)

A) hQ

x

of ele­ of

H

for each

if

x

In

Is the set of all real-valued functions on

D^,

ordered as just described, then a

directed system (ha :oe in o-convergent to an element

A) hQ

of elements of H is of H if and only if

it is "eventually dominated” (that is, there exist elements for all

h' a

and

h”

of

H

beyond a certain

verges to

hQ (x)

for all

x

such that a 1)

h 1 < ha < h”

and

Ha (x)

con­

in Dg.

The second statement is a specialization of (3«*0first, if we take

h1

Again, let

F

ordered product of

n

a l 9 ’‘‘9 ments of

L

an ln F. Then

is isotone. (3-6)

constantly

-»

and

h"

constantly

be a lattice, and let copies of

F.

Let

Fn

So is the +».

be the partially-

L = (L(a1, ..., an ):

be a lattice combination defined on n-uples of ele­ maps Fn on F. It is easy to see that the mapping

We now prove THEOREM. With the notation above, if

(a1 (oe) :a

in

A ) , ..., (an (a) :ot in

A)

elements with the respective o-limits the net (L(a1 (a), ..., an (.a)) :a vergent to L ( a ‘1, ..., a fn ) .

in

are nets of a^,

A)

..., a'n ,

is o-con-

18

I.

PARTIALLY ORDERED SETS AND SYSTEMS

For each number directed by each

mj

>,

in

j

in the set

Mj

and each

n^

in

1, ..., n,

there are sets

\/Mj = /\Nj = a 1 . and such that for

respectively, having Nj

it is eventually true that

m^
y,

and

a - x

If

x ^ a

then

a + x + b)>a + y + b

+ b < a - y and

The four statements -y + x y B> -y > -x

If a - y + b. -x < -y;

x > y,

With

by setting

a = b = B,

replacing

x

by

z = y

and

y

then

x + y > a + b.

x > y, x - y > 0 , are equivalent.

in (6 .ib> we obtain

this implies -y

+ b.

y ^ b,

by

-x < -y. -x

Now

a - x + b
a + b

and

x + y > x + b,

whence (ii) follows; and (Iii)

is an obvious consequence of (Ii). (6 .3 )

COROLLARY. the pair F

Let

a,b,x,y

be members of F + .

If

(x,y) has either a supremum or an infimum In

which is a member of

F ,

then all four pairs

( x, y, ) ( - x , - y ) , (a + x + b, a + y + b,) [a-x+b, a-y+b) all belonging to F + ,

have both Infima and suprema, and the equations

a +

(x V y) + b = (a + x + b) V (a + y + b)

a -

(x V y) + b = (a - x + b) A (a - y + b)

(-x) V i-y) = -(x A y )

28

I.

PARTIALLY ORDERED SETS AND SYSTEMS

and their duals are satisfied.

Assume first that is a lower

is a lower bound for {x,y}

is

x V y

exists and is in

bound for -x

and

a - x + b and

is

< a - (x V y )

x+b) A (a - y

(a

-

we

findthat

x A y

+b .

a-(x V y )

+ b

Every upper bound for

lower bound for

That is,

{a - x + b,

a - (x \/y) + b =*

+ b) . In particular, if we set y - ix V y) + x,

exists and is

F+ . By (6.2111),

-y, so by (6.2i)

a - y + b.

) x V y , so by (6.1b) every

a - y + b)

F+

x V / y + x A y ^ x + y.

is a commutative group,

-(x V y )

In particular, if

a = y

and

b = x,

being therefore in

F . When addition is commutative this yields the last equation in the conclusion. that

Since

(a - x + b) V

a = b * 0

x Ay

exists, we can dualize the proof above and show

(a - y + b)

exists and is

permits applying the above proof to and

y

a - (x A y )

-

Letting

gives us the second-last equation in the conclusion.

by

-y

-x

and

-y;

replacing

This

x

by

-x

in the equation just established furnishes the remaining

conclusion. The equivalents of

f+ , f“

and

|f|

as usually understood in

function-spaces or in lattices do not always exist in partially ordered groups,

butwe follow the

(6.10

usual path when this is possible.

DEFINITION.

Let F+

partially ordered space such that F+ ,

f V 0 and

be a group embedded In a F.

If

f

(-f) V 0

Is a member of

F+

both exist and are in

we define f

f * f \/ 0 ,

f’ - (-f)V 0 , If I - f+ + f REMARK. (-f \/ B)

By (6 .3 ),

exists and belongs

(6 .5 )

COROLLARY.

if either of the suprema f\ / 0 toF+ ,

When f+

and

so does the other. and

f”

are defined, the

relations f+ > 0, f” > 0, |f| > 0, f+ - f” = -f“ + f+ = f are satisfied. The inequalities are obvious. so by (6 .3 )

f + (-f)V 0 + 6

By hypothesis,

= (f + [-f] + $) V

(-f)\/ 0

(f + 0 + 0),

or

f + f" - f+ . Also by (6 .3 ), S+ (-f)V 0 + f = (0 + [-f] + f ) V (0 + 0 + f), or f" + f = f+ . These imply the conclusion.

exists,

6. (6.6)

THEOREM.

PARTIALLY ORDERED GROUPS Let

F+

29

be a group under

contained in a partially ordered set of

elements of

F

ftnot necessarily in F+ ),

+

F.

and be

If each pair

F+ has a supremum and an infimum in (6 .1 b) is equivalent to

the statement If

(6.1b*) (a

a,b,x,y,z

are In

- x + b) < (a When

F+

F+

x > y A z,

and

y + b) V

(a - z +b ) ,

Is a lattice embedded in

then

and dually. (6 .1 b) is

F,

equivalent to If

i6. 1b") a - x

a,b,x,y

+ b < a - y

are in

F+

and

x > y,

then

+ b.

Clearly (6.1b) implies (6.1b1), and it implies (6.1bn) by (6 .2 ).

y A z

If

exists, the hypothesis s = y A z)

implies (with

that

"s < y x ^ y

A

s < z

and

z,

implies

s < x "

so by (6 .lbr) half the con­

clusion of (6 .1 b) holds; the other half is established dually. a

lattice and (6 .1 b") holds,

c

-

(x A y )

for

+ d>c-x + ' d

c - ( x A y ) + d > ( c - x

and

eachc,d^x,y c-(x A y) +

+ d ) \ / ( c - y + d).

Is similarly established, we first replace a

-

y + b,

b

-

(a - x + b) V

then replace

a - (x A y) + b. equality holds. satisfied.

c

by

is

we have

d ^

c - y + d,so

x

by d

a - x + b by

a;

or (a - x + b) V

and

y

by

we

obtain

(a

- y + b) >

The reversed inequality has already been established, so Then the statement in (6 .ib•) concerning Infima Is

The other statement is established dually, so (6.1b*) holds,

and therefore so does (6 .1 b). (6 .7 )

F+

F+

In the dual statement, which

band

(a - J + b) + a _ 0

-b*

+ f2< 0.

If we define

b
0, so

b =

have an upper bound fj - b» < 0

f^ - b* + f2 , wehave

b < f1

and

To establish (ii) use induction.

Suppose it valid for

n = k - 1,

b*

and By (i),

and let

**19 **’9 **k e^einen^s su°h that each pair of consecutive members has an upper or a lower bound (hence, by (I), has both) in F . By the induction hypothesis, the set bound bound

(f1, ... f ^ )

has a lower bound

c

and an upper

c 1 in F+ . Also by hypothesis, the pair tfk-1,fk ) a lower d and an upper bound d' in F+ . Then f^_1 is an upper bound

for the pair Clearly

(c,d),

so by (I) the pair has a lower bound

h < c < f^

lower bound for

(1=1,

..., k - 1)

(f1, ..., f^).

and

b

b < d < f^,

in F

so

.

b is

a

The existence of an upper bound Is

established dually, and by Induction (Ii) Is valid. The connection between bounds and the group operation Is inter­ esting enough to justify a digression, even though the results of the next few paragraphs are not essential to later proofs. For each such that and

F^

Fa ,Fk

a

and

a b

In

F+

let

Fa

be the set of all

in F+

have a common upper or lower bound in F

have a common element

c,

and

. If F0 + s are arbitrary members of

a f,bT

respectively, by (6 .7 ) the five elements

common upper bound In

b

F+ ,

so

a*

is in

F^

a',a,c,b,b' and

b1

in

have a F& . Hence

F, = F 0 . Thus the partially ordered group F^ Is classified into D o # + mutually exclusive classes F& . If a is in the class F^ containing the pair (0,a) Then

has

bf

and a lower bound

a = -[b' - a] + [b*] = [a - b] - [-b],

brackets being

^ 0*

either of the forms versely, If

x

0 , so

x

and

an upper bound

.If such that a - b,

So every element in

a

can be represented in and

has either of these forms,

c

Is an upper bound for

- b is

d

are

]> 0.

Con­

F^,

there are elements

in F ^ .

Then

c

It follows that

It is in fact an invariantsubgroup.

For if a

c,d

- d > 0

In

and

F+

c -d >

F^

Is a subgroup of

Is

in F^

and

b inF+ such that b > 0 and b ) a. and -x + b + x > -x + 0 + x = 0, so

-x + a + x

For each set

is again in F ^ .

F& ,

if

b

and

F e they have a common upper bound d in F , d - b and d - c cL + > 0, and so b - c = -(d - b) + (d - c) is in F,^. Conversely, if is

in F^, b - a and

0

and

andtherefore b

the sets

Fa

subgroup

F^.

a —< c + a,

have a common upper bound

are the cosets into which

F+

is in

in

c are both

In

b - a

F+ .

x is

F , there exists an element Then -x + a + x < - x + b + x

b < c + a

x

F^ b are in

c > a, c > 0, d < 0, d < b.

so

the quantities in square

F^ c

is in

0,

F .

where

a and

c - d, -d + c,

b in

c

in F+ ,

are so

Fa .Thisshows that is classified by the Invariant

6.

PARTIALLY ORDERED GROUPS

31

The next theorem shows that (6.1c) follows from (6.la,b), provided that

F+

(6.8)

has a certain closure property.

THEOREM.

Let F+

be a group contained In a

Dedekind complete partially ordered set satisfying (6.1a) and (6.1b). and F+

b

are in

and

F+ and

a < x < b)

holds, so that over, if by

^

set

S1

a < b,

and

the set

is Dedekind closed.

(x : x

and

S2

are subsets of

and having upper bounds in (-x : x

S1

in

and

More­

directed

and

in

S 1}, then

F.

F+

F+ , x2

a

in

Then (6.1c)

F + isa group embedded in

(x., + x2 : x 1 in

is the set

F

Assume that whenever

S

is the

S2 )and

S_

V S j , V S 2 ,V S

and

A S _ all exist, and

Vs - V s 1 + Vs 2 , As_ = - vs 1 . It is clear that (6.1c) follows from the statements in the last sentence, so this is all we need to prove. The sets directed by and

S2

and

b 2 are upper bounds in

is an upper bound for

S

VS.,, V S 2 , V s VS., + V S 2

belong to

and A s _

and

is

F+

for

and

all exist in F

is an upper bound for

S 1; for all

a2

in

S2

S,

V S ]> V S 1 + V S 2 -

The

established, so equality holds, and

so

we have

whence’ -a1 + V S > a2 , and -a1 + V S > V s2 * T*1®31 so V S - V S 2 ^ a., . This holds for all a 1 in S 1, so VS.,,

S_

S^

-b1

is a

S _ . By the hypothesis of Dedekind completeness of

Dedekind closure, to

S 1, S2

and

F

and

belong

V s a 1 + a2,

> a i + \/S2, \/S - V S 2 >

reverse inequality was previously the first equation established. The

second is proved analogously. The equations in the last conclusion of (6.8) can be established without assuming the Dedekind closure of the sets a < x < b)

provided the sets

S1

merely the greatest element in

and

S^.

S2

(x : x

in

F+

are finite, for then

and VSi

is

This and (6.8) might lead to the

suspicion that (6.1c) is always a consequence of (6.la,b).

This is not so;

if we omit the hypothesis concerning Dedekind closure in (6.8) the equa­ tions in the conclusion may fail even when all the infima and suprema mentioned in them exist.

For example, let

reals under addition, and let plus two elements

Q_«|>Q2

F

F+

consist of the group of

consist of the reals with the usual order

wltl1 the properties

q_1 > -1, q_1 < x

whenever

32

I.

PARTIALLY ORDERED SETS AND SYSTEMS

x>-1;

q2 < 2, q2 > y

whenever

and let

S2

1

consist of

y < 2.

alone.

Then

Let

S1

consist of all

x < 1,

V S 1 - V S 2 = 1, V S =» q2 < 1 + 1,

A s _ = q1 > - 1 • (6 .9 )

COROLLARY.

If

P

is a Dedekind-complete par­

tially ordered group, the operation ous on P)

P x F

is also

to

P,

+

is

and the function

o-continu-

(-x : x

in

o-continuous.

This follows at once from (5-7)* $ 7*

LINEAR SYSTEMS, MULTIPLICATIVE SYSTEMS AND ALGEBRAS WITHIN PARTIALLY ORDERED SETS

As usual, by introducing an operation of "scalar multiplication" in a commutative group we produce a "linear system."

We shall always use

the real numbers for our scalars. (7-1)

DEFINITION.

A system consisting of a set

objects, a binary operation

+

P

of

and an operation of

scalar multiplication, Is a linear system If

(F,+)

is

a commutative group, and also: (

i)

To each real

a

and each

a

in

P,

there

corresponds a unique scalar product, de­ noted by

a a or

act; and, for all real

a,p and all a,b (a + p) a = ota + p& ,

numbers ( ii) (Iii) ( (

a(a + b)

iv) a (/3a) =

in

P,

= ota + ctb , (a/3)a ,

v) 1a = a .

It follows readily that

(-1)a = -a

and

Oa » $

for all

a

F. To define a linear system embedded in a partially ordered set, It remains to relate the operations to the order-relation. (7 .2 )

DEFINITION. A linear system F+ is a linear system embedded in a partially ordered set F If (a) (F+ ,+) (b) if a

a y_ 0 If

F+

is a group embedded in P, and is in F+ , and ot is real, and and

a > 0,

coincides with

then P,

act > 0 .

we say that

P

is

in

7-

ALGEBRAS IN PARTIALLY ORDERED SETS

33

a partially ordered linear system. The following theorem is an obvious corollary. (7 .3 )

THEOREM. system, and and

a

F

is a partially ordered linear

is a subset of

F

having a supremum,

is a positive number, then the set

Goia: a

in

{-aa: a (7 A)

If S

S)

in

has supremum

S)

THEOREM.

«(VS),

has infimum Let

F

and the set

-cc(VS) .

be a Dedekind

of

F.

-complete par­

a

tially ordered linear system, and let

a

be a member

Then the following three statements are

equivalent. (

i)

o-lim n ^ a

exists,

n —* 00

( ii)

o-lim n ”1a n — > 00

(iii)

a

exists and is equal to

0

and

0,

have a common upper or lower

bound. If (i) holds, by (3-1), (3-9) and (7-3) we have o-lim n ”1a = o-lim [(2 n)” 1 a] = o-lim n" 1 (a/2 ) = 2 ~ 1 [o-lim n“ 1 a] , so (ii)

holds. If (ii) holds,

let M,N

(3-1)-

Let x

let n

integer

such that

0 < nx,

establishing (ill).

be in

N,

and

n “1a ,

V

or

A,

supremum, it shall be with reference to

or

thewords

>, never

». POSTULATES. (a) F is a [ a ]-complete and infinitely distrib­ utive lattice under the partial ordering >. (b)

»

(c) G

is a strengthening of

>

(cf. (1 •3 )) •

is a Dedekind complete partially ordered

set, such that for each two elements 36

with

of

G,

in

G

8.

THE POSTULATES

if

g1

and

g2

37

have an upper bound

they also have a lower bound in

G,

and vice versa. (d)

IQ

is an isotone function whose domain is a

subset (e)

E

of

tained in

G•

E

[countable] subsets

directed by

having

», «

V S 1 > A S 2,

V I q CS.j )

and

inequality

A l Q (S2)

VIqO

respectively and

and such also that

j)

exist in

> A i 0 (S2)

G,

the

holds.

If e1 and e2 are in E, and I0 (e^) and IQ (e2) have a common upper or lower bound in G,

there exist elements

such that (g)

and whose range is con­

For each pair of

(f)

F

e* «

e^ «

e'

and

e”

of

E

e", i = 1, 2.

If 6*1^2 and e3 are in and IQ (ei) and IQ (e 2 ) Lave a common upper or lower bound in G, then for every f in F such that E

f »

mid(e1,e2,e^)

such that

f »

there is an

e

e >> mid(e1 ,e2,e^);

in

and

dually. REMARK. (6 .7 ) •

(8.1c) holds when

It clearly holds if

particular when

G

G

I

is a partially ordered group, by

has a greatest and a least element, in

is a complete lattice.

We now list some easy but useful consequences of the postulates. (8.2)

THEOREM. (a) (b)

If (8.1

for each

e

[.*)) holds: In E,

such that

there exist

in

E

e •«

e «

if

e 1, ..., en are elements of

e*

and

E

such

e”

e";

that each pair ^ 0 ^ei-1 ^ *^o^eI ^ 1x8,8 an upper or a lower bound, there exist e* and e"

In

(1=1, (c)

If

E

such that

e* «

e^ «

e”

..., n);

e1, e.j 1, ..., en , e*

such that the

I0 (©^)

are elements of

E

have an upper or lower

bound, and e^ > e ^ »

and the set of three elements IQ (e2)> also true. For (a) holds for

we take

n = 2.

is bonded.

e1 = e2 = e

I (e^),

The dual is

in (8.if).

Assume it true for

e^

e^, i = 1, 2;

By

(8.1 f ) statement(b)

n =*k - 1, and let

e1, ..., ek

satisfy the requirements of (b) . By hypothesis there are elements e" 1

such that

ments and

e2 *,e2”

e^ «

< < 6 ’^ (i - 1, ..., k-1),

such that

e2 *

and

I0 (©2 ),

(8.1 g) holds:

LEMMA.

(8.3)

If (8.1 [ e2; S* being directed, there is an element e* in S* such that e* » e 1 1 and er »

ed *;

and finally there

is an

e in SQ131Y ^ such that Up

e »

e* . The

proofs of the other statements in (b) are similar. (1 0 .3 )

Let (8.1[a]) hold.

COROLLARY. are summable and

U-elements, and

S2,..., Sn

are associated with

respectively, there exists a set u1

such that if

The set

is an

e »

e± , i = 2, ..., n.

Let

S1

(10. 10

u 1.

e-j_in

exists aset e*

Let

S in

e

is

associatedwith S

If

u

is an upper

and (10.1) applies.

of elements Then

... «

formed by (10.1) from

S1

and

e* »

e 1,

\/S2 < u. S2;

then

u

is a sum­

e «

u,

there

and such that for

be any set associated with S2

e.

u.

By repeated use of

e2,...

of

Let

be the set

e1 »

S e

E

for all

such that e*

in

S,

V ( V S 1 , V S 2 ) = u.

(10.5)

THEOREM. summable

Let (8.1 [cr]) hold.

U-elements and

summable For By (8.1c) and

each

If

u^, . .., uR

I1 (u^), . .., 1 ^ (un )

common upper or lower bound,

which is

I 1(u1)

and

it is true that

u.

VS =

Then

in E

(8.2d) we select a set

and

such that

i = 1, ..., n,

e «

e2«

associated with

Let (8.1 [a]) hold.

COROLLARY.

e 1«

u 2, ..., un

has the desired properties.

mableU-element and every

..., uR

in S2, ..., SR

e in S

be associated with

I0 (e^) S = SSUp

S

e2, ..., en are

respectively, there

bound for

If

u 1 > u^ (i = 2, ..., n ) ,

are

have a

A (u1, ..., un ) is a

U-element. u •,

let

S(uj)

be associated with

, ..., I-| (un )

(6.7), the elements

also an -upper bound for

IQ (ej), e.

in

S

u •, j = 1, ..., n. have an upper bound,

j= 1, . .., n.

So

construction (10.1) can be applied to define Sinf. By (10.2), V S irrp = A (u1,..., un ), so this is a U-element. Being < u 1, it is summable.

(1 0 .6 )

THEOREM.

Let

K

be a [countable] collection of

kk

II. summable

DEFINITION OF THE MAPPING

U-elements such that for each pair

elements of

K, I 1(u1)

and

upper or lower bound. For each finite subset that

u

in

K,

u 1, ..., uR

I1(u2)

Then let

VK

is a

S(u)

of

U-element.

be associated with

of elements of

I1(u1), ..., I1(un ) have

u^,u2

have a common

K,

u.

For each

by (8.1c) and (6.7) we see

an upper bound.

So we can apply (10.1)

to

the sets

S(u.), ..., S(uII) to construct Sa___. I o Lip Let S be the union of all the sets S3Up thus constructed. [Then S is countable]. If e^ and

e1 *

are in

S,

we can find two finite sets

u 1, ..., un

and

U 1 *9 ^m* elemerrts of K and elements e 1, ..., em * of ..., S(‘um r) respectively such that e « V l e ^ •••> en ) and e* « S same element as now

S

em ,^‘ ^ an^ U 1 ^-s also an u j ,, we can clloose the e^ and e j f> since the S[u] are directed by » . But

contains

directed by» . VS(u) = u.

e*In

S

e

in

(10.7)

e 1, . .., eR , e ^ 1, ..., em f,

Is a

such that S

eT »

e, K,

there are sets Hence

the U-element THEOREM.

U-element.

u in

of those sets such that

V {u1, ..., un ) < VK. is equal to

e" »

VS

This holds for all

hand, for each ei, ..., eR

an element Therefore

there is an

S(u1),

If

e

so

is in

so

V S > e,

so

V S > VK.

whence

S

is

S(u), VS >

On the other

S(u1), ..., S(un )and members

e «

V S < VK.

V ( e 1, ..., eR ) < This proves equality,so

VK

VS. Let

K

be a [countable] collection of

summable

U-elements directed by

summable

U-element if and only If the set I1(K)

an upper bound in

G;

>.

Then V K

and in that case

is a has

I1 (V K ) =

(K) .

I 1 (u), u

If

VK

in

K.

is summable,

I^VK)

is obviously an upper bound for

So I^VK)

:> Vlj(K)

.

Conversely, suppose there is an upper bound forI1(K) . Using the of (10.6),we construct the set S. For each e in S there are

notation elements

u v ..., un of K such that e V(u.j, ..., . Hence e « u, and IQ (e) < I1 (u) < V l 1 (K) . VS Is a summable U-element, and also

Then by definition

I1 (VK) = I 1 (VS) = V l I Q (e): e < Vl^K)

.

in

S)

VK =

11. DEFINITION OF THE MAPPING OR INTEGRAL

^5

The reverse' inequality was established above, so equality holds. (10.8)

THEOREM. are

Let (8. [cr]) hold.

U-elements and

mid (u1,u2 ,u^)

u 1 < u2 ,

If

and

is a summable

u 1, u 2

u2

and

u3

is summable,

U-element.

Let S 1 and be sets associated with u 1 and By (10.3), there is a [countable] subset S2 of E

tively. X>

having

V S 2 = u2 ,

e2

in

for which

S2

and such that for each e2 »

e1 .

So

summable

mid (u1,u2,u3)

in

S1

there is an

Then by (10.1) we can construct

This is [countable and] directed by (10.2).

e1

u^ respec­ directed by

is a

»,

and

V S mld = mid ( u ^ u ^ u ^ )

U-element.

Since it is

< u 2,

by

it is a

U-element. $ 1 1 . DEFINITION OF THE MAPPING OR INTEGRAL

Again we shall assume (8.1) or (8.1a) to be satisfied. frequently need to refer to the set of all summable given

f

of

F,

We shall

U-elements above a

so we shall provide a symbol for this set, and also for

its dual. (11.1)

DEFINITION. of all summable L[< f]

LEMMA. L[< f] form

isin

F, U [>f]

u

is the class

satisfying

is the class of all summable

satisfying (11.2)

If f

U-elements

u > f,

and

1

L-elements

1 < f. Let (8.1 [cr]) hold.

If

f

is in

is not empty, the set of elements of 1^ (u) (u

in

U[> f])

is directed by

F

and

G

f]

suchthat

1^1) < g-|,g2 * clearly

g1

> f,

and

g2 are in the class, there areelements

g 1 = I1 (u1)

So by (10.5), so it is in

and

g2 = I1 (u2) . For

u 1 A u2

U[> f ]•

both

any

1

in

(i = 1 , 2) .

DEFINITION. If f is an element of F such that U[> f] and L[< f] are non-empty, we define the

upper and lower integrals [or images] of respective formulas

f

of

L[< f],

U-element, and it is

Also, by (9 -5 )

I 1 (u1 A u 2) < I1 (u±) = (11 .3)

is a summable

u^,u2

by the

k6

II.

DEFINITION OF THE MAPPING 1(f) = Al,(U[^ f]) , 1(f) = VI, (L[^ f])

These surely exist; for if L[< f],

-chen

I^u*)

lover bound for

ur

•

is in

is an upper bound for

I, (U[> f] ),

U[) f]

1'

and

I1 (L[^ f])

and

in

I (lf)

is a

and by (11.2) and (8.1c) the right members of

the above equations exist. If former Is Thus

IT[> f]

>

I 1 (u)

L[< f] 1

I 1 (u) > 1 (f),

Then

so must be
f]),

and

each element

I1 (L[< ?]) > and

its infimum,

1(f)

of the

must be

>

its

1 (f)

isa lower bound for

1 (f).

So:

Let (8.1 [a]) hold.

and

u

I1 (u) > 1 ^(1 ).

If

f

is in F,

exist, they satisfy the relation

1(f) < 1(f). Another corollary of the definition is the following: (1 1 .5 )

Let (8.1 [a]) hold.

THEOREM. in

P

If

f1

and

fg

are

and both have upper and lower integrals, and

f 1 < f2 ,

then

T ( f . ) < T ( f 2)

For then

U[)> f 1 ] ^ U Q

f2],

and so

K f , ) < Itf,).

I1 (U[^ f 1 ]) => I1 (U[^ fg]),

the former set has the smaller infimum, that is

Iff^ < I(f2) .

The other

conclusion is established analogously. (11.6)

DEFINITION. ment of

F

Let (8.1[a]) hold.

for which

1 (f)

and

If

1 (f)

f

is an ele­

are defined and

equal, we define the integral, or image, of 1 (f) = 1 (f) = 1 (f).

In this case

f

f

to

be

is called a sum­

mable element. We have already used the adjective "summable" in (9 -1 ); so we shall show that there is no contradiction between the two uses. (1 1 .7 ) is a

THEOREM. Let (8.1 [a]) hold. If f is in U- or L-element, it is a summable U- or

P and L-ele­

ment as defined in (9*1) if and only if it is summable as defined in (11.6). In this case 1 (f) = I.,(f). We consider

U-elements only; the case of

and

L-elements can be

11. DEFINITION OF THE MAPPING OR INTEGRAL discussed dually.

Suppose first that

sense of (11.6). exists a For all Thus

U-element e

in

I0 (S)

f is

Then there is a set u

a

l*-7

U-elementsummable in the

S associated

with f,

and there

summable in the sense of (9 -1 ), such that

S we have

e «

VS = f
f] contains

f

is a summable f.

L-element itself, so

in

S, by (9 *7 ) there is a summable

so

I0 (eQ ) < V l ^ L K

f]) = 1 (f).

eQis in

«

eQ ,

f

1

then

L[ 1Q(e^ + I0 ^e2 ^ whence I 1(u1 V u 2) + 11(u1 A u 2) 2, (u i) + I-| (u2) . This completes the

§ 1 3 - LATTICE PROPERTIES OF THE CLASS OF SUMMABLE ELEMENTS

Postulates (1 2 .1 [a]) are not sufficient to cause the class of summable elements to be a sublattice of

F.

However, we can establish some

closely related properties. (1 3 .1 )

THEOREM. Let (12. 1 [

f^

are summable and that

U-elements such that also a summable

mid (u1A u 2,u2 ,u^)

(10.8),

f 1.By

and

u^ ^ f^,

U-element, and

is a summable 1^

and it

is clearly > mid (f1,f2 ,f^).

i = 1,

2, 3 , mid (1 ^,1 A.t ) I I, V L C J is a summable L-element and is > Thus the upper and lower integrals of mid (f1 ,f2,f

mid

Likewise, if

U-element,

is in

L[
f 1 ]

I1 (u^)

and

and

u2

in

U[^ f2 ].

so by (1 2 .4 )

I1(u2),

By the preceding proof,

f 1 V f2 **

f 1 A f2 is summable.

Similarly,

be any three summable elements such lower or upper bound. Then

f 1 V f*2

f1 A

f*2 are summakie > and first part of this proof mid (f^ A f2 ,f1 V ?2’f^ is summable. By (2.13), this is the same as mid (f1,f2,f3) . (13.2)

THEOREM. and

f2

and

I(f2)

f 1 A f2

Let (12.1[a]) and (12.7) hold.

are summable elements of

F

have a common bound, then

such that f ] V f2

are s-ummable, and

Kf,V f2)+ i(f1 A f2)= i(f1)+ i(f2)

If

f1 K f^)

and

that and

14.

We already know that ^

belong to

U[^ f ^ ,

U [ > f 1 V f 2i

and

f 1 V f2

I = 1, 2.

u,Au2

in

f 1 A f2

and

Then by (12.8),

A f 2l,

U[) f,

I(f, V f2 ) + I(f, A Since

55

MONOTONE SEQUENCES OF SUMMABLE ELEMENTS

f 2)

is an arbitrary member of

are summable. V

Let

Is In

and

< I,(u,) + I,(u2) .

U[^ f^],

this implies

I(f, V f 2) + I(fl A f2) < I(fl) + I(f2) ' The reverse inequality is obtained by considering elements of i = i, 2,

L[< f^],

which completes the proof. As an aid in verifying one of the hypotheses in the two preceding

theorems, we establish a lemma. (13.3)

LEMMA. f1

If

andf2

E

is directed by

are summable, then

>

or by

Kf^)

and


f^] e

and

e^ «

u^

satisfying either

e < e^ (i = l, 2).

Hence

lower bound in

by (6.7) they have a common upper bound.

G;

IQ (e1)

and

e = 1, 2.

I0 (e2 )

consecutive elements of the finite sequence I^Ug), I(f2)

has an upper bound in

G,

By hypothesis,

e > e^ (e - 1, 2)

or

have a common upper or Each pair of

I(f.,), I 1 (u1), IQ (e1), IQ (e2)>

so by (6.7) all six have a common

upper and lower bound. 4 14 .

MONOTONE SEQUENCES OP SUMMABLE ELEMENTS

In order to establish convergence theorems of adequate strength, we apparently need

to add something to postulates (12.1).

Theconvergence

theorems will rest on theorems concerning monotone sequences, and in this section we establish two such theorems.

The first is of less general

applicability, but is not entirely without interest. Our first strengthening of (12.1) is as follows. (1L.1)

POSTULATE.

G

is-a group under

+

with the

property that whenever S 1 , S2, ... are countably many subsets of G directed by < and having A s ^ = 0 for

1=1,

2, ...,

it is possible to choose elements

g-l > g2> ••• such that g^ is in Si and the sums g 1 + ... + g^ (k = 1, 2, . ..) have an upper bound.

56

III.

CONVERGENCE AND MEASURABILITY

For example, the reals have this property; but the group bounded

real functions on an infinite domain

D

of

lacks the property, as we

see if we take each S^ to consist of all the functions which vanish on finite subsets of D and are 1 on the rest of D. (14.2)

COROLLARY.

and (1 4 .1 ) hold.

Let (12.1[a])

S^

S2, ...

by




fQ],

and so

Sj + 8 •

J

By (1 4 .2 ) and (6 .8 ),

T(f0 ) i g • On the other hand, since I(fn ) < I(fQ ) , Hence

so

f

< f

for all

n,

g = lim I(fn ) < I(fQ ) .

is summable, and I(f0 ) = g = lim • The somewhat restrictive requirement (1^.1 ) can be replaced by the much weaker assumption of normality of G (cf. (5 *9 )), provided that we strengthen the hypotheses on the sequence (fn : n = 1 , 2 , ...): (A.M

fQ

THEOREM.

Let (12-1 [cr]) hold, and let

G

be

a

58

III. normal.

CONVERGENCE AND MEASURABILITY'

Let

f 1,f2, ...

be an Isotone sequence of

summable elements such that there exists a summable U-element Let

f

u

1(f) =

which satisfies

11m fn . n — >oo

=

llm

Then

u > fn , n = 1, 2, ... .

fQ

Is summable, and

I(fn).

n —> oo As before, we define _ 1(f)

g

are defined, and by (11-5) For each

U

fn ] and

l'n

to be

llm fn n — > oo 1(f) > g-

n,

the elements

in

L[ f ] SI II

1* II

and

in L[< f„] 2S. H

so that

(*)

CMu'O

J-1

- I1(11_j)] + g) < r(g) + €

J

for all positive integers

n.

J Without loss of generality we may assume

u *n ^ u **or each n > since we may replace u*n by u'n A u . Next we define un - Vtu'j, u'n ), ln - Vll'j, ..., l'n JThen

uR

is a

U-element by (10.6), and is

summable because it is < u;

and

1

^

fn < un (n = 1’ 2 ’ ••’J* As in the proof of (1U.2), by induction we prove

is asummable

W Since

r

S

2

L-element by the

;

,

dual of(10.5).

[Ii(u’n> -

is isotone and

+ W

■

this implies

^

r(I l(un ))
Uj > f . for each

since fore in

U[^ fQ ] .

Since

r

j,

59

It is summable, being V nun ^ V f j = fQ ,

< u;

and

and it is there­

is smoothly rising and consequently Is

o-continuous, r ( I , < V n»n »

- V nr(I,(»n )) < r(g) + € •

But

V nun

is in

U[> fQ ],

so

1 ^ V nun ) ^ I(f0),

and

r(I(f0)) < r(g) + € .

e

Since

Is arbitrary,

r(I(f)) £ r(g) . Here

so by the normality of G (cf. (5 *9 )) previously established inequality 1(f)

r

Is any member of

R^,

1 (f) < g. With (11A) and the > g, this implies 1(f) = 1(f) = g,

and the proof is complete. It would seem reasonable to expect that the hypothesis of the existence of a summable have an upper bound in

u > f G.”

by the following example. [0,1],

G

F

could be weakened to read "the

consists of all extended-real functions on

of all (finite) real functions on

functions on

[0,1].

is the same as

>.

The partial order > IQ

I(fn )

The impossibility of such a weakening is shown [0,1],

E

of all continuous

has the usual meaning,

is the identical mapping.

We find that

and u is

» a

U-element If and only if it is lower semi-continuous and bounded below, and dually for

L-elements; a

U- or

L-element is summable If and only if it

is finite-valued. Now let the rationals In sequence

t1, t2, ... ,

1 = 1 , ...,

and to be

and let 0

f

elsewhere.

[0,1]

be arranged in a

be defined to equal Each

f

I

is a summable

at

t^,

L-function,

and I(fn ) = “ ^n * The supremum of I(fn ) is the element f, defined by the equation f(t^) = I (i = 1, 2, ... ), f(t) = 0 elsewhere. But ment

V nfn

is not summable.

u > V nfn -

This

u

If It were, there would be a summable

U-ele­

Is lower semi-continuous and finite-valued, but

Is unbounded on every interval.

This contradicts a well-known property of

semi-continuous functions. § 15.

DOMINATED CONVERGENCE

The results in the preceding section allow us to establish a generalization of the Lebesgue dominated-convergence theorem. (1 5 -1 )

THEOREM. Let (12.1[a]) hold, and let G be normal. Let (f(a):a in A) be a net of elements

60

III. and

f

CONVERGENCE AND MEASURABILITY

an element of

A

with the following

properties. (

i)

For each

C il)

a

in

A, f(a)

is summable.

There exist summable elements F

such that

In (iii)

f 1 < f(ot) < f"

f^f"

of

for all

a

A.

The net

(f(a):a

vergent to

In

A)

is

I0 (eQ ).

e* » If

eQ ) e*

is di­

Is in

S',

15. by (9 -7 ) there is

a

DOMINATED CONVERGENCE

U-element

u*

such that

63

eQ «

u» «

e* . Then

£ I i(u ^ i Io (e')' 30 T(eo) ^ V e>)- 811(1 because e * is an arbitrary number of S 1, i(e0) ^ A l 0 (S*)« Dually, if S is the set {e: e

in

E

and

e «

eQ ),

then

I(eQ ) ^ V l Q (S) " A I

q

CS1),

V l 0 (S) < I0 (e0 ) • These inequalities establish the conclusion.

and also If

»

is

reflexive, both S and S* contain eQ , so A l 0 (S') = ^o^o^ = V l Q (S), and as just proved this establishes that I(©Q ) exists and is equal to w

• We now return to sequences of functions, and establish some easy

theorems, the last of which is a generalization of Fatou's lemma. (15.7)

THEOREM. Let (12.1 [a]) or let (1^.1 )hold.

If

hold.

S

i(As) = A

k s

f*,

then

^

and

As

1, 2, ...),

the conclusion holds by (15-2).

such that

A nfn = A s .

THEOREM. Let (12.1 [ fj > f», (j *

Let

G be

normal,

is a countable set of

sum­

>

f*,

a summable

element

is summable.

S*

are summable.

S*

f, f 1, f2 , f^, ....

such that

fR > f (n =

by

By (13.1)

f2 > f^ > ...

S

As

S

).

By (2 .^0 , there is a sequence

then

normal,

is summable, and

elements of

(1 5 *8 )

G be

is a countable set of

summable elements directed by mable lower bound

Let

Then

fR )< g*;

lim inf fn

is sum­

in particular, if

then I(lim inf fR )
m

we have

Kf-^) < g»,

and

6k

III.

moreover f*21 ^

CONVERGENCE AND MEASURABILITY’

I(fm 9) ^ lim inf I(ij)

•••f ty (1^. 3 )

if* this last exists.

V m?m ' is summable, and

and KN/jn^jn') < lim infn I(f*n ) so the proof is complete. §16.

Since

K V / ^ )

if this exists.

f1* ^

= V ^ K ^ ' ) ^ g»,

V m?m ' “ lim inf fj,

But

STRUCTURE OF THE SET OF SUMMABLE ELEMENTS

The information about summable elements obtained In the preceding sections can be combined into a theorem on the structure of the set of summable elements of

F.

Recalling the discussion following (6.7), we note

that each element

of

G

all elements

g*

g

determines a subset

such that

g

and

g*

GCT of G consisting of o have a common upper or lower

bound; two such classes are either disjoint or identical, and each is in fact a coset relative to the invariant subgroup

G^•

Correspondingly, we

adopt a notation. (16.1)

DEFINITION.

Let (8.1[a]) hold.

F gun]

to be the set of all summable elements of g

in

G,

F

such that (16.2)

For each

is the set of all summable elements

1(f)

t THEOREM. Then

is defined F.

is in

f

G . o

Let (8.1 [cr]) hold and let

G

be normal.

F gum is the union of disjoint classes

F^

with

the following properties. (

i)

If

f,ff belong to distinct classes

F ,,

the pair

{f,f *3

upper bound nor a lower bound in ( II) (Hi)

F ,

has neither an F gum.

F sum is Dedekind o -complete. Each class Fg is a conditionally

-com­

plete lattice. (

iv) If (1 5 *^) holds, for all IQ (e)

( (

v) I

Is a

e

in

If and

f

I(f')

F_,l rri contains oUIu the equation

E,

(t o -

00 *-continuous

continuous and

map of F gum into G. vi) If E is directed by > or by is only one non-empty class F^,

and

f* F . o

f" ].

changes, shows that

mid

= mid (f^f" *niid (f1,f2 ,f^)).

(13-1) and (17*3) it is sufficient to prove that

So by

mid (f1,f2,f^)

is

summable. Let

S ^

(i = 1, 2, 3)

the first directed by

«

be three [countable] subsets of

and the other two by

»,

such that

E,

A s ^

=■

f -, V S (2^ = fQ, V S (3) = f,. For each pair e ^ and of elements i /j \ fi 1 (i) of S^ ; such that e; ; « e^ ;, we choose exactly one U-element and (!) (i) exactly one L-element between e : ' and e;, this is possible by (9 *7 ) (i) The chosen U-elements form the set S^j ' : the chosen L-elements form the set and

S-^ .

u, A u 0

(2.13),

If

u 1 ,u2

S^1\ S^2^ respectively,

are in

are summable

U-elements by (12.4 ) and (10.5).

mid ( u ^ u ^ u ^ ) is a summable

U-element whenever

(i = 1 , 2, 3 )-

Likewise, whenever

1^

mid (l^lo,!,)

and

are summable

Let tively.

mid (f-,l0,lx)

3

i

is in

(i = 1 , 2, 3), both L-elements.

("i)

X 1,l2,l^ be arbitrary members of

(2)

, S^

S ^

(i = 1, 2, 3)

exist elements

^ 2^ 3

wlth

such thatu 1 < X 1 ,u2 > l ^ u ^ > I3 X^

in

S^1 ^ such that

respec­

< f2 ^ 1 2 ,

Since

it follows that

mid (fn,1 2,1 ^)

I 1(f2),I1(f1),I1(mid is in

and

L[^ mid

f2, so the ele­

f 1 £ mid ( f ^ f g , ^ ) , (f 1 ,f2, f ],

)> f 1 ,

by the distributive

law

[mid (f 1,x2,x^) ]\/x1 = [(f, V

X 2)

.

A(f,V ^ 3)/\ (X2V ^ J I V X ,

= (X, v V A t x ,

V * 3M

> mid (X1,X2 ,X5) . Hence, using (12.6),

G.

so

and

I1 [mid ( f ^ l g , ^ ) V f ^ £ I[mid ( f ^ f g , ^ ) ] X

Then

.

have a common bound in

mid (f1,l2 ,l5) £ mid (f 1 ,f2,f^) [mid (f 1 ,12,15) ] V f 1

and- there

X^ > ui -

mid ( f ^ l g , ^ ) < mid ( u ^ ^ , ^ ) < mid ( X ^ X ^ X ^ )

Since

(i)

By the manner of definition of these sets, there exist elements

in

ments

u.j V u2

By (10.8) and (i) u^ is in S^ '

(xi \ A 2 \ A 3)

Also,

17 • MEASURABLE ELEMENTS

67

I 1 [mid ( u ^ u ^ ) ] < I 1 [mid (X1,X2 A 3 )1 < I, [mid (f,,X2 ,X3)yX,] < I 1 [mid ( f ^ X g . ^ J V f , ] + I^X,)

- I, (f,)

< I [mid (f,,f2,f5 )] + I,(X,) - I,

(mid ( u ^ U g , ^ ) : u2

and has

mid ( u ^ f g , ^ )

ln

(2)

Sg }

as supremum.

and

u^

ln

Hence by

( 1 0 .6 )

I1 (mid (u1 ,f2,f3)) < I[mid ( f ^ f g , ^ ) ] + “[I1(X1) But

mid(u1,f2,f3)

is

ln

.

U[> mid ( f ^ f g , ^ ) ] , hence

I (mid (fv f2 ,f3)) < I[mid ( f ^ f g , ^ ) ] + [I1 (X,) - I1 (f 1) ] . The set of values of the quantity in brackets corresponding to all S^^

has infimum

0,

I [mid ( f ^ f g , ^ ) ] < I [mid ( f ^ f g , ^ ) ] This proves that

I[mid ( f ^ f g , ^ ) ]

in

If

(f(o)sa

is a net of measurable elements and is convergent to a limit

.

exists, and the proof is complete.

THEOREM. Let (12.1 [a] hold.

(17-6)

X1

so by (6 .9 )

fQ

in

F,

in

>' ,E' ,G' ,I0 ‘)

(P, » ,

E,G,IQ )

and

be two systems satisfying (8.1 [

is an

fl-summablllty may be ignored, because

subset by using only satisfy

I*-summable

I-summable element of

^ ( u f)

greatest element and all U*[f*]

V ^ 0 (S')

- \Zir\IQ '(e1)): e*

* Vl^^CS'))

r(I ^1(u* 1)) . The dual holds for Now let f*

SO

71

I(J>(f))

and

I(4 (f'))

I(£(f *)) ^

I*(f*) is

also in

r(I* (f •)) .

It follows

both exist and are equal to

and (iii) is established. Finally, suppose that

9Q

I — summable and is ln the domain of U-element such that directed by »

u* ^ f 1,

andhaving

and the set consisting

off'

is smoothly rising and that

,Qq . Let

and let

S'

u*

be any

f•

is

I — summable

be a [countable] subset of

V S ' = u * . Because alone is directed by

issmoothly " ^i(u ') • r ^Cie discussion in the preceding paragraph remains valid with 0o (f') in place of $(f')> and shows that K t f ^ f 1)) ^ r ’d'ff*)) and

I(^^Tf*)) ^r(I'(f')).

r( 1 1(f ')).

Hence

$Q (f)

is

I-summable, and

I($0 (f) *

This completes the proof.

h.

summable. Let

e2 in S2

-

g < h,

U-element

e1 in S 1

(-mid

,f))]

/ \ [ \ / mid ( e - ^epSf)] -e2 * in S2 -e1 1 in S1

= - mid (-l,-u,f) , which is summable by (19-3) and the definition of measurability. (21.2)

THEOREM. satisfied. longs to

Let the hypotheses of Theorem

Let

f

(19-1*) be

be a measurable element which be­

F+ ,and let

c

be a real

number.

Then cf

is measurable. As in the preceding proof, we need only show that is summable when and

u > 1.

Let

S^,S2

1

is a summable

(c^e: ein

L-element which we call

mid

u

mid (l,u,cf)

a summable

Also, by virtue of (21 .1) it is enough to consider -i be associated with l,u respectively, and let c

set we call

L-element and

u_, c

S^)

= 1, 2 ).

(e

lc , and

Then

V c ~ 1S2

U-element c > 0. be the

A c ” 1S l is a summable

is a summable U-.element which

and

(l,u,cf) =

\ /

[

e2 in S2

mid (e.,e2,cf)]

/ \

e1 in S 1

\ / 1 t / \ e2 • in c S2 e 1• in c = c

\ / e2 * in c

1 mid (ce- • ,cep »,cf)]

S1

[ S2

mid (e»,e »,f)] e^

in c

S1

- c mid (lc ,uc ,f) . This last is summable by (1 9 .^). To establish the measurability of the sum of measurable elements

21 .

OPERATIONS ON MEASURABLE ELEMENTS

79

it seems at least desirable and probably necessary to add a hypothesis, assuming a strengthened form of normality for

P.

Let us first introduce

a definition.

(21 . 3 )

DEFINITION.

If

tially ordered set

F+

P,

is a group embedded in a par­

Rp +

is defined to be the set

of all smoothly rising extended real-valued functions on

P

which are finite on

tion

are in (21 .10

F+

and satisfy the equa­

r(f1 + f2) * r(f.j) + r(f2)

whenever

f1

and

f2

F+ .

THEOREM.

Let the hypotheses of (19-M*]) hold.

Assume further ( i)

if

f.j

P,

there exists a function

and

such that (ii)

if

f

F+ ,

so is

r f

are distinct elements of

r(f1) ^ r(f2),

is in

all Then whenever

f2

in

P

and

Rp + ,

and

g

r(f)

then

r

in

f

is finite for is in

u

f + g.

a summable

mid (l,u,f + g) 1

and

u

U-element and

is summable.

respectively. Let

e * - ( e " - e ') - ... - (e " o 1 1 m S* and e^', ..., em "are in of

the form

are in

S”

e0” + (e^1 - e ^ ) and

e^,

..., en *

We easily verify that and that mable in

1^ s A & m *

U-element.

SM ,

for each

Let

S'

Rp +

and

SM

1

is a summable

the element

be sets associated with

- e •), where e •, e » , ...# , e • are in m 9 o 1 7 m S"; let Sn" be the set of all elements + (enfl - en *)> where are in Sm *

is a summable in

1 < u,

Sm ' be the set of all elements of the form

e0", e ^ 1, ..., en"

S f.

is directed by

« and

L-element and s V s nM

Also', if eQ ', r

F+ .

are measurable members of

As before, it is enough to show that when L-element and

R„ ^ r ,+

and

em '

are in

S*

and

Sn" by

»,

is asume^n ,

..., em"

we have

r(e0 » - [e1M -©-,•] - ... - [em” - em #]) - r(eQ ) - r ( e }") + r(e1 ') - ... - r(em ") + r(em t) , and since

r

is smoothly rising this implies that r d m ) - r(l) - m[r(u) - r(l)] ,

where

r(u) - r(l)

Likewise

is

^ o and may be

r(un ) » r(u) + n[r(u) - r(l)]

+ »,

in which case

r(lm ) * - ».

80

IV. We now define

r

ALGEBRAIC OPERATIONS

fm>n = mid ( l ^ u ^ f ) ,

- mid ( l ^ u ^ g ) .

Since

is isotone,

p(fm,n) and analogously for r(ln )

be

+ oo,

hypothesis,

n ) • But

and r(f)

f ■

Since

is in P

For

r

F+ ; andsimilarly, so is

and

n

lim ( lim [r(fm n ) n —> oo m — > oo 9

r(fm

n-

+ rfg^ n )]) . 9

If

r(u) - r(l) > 0,

r(un )

> r(g) > r(ln ) ,

for all

we have

r(uR ) > r(f) > r(ln ) so

By

we have

We now distinguish two cases. m

nor can

lim ( lJjn t f ^ n + g ^ n l ) n —> oo m —>oo 9

+,

r(s) -

large

- oo,

r(fm n ) isfinite.

is a complete lattice, the limit

in Rp,

(*)

r(un ) can never be

is finite, so

s =

exists.

(!*(!„,)'r(um )'r(f 5} ’

n)

=

r(f)

and and

* ( 8 ^ ) = r(g) •

Thenby (*)

we

r(s) - r(f) + r(g) - r(f + g) , whence (**)

r(mid (l,u,s)) - r(mid (l,u,f + g)) .

If ©n the other hand

r(u) - r(l) * 0,

all real numbers

and (**) continues to hold.

r

in

R™

r ,+

.

z,

then

mid (r(u) ,r(l) ,z) - r(u)

for

Thus (**) holds for all

whence mid (l,u,s) = mid (l,u,f + g) .

But lim 11m mid (l,u,f m — > oo n —> oo 9

+

) 9

- mid (l,u,s) , and the elements and

u.

mid (l,u,fm n + ^

Hence by (15*1)

summable, and

f + g

mid (l,u,s)

is measurable*

n)

are summable and are between

is summable; so

mid (l,u,f + g)

1 is

find

C H A P T E R

V

REAL-VALUED FUNCTIONS

$ 22.

INTEGRALS OF REAL-VALUED FUNCTIONS

An important special case of the preceding theory is that in which and

F

F+

consists of all extended-real-valued functions on a domain consists of all (finite) real-valued functions on

T,

the defini­

tion of order, addition and multiplication being the standard ones. sets

F

and

F+

satisfy all the requirements placed on sets

in preceding theorems, so (with appropriate hypotheses on all preceding'theorems are available. G

F

G, E, IQ ,

G.

Accordingly, we assume the following

postulates. POSTULATES. (a)

F

consists of all extended-real-valued func­

tions on a domain on t (b)

F+

F, f 1 ^ f 2 in

T;

means

and for all

f 1, f2

fj(t) > f2 (t)

for all

T.

consists of all (finite-) real-valued

functions on F+ , fj + f2

T;

if

f1

and

f2

are in

is the function

(f1(t) + f2 (t): t

in

T),

andscalar

multiplication and binary multiplication are analogously defined(c)

»

(d)

G

is a strengthening of

(e)

ordered linear system. IQ is an isotone function whose

is a normal Dedekind-complete partially

subset E tained in (f)

of G.

F

domain is a

and whose range is con­

For each [countable] subset 81

F+ etc.)

We do not

to iDe the real number system, but we do wish to

perform linear operations in

(2 2 .1 [a])

These

and

However, one new idea enters, namely

the concept of a measurable set, and this we must investigate. wish to restrict

T

S

of

E

82

V.

REAL-VALUED FUNCTIONS

directed by IQ (S) (g)

If

e1

IQ (e2 ) G, (h)

«

and such that

is bounded below, and e2

are in

E,

there exist elements e' «

If

and

e1 ,e2

and

I (e 1) and

e.^ « e^

e*

and

e”

of

E

e", i * 1, 2.

are in

E,

and

I0 (e2 ) have a common upper or lower

bound in that E

and

have a common upper or lower bound in

such that and

As < 0

/\l0 (S) ^ > e »

f

in

F

such

there is an

e

mid ( e ^ e ^ e ^ ) ;

in

and

dually. (i)

Whenever and in

c2 E,

c 2 I 0

e1

and

e2

are

in E

are real numbers,c ^

and

c1

+ c2e2

is

and IQ (c1e1 + c2e2) = c 1IQ (e1) +

( e 2 ^ 5

e

8 ,1 1 (1

^

e

^

,

^

8 1 1 (1

e 2

*

a r e

in E and e^ « ei 1 (i = 1, 2) and c < 0, then e1 + e2 « e1 * + e2 * and ce1 » ce1 *. It Is obvious that if we assume ( 2 2 . 1 [a]) we at once obtain the conclusions of (19-3) and (19*^); and if we also assume hypotheses (I9*5d,e,f) we obtain the conclusions of (19«5)*

However, we can establish

stronger statements. As usual, let us define » + x ~ x + « > » « If x > - o o , - oo + x = x + ( - o o ) = - o o if x < + oo. Then we can define f 1 + f2 t

in

is F+ .

to be

(fj(t) + f2 (t): t

In

Tat which one of the values

- oo. + oo,

so

f2(t)

U-function is ever

f 1 + f2

U-functions or both (22.2)

provided only that there Is no

This agrees with the definition of +

In particular, no

ever

T)

fj(t),

- oo,

Is defined whenever f 1

«

and no and

and theother

f1

and f2

THEOREM.

Let (22.1 [a]) hold. Let

2),

and let

Kf^) s

and

f1

iCf^)

areboth

and

f2

are defined

be a function such that

s (t ) — (t ) + f2(t) whenever the sum Is defined, s(t) being arbitrary (finite, + oo or - oo) else­ where in

T.

Then

I( f ,) + I ( f 2 ) < 1(8)

£

1(8) < T ( f 1) + T(fg) ,

T(-f,) - -Kf,), K-f,) - -T(f,) • In particular, if

f1

and

f2

are in

L-function is

f2

L-functions.

be functions such that (1-1,

is

when

are summable, 3 0 are

22. and

83

INTEGRALS OP REAL-VALUED FUNCTIONS

-f,,

and

I(s) = K f , ) + I(f2 )

and

I(-f) =

-I(f,) If

la in

defined for all tively, then u 1 + u2

t

in

U[^ f± ] T.

(1=1,

S1,S2

If

u 1 + u 2 = V ( e 1 + e2 : e 1

is a

1),

then

u ^ t ) + u2 (t)

are associated with

S1 and

ln

u,,u2

e2

ln

S2),

and

e2

in

Is respec­ so

U-functlon and

I1(u1 + u 2)=

V ( I 0 (e1 + e2): e, ln

-

S,

S2 )

+ V i 0(s2)

V W

= I 1(u1) + I1(U2 ) . Whenever

f ^ t ) + f2 (t)

is defined, we have

s(t) * fj(t) + f2(t) < u 1 (t) + u 2(t) . Where

f ^ t ) + f2 (t)

Then

u ^ t ) is

+ * > s(t) .

isundefined, one summand

+ », while

Hence

u 2(t)

u 1 + u 2 > s,

cannot

be

(say

f^(t))

-

so u ^ t ) + u2 (t) *

is

+ «.

and

I(s) < I1(u1 + u 2) = I1(u1) + I1(u2) . This holds for all

u 1 ,u2

in

T(s) i

U[^> f 1 ], U[> f2]

Ai/uD

respectively, so

f 1 ]) + A i , ( u q f2 ])

= i(f1) + i(f2) . The statement concerning -f1

I(s)

is established dually, and those concerning

are easy to establish.

(2 2 .3 )

THEOREM. and 1(f) |fI

and

Let (2 2 . 1 [a]) hold. 0^

If

f

is summable,

have a common upper or lower bound,

is also summable and -I(If I) < 1 (f) < I(If I) .

For then by (2 2 .2 ) and (6 .7 ), I(-f), 0 q and 1(f) have a common bound, and by (1 3 -1 ) If I is summable. By (1 1 .8 ) the inequalities hold. t

in T)

For each fixed t in T, is smoothly rising on F

the function r^ = (r^.(f) * f (t ): and is finite and additive on F+ ,

so

Qk

V.

it belongs to

RF + .

REAL-VALUED FUNCTIONS

It is obvious that hypotheses (i) and (ii) of (2t.3)

hold, so by this and (21 .1) and (21 .2) we have (22.U)

COROLLARY.

Let (22.1 [a]) hold.

If

f1

c

is a real

are finite-valued and measurable and number, then REMARK. and

fj(t) + f

n

and

cf1

are also measurable.

This can be improved; if

f2 (t)

is defined for all

f2

t

f1

and

in T

fg

then

are measurable f 1 + f2 is

The proof is essentially that- of (21 .3 ); the chief change is

measurable. that

f 1 + f2

and

and

gm n

may not be finite-valued, but in any case their sum

is defined. Given any set

M,

we shall denote by

function, that is the function whose value is is

0

x^ 1

its characteristic

at each point of

M

and

elsewhere.

(2 2 .5 )

DEFINITION.

If

finite measure

if

the measure of

M

x^

M

is a subset of is summable,

D,

M

and

has inthis case

is

= I(xM) ; M

is

element of

the set

F.

a measurable set if x ^

If for a sequence

E 1, E 2, ...

is a measurable

of sets we denote by

the set of all points which belong to infinitely many lim inf En

En

lim sup ^

and by

the set of points which belong to all but finitely many of the

En* we limit of the

that the characteristic function oflim sup ER is the upper characteristic functions of the En , and likewise for the

lower limit. (2 2 .6 )

COROLLARY. E 1, E 2 , ... lim sup En For

by (17-8). (2 2 .7 )

and

X tj -p = nn

mE ^ 0q .

0^.;

lim inf En

Then

U nEn , D nEn ,

are measurable sets.

duallY* whence the conclusion follows

COROLLARY. measure

Let (22.1[a]) hold, and let

be measurable sets.

Let (22.1 [ f"; and Let

in

N

h

N be the set

then by hypothesis u - 1

mN - 0q . Since

L-function

is defined and

1

f•

0

elsewhere.

and a

is a non-nega.tive

be the function which has the value

and is

f»

u(t) -

For each positive integer

n, I(nxN ) - nI(xN ) - n0Q - Qq , so using (13-1), O q ^ K l n x ^ l A t u - 1]) < I(nxN ) By (U.it), 1(h) - lim I([nxn ] A f u - 1]) - 0^. Let s be the function which has the value f 1 (x) + h(x) whenever this is defined and the value

u(x)

elsewhere.

By

(22.2),

I(s) < I(f1) + T(h) = K f ^ But where case

.

f ^ t ) + h(t) is defined we have either t

f2 (t) = f ^ t ) * f ^ t ) + h(t),

or else

t

in

not in N,

N, in which

in which case

23f2 (t)

f ^ t ) + [u(t) - 1(t) ] «■ f ^ t ) + h(t);

undefined we have

f2(t) < u(t) = s(t).

Interchanging the roles of so N

f1

and

So

f2

f2 ^ s,

f ^ t ) + h(t)

and

and

0

elsewhere. U-function

By (1^-3), 1(h)

u1 ^

I(f2) £ K f j)-

= lim I(nx^) = 0q ,

u is in

^

1 ’ ui ^

(2 2 .1 6 )

THEOREM. Let (22.1 [a]) hold.

(i *

let

2),

such that

I(f) .

g

are equivalent.

Then

N

f

and

be the subset of T

elsewhere. Nk

Then

be the set

d]> o, (t : t

T and

f

and

23-

(2 3 .1 )

1)

there is a

g

be

and

g

are different.

for

t

on

N

and

For each positive integer

d(t) > 1/k).

Then Since

MEASURABLE FUNCTIONS AND LEBESGUE LADDERS

A function (of degree

on

is in

I(g') ®

0 £ ( 1 A ) X N < d, so I([lAl X N ) “ and mNk - 0^. k k N » mN = 0^, and f and g are equivalent. j

so

1

and

to g(t) - f(t)

and 1(d) » 0^. in

f

g ^ f

on which

and

»

and hypothesis (a) is

Let

F

d be the function which is equal o

N[> f 1 ]

summable members of

Let Let

^ u + ui

is

establishes the reverse inequality,

L[< then verified.

k

and where

I(f2) = T(fj). The other equation is established dually. If hypothesis (b) holds, let h be the functionwhich is

summable

to

87

MEASURABLE FUNCTIONS AND LEBESGUE LADDERS

if

0

on a vector space is called positively homogeneous

0(kx) = k0(x)

when

k > 0.

THEOREM. Let (22.1 [a]) hold.

Let

0 *

(^(y.,, •••, yn ): 7± real, i - 1 , ..., n) be a func­ tion of Baire defined and positively homogeneous on n-space. functions. (^(f^t),

Let

f 1, ..., f

be finite-valued measurable

Then the f■unction ..., fn (t)): t

in

0(f) = T)

is measurable

By (17-7), (19*3) and (19 •**■), if 0 is a homogeneous linear combination of the y^ or is a lattice combination of such linear combina­ tions, the conclusion holds. on the set

Zly^l = 1

By the Stone-Weierstrass theorem [Stone 2],

every continuous function can be uniformly approxi­

mated by such lattice combinations. Hence if 0 is positively homogeneous and continuous, we can find a sequence of lattice-combinations 0 1, 0 2 ... such that limn 0n (f (t)) = 0(f(t)) for all t; and by (17*8) measurable. The extension to Baire functions is immediate.

0(f)

is

88

V.

(2 3 .2 )

Let (2-1 . 1 [a]) hold and let the whole

THEOREM. space

T

REAL-VALUED FUNCTIONS

be a measurable set.

Let

0 =

(^(y1 > • ••> yn ): r eal, 1 * 1 , ..., n) be a Balre function on n-space. If f 1, ..., f are finite­ 0 (f 1, ..., f )

valued and measurable, If

0

is measurable.

a + b 1 y 1 + ... + bnyn ,

is a linear function

1

sion is valid by (22.k), since the function identically

the conclu­

is measurable.

All lattice combinations of such functions are measurable, by (1 7 .8 ). the Stone-Weierstrass theorem, If Integer

m

0

It can be approximated to within

{y : -m < y^ < m, 1 * 1 ,

..., n}

By

Is continuous for each positive l/m

on the interval

by such a combination

lim 0 (f(t)) = 0(f(t)) for all t, n->oo of all functions for which 0 (f)

and

0(f)

0 m . Hence

is measurable.

The set

is measurable contains all continuous

functions and is closed under passage to the limit, so it contains all Baire functions. (23.3)

Let (2 2. 1 [

an = 00,

llm

llm a_n = - ». n —*00 n —»00 By the "mesh” of the ladder we shall mean the supremum of the numbers

an - an _1, n = o ,

+i,+2,

... .

If we try to follow the familiar "ladder” method of defining the integral with the help of the measure, we find a difficulty when

T

lacks

finite measure that can be overcome, but by a device which we would prefer to avoid using when

T

has finite measure.

We succeed in treating each

case in the desired manner by tjie following artifice. Lebesgue ladder. all integers.

If

If

T

T

n

Is at least equal to the length N(A) • Let f be in

(t: an 1 < f(t) < an )

Let

F.

L

*

A

be a

shall mean the set of N(A)

such that the distance from

A summation designated by a sign subset of

N(A)

does not have finite measure,

of all those Integers [an 1,an ]

has finite measure,

an - an_1

0

shall consist to the Interval

of the interval.

shall mean a summation over some

If for each

n

in

N(A)

the set

has finite measure, and the finite sums

^ ' n = - h bn “ {t ! an-1 < f(t) i an } converge to a limit as associated with

f

and

h, k —* 00, A,

this limit Is called the "Lebesgue sum"

and is designated by

X;g that

exist; otherwise

in

G.

m*E

m*E

is not

»

such that

^ g

for all

Then for each subset

to be

m*E = 00.

m*(E1 U E 2) < m*E1 + m*E2

Also, if

»

I(Xg)

If it

E

ii summable

of

in we

U-functions

We readily prove, by use of (22.2),

for all sets E 1,E2

contained in

it is the infimum of m E f for all sets

finite measure which contain

g T

T. E'

of

E.

Starting with the exterior measure, we can construct a theory of measure according to Caratheodory1s procedure.

The relationship with our

theory is as follows. (23.8)

THEOREM.

Assume that (22.1[a]) and hypothesis (i)

of (23.6) hold, that a subset of E*

of

T

T.

T

If

E

is measurable, and

that E

is

is measurable, for every subset

the equation

(*)

m*(E* f\ E) + m*(E' - E) = m * E ! is satisfied; and if (*) holds whenever measure,

E

E'

has finite

is measurable.

As already remarked, the left member of (*) is prove the first conclusion, it remains to show that if the reverse inequality holds. Since

E

and

be a summable I(u A x e )

and

1

xE

and

1 - Xg

exist; their sum is

are measurable.

Let

I(u) .

But

. “ x A x < u A x > E'AE E' E = E

and similarly

x E'-E “ xE' A[ 1 - xE ^ =< uAti - xE 1 > so

m*(E' A E) + m*(E* - E) < I(u) . From the choice of

that the left member of (*) is of the first conclusion.


f2

E,

= Jo {ev

to mean

f 1 3 f2,

while

f° 3 f2 . Let

G

be the real number system.

easily verified and

G

is normal.

f1 »

f2

shall

Then (12.ia,b,c,d) are

In order to Investigate (12.ie) (which is the same as (8.1e)) we first prove a lemma. (25.3)

LEMMA. If S is a subset of F directed by » , then V S = V(f°: f in S) ; if S is a subset of F

directed by « Let

S

then

be ordered by

A s = A ( f C: f »,

and let

95

in p

S) .

belong to

VS;

then

p

96

VI. APPLICATIONS

belongs to some in

S

f

such that

\/{f0 : f

in

In

S.

g »

S),

Since S

f .Then

p

is directed by » , is in

f

V S C V(f°: f

so that

and

in

there exists

g

f C f° C g° C

S).

The reverse Inclusion

is evident, so the first statement Is established. If ordered by

S

Is ordered by

».

VtT - fc : f

the set S' = {T - f: fin

Hence, Vs* = V((T - f)°: f S 3 = T - A(f°: f

in

V t T - f: f

«,

Since

S 3 = T - /\if: ** inS 3 = T - /\S,

in

is

S3 =

in

in S 3 .

S)

\/S» = this proves the

second statement. Now let

S^Sg

tively, such that compact,

be subsets of

A S 2 < V S 1 . Each

by (25.1 iii).No point can

E

directed by

set

» , «

e2 - e° with

ei

respec­ In Si

is

belong to all sets

e2° - e^0 (e^ in S^); hence finitely many such sets e2°j “ e i°j (j == 1, ..., k)have an empty intersection. Choose e2 in S2 such that e0 «

e0 . (i = 1, ..., k); choose e- In S- such that e. » e. . ^ c o 1 1>J (j = 1 , ..., k ) . Then e 2 - e 1 is contained in all the sets

e°2 • - e°1 y

hence is empty.

Consequently

!0 (e2) < IQ(e}) . Hence A l Q (S2) ^ without the restriction to countable sets If k «

h,

k

is In

F

and

k

Is

h°.

their union Is an are in

e

F,

in

E

Is In

ii) interior to an

Finitely many such

e

such thatk «

and

S^.

compact, and h

each point ofk is by (18.1

closure is in

e°2 C e°1, e2 C e 1,

and ( 8 .1 e) isestablished,

,

e

F, and in E

whose

have Interiors which cover e «

h.

Since the

k;

empty set

and

T

this implies (8.if); and since mid

(e^e^e^)

E

d 12

e^»e2 anc* are, this implies (8 .1 g) . Let e^,e2,e^ belong to E. Write d 1 for for e 1 \J e2 - e^, and ^ 12^ for e1 A e2 f) e^,

the other

d^j

being analogously defined.

is

in

when

likewise for

e2

and

ey,

Then

and also

e1 - e2 \j e^,

e 1 = d 1 + d 12 + d ^ e 1\J e2 \J

d^

+ d 12^,

and

and

= d 1 + d2 + d^ + d 12 +

d 13 + d 23 + d 1 2 3 ' and mld (ei>e2 »e3 ' = d 12 + d 23 + d 13 + d 1 2 3 * By (25.1), r 1£ 1 ^(©i) * I0 (e ! u e2 u e3^ + (e1,e2,e^)) + IQ (e1 fl e2 A e^) . From this we see at once that (12.ih) is satisfied. Moreover, by a similar but easier argument we show that (12.7) also holds. We can now apply the results of the ^theory previously developed. First we shall look at the structure of sets which are U-elements. (25.M

L-elements or

THEOREM. An element f of F is a U-element if and only If it is an open set; it is an L-element if and only if it is a compact set. Assume

directed by

f

to bea

» such that

U-element; V s = f.

let

S

By (25.3),

be a subset of f = V(©°: ©

E in

S 3,

25. hence is E

open.

Likewise, if

so is compact. in

E

and

e« e

p

Thus

VS.

compact. E

If

p

e

of

p*

of

- f,

and

in E

e

»

and

T - f

If p

pis in

isa

to

and

be a subset of in

S},

open. By (8.2d), the set

that

e°

U-element.

each p

in

is in

and

ofE

cover

f,

there is

ec C f° =

Next assume

f, and e »

p*

f)

is not in

e. «,

hence

e

in

finitely

their union is an

is directed by

AS;

is not in

f,so

f

f corresponds an

f;

and

S =

ec C T - Ip*).The interiors of

e 1, ..., en

E,

S = te: e

point

T

is in e°

many such elements set

such

f = VS, and

p*is in

such that

element

E

S

f = A(ec: e

f ) is directed by » .

in

97

L-element, let

Conversely, assume f

by (25.1ii) an is in

f is an

such that A s = P.Then

> 1 .

for which

1 ,1 , 1

The symbols

E e

and of

e «

E

u) .

Thus

for which

I-j(l)

I 1 (u)

e«

Is an L-element (that is,

is necessarily summable, and of

IQ (S)

u.

if

is

has the same value for all such sets ^S,

(e: e

supremum of

e

for which

in

is the This is

acompact set), it

Is the infimum of

IQ (o)

This is the same as

1 (1 ) .

will be replaced by

m,m*,m#

for all sets

respectively

(read measure, exterior measure, interior measure respectively). According to the foregoing, the measure of an open set is the supremum of the ele­ mentary measures

IQ (e)

of all sets

e

of

E

whose closures are in the

open set, provided this supremum is finite; the measure of a compact set is the Infimum of the elementary measures whose interiors the compact the Infimum of

mu

I0 (e)

1

e

of

set lies; the exterior measure of a set

for all open sets

u

contained in

f.

If

E

In

f

is

of finite measure containing

if such sets exist; the interior measure of all compact sets

se"^s

f

m*f

is the supremum of and

common value (necessarily finite) is denoted by

mf,

m*f

f

ml

for

are equal, their

and

f

is said to

have finite measure. If so are

f1

and

f 1 U f2 and

It is easy to show o.

So if

f1

and

f2

are sets of finite measure, by (1 5 -1 ) and (15-2)

f 1 A f2,

and

m(f1 \J f2) + m^f1 fi f2) = mf 1 + mfg.

that the empty set has finite measure, namely measure f2

are disjoint, the above equation yields m(f1 U f2 ) = m f 1 + mf2 .

Condition (1^.1) obviously holds. By (1 h .3 ), if Is a sequence ofsets of finite measure such that union

f 1, f2 , f^, ...

f 1 C f2 C f^ C ...,

Unf n is a set of finite measure If and only If the numbers

the

98

VI• APPLICATIONS m( \Jn?n) = V^mf^ = lim mf^. are necessarily i — » 00

mf 1 , mf2 , ... are bounded, in which case If f 1 f2 zd f^ o ..., the numbers mf^ bounded, so if

flIIfi l in

A)

is of finite measure, and

m( fLf-J = Ii Ii

•

lim

is a directed system of sets of finite

all contained in a set

f”

mfL1 . Also,

00 measure,

a o*-convergent to a set

of finite measure and

f , by (1 5 *1 ) f~Q has finite measure, and mfQ = lim mf^. (We choose f' to be the empty set in hypothesis (ii) of a Ink (15.-1)). By (1 7 -1 ), a set

f

sets of finite measure with f] U (f2 0 f) f2 f\ f

the set

mid (f1,f2,f) =

has finite measure.This clearly implies

finite measure, since we can choose if

f.j to be the empty f 1 U (fg 0 f)

has finite measure, so does

finite measure.

°^

is measurable if for all pairs

f 1 C f2,

that

f2 f\ f

set.

Conversely,

whenever

So a necessary and sufficient condition that

measurable is that

f2 fl f

have finite measure whenever

f2

f

f1

has has

be

has finite

In particular, by (1 7 *5 ) all open sets and all compact sets are

measure.

measurable. We now stopto prove (25.5)

LEMMA. g

a lemma.

Let f

be a subset of

be a set of finite measure and let f.

Then

m*(f - g) =

mf - m*g and

m*(f - g) = mf - m*g. Let€ and a compact 1 - u

be a positive number. set

1 C f

There exists an open set

such that mu< m*g + e

is a compact subset of

f - g,

and 1 C\ u

m*(f - g) > m(l - u) = m(l - 1 0 u)

and

u o g

ml > mf - e .

Then

has finite measure,

so

= ml - m(l fl u) ^ ml - mu >

mf - m*g - 2 6 . Hence

m# (f - g) ^ mf - m*g.

such that that tains

mu < mf + €,

m l 1 > m*(f - g) g, and

Next, let

and let - e. Then

1* be l 1 C u,

m*g < m(u - l 1)= mu - ml*

u be an open set containing a compact subset of f - g

f such

and

u - 1 * is open and

< mf

- m*(f - g) + 2€.Hence

con­

m*g < mf - m*(f - g) . With the previous inequality, this establishes the first conclusion. The second is obtained by merely interchanging g and f - gAs a corollary we have (25.6)

THEOREM. has If

f - g; g C f

If and if

fand

g have finite

measure, so

g C f, m(f - g) = mf - mg.

the preceding lemma shows that

m*(f - g) =

25.

MEASURE IN LOCALLY COMPACT SPACES

m*(f - g) = mf - mg,

establishing the second conclusion; if f f) g,

finite measure so has f - f H g, ment

f2 - f 2 Pi f,

For let

f2

f

f2

bas.

have

is measurable, so is its comple­

have finite measure; then

which has finite measure, and so

measure whenever

f,g

and by the second conclusion so has

which is f - g. Prom this it follows that If

T - f.

99

f2 fl(T - f) =

f2 A (T - f)

has finite

Now the family of measurable sets contains all

compact sets and all open sets, and is closed under complementation and under formation of countable unions and intersections. tains all Borel sets in

Let us extend the definition of ever

f

It therefore con­

T. m*

by setting

m*f =

in

H;

clearly

To show that it satisfies (1 .1b), (that is, is proper)

we make a simple calculation: (28.1) H,

LEMMA.

If

B

is a bounded linear operation on

and

and

y

are in

x

H,

then

2 (Bx,y) = (B[x + y],[x + y] ) + i(B[x + iy],[x + iy]) - (1 + i) [(Bx,x) + (B y ,y )] . Now if

B" ^ B *

and B 1 )

(Bx,x) = 0

for

all x

x

in H,

whence

and

(28.2)

y

LEMMA.

in

Let

B", then the difference

B = B" - B * satisfies

H, so by (28.1) we have (Bx,y) = 0 B = 0. S

Thus

>

for all

is a (proper) partial ordering.

be a subset of

G

directed by

>

1 09

2 8 . REMARKS ON OPERATORS IN HILBERT SPACES and having an upper hound

B*

in

G.

exists a hounded hermitian operator (

I)

(Cx,x) « \/((Bx,x) : B

( ii )

in H; lim (Bx,y) = (Cx,y) B in S y in H;

(ill)

c - Vs.

Then there

C in

such that S) for each

x

For each

x

in

H,

((Bx,x)

:B

for each

in

S)

x

and

is an isotone net of

real numbers, so has a limit which is its supremum.

Each term in the

right member of the equation in (28.1) has a limit, so for each in

H

the net

B ” < B < B f. BQ

of

in

H.

G,

(Bx,y) Let

M

I|Bq ||

he the larger of x,

I|B| I < M.

eventually

Thus for all

I(Bx,y) | < M * |x| •|y| , the unit sphere. in

y,

y

H.

(C[kfx* + H

x

x

and

H

we eventually have

(Bx,y) x

x

(B*x,x) < M,

is hounded for x,y

the function(y,Bx)

in

is linear

Cx.

Thus

with some element of

lim (Bx,y) = (Cx,y)

If in particular we choose so (Cx,y)

y

y = Cx, y

the function

is also linear in

] - k*Cxf - k"Cx", y)

for all

x.

We have already shown it hounded.

and

(Bx,y)

is

Thus

vanishes for all elements

k',k”, whence

x

the previous estimate shows

C

is a

x*,x",y

linear trans­

Also, for all

a and

y

H, (Cx,y) = lim (Bx,y) = lim (x,By) = lim (By,x) = (Cy,x) = (x,Cy)

so

C

is hermitian.

,

This completes the proof of conclusion(ii) .

Con­

clusion (i) follows at once, by the second remark after (3-1)* By (i), an upper bound for

C S

is an upper bound for in

G-

Then for all

S x

in in

G. H

Let C* he also

we have

(Bx,x) for all B in S, so by (i) (C'x,x) > (Cx,x), This establishes (iii) and completes the proof.

and

(C*x,x) > C* > C.

This lemma allows us to prove that in G, o-convergence is equivalent to eventually-bounded strong convergence: (28.3)

y

eventually

for all unit vectors

the inner product of

and all complex numbers

formation. in

S,

the same is true of its limit; being hounded on the unit sphere,

linear in x, of

x,y in

|Cx| < M -1 x| . Also, for each fixed

that

is in

-M (B"x,x) < (Bx,x)
Jn

this approaches

6

>

0 . Hence the 0 . The result

is some fixed element

... J . dn (k,)'3' •••( V ' 3” jn= 0

o = 0’

IQ each such factor

as

. . s

k 1, ..., kn

■

Increase

31 •

RESOLUTION OF REAL ALGEBRAS

129

(cf. (7-4)), so IQ(e) ^ and (2 2 .le) is established. The proof of (2 2 .if) is essentially the same as in, say, Section 2 6 , so all of postulate (2 2 .1 ) holds. Hence we can define an integral I on a subset Fsuln of F, having all the properties established in Section 22 and earlier sections. In particular, (1 5• ^) holds for each eQ in E; we need only take e| = eQ - 1/1, e£ = eQ, e'^' = eQ + 1 /i, i = 1, 2 , 3 , ••• • So 1(e) = IQ(e) for every e in E, which implies conclusion (vi). Conclusion (I) follows from (22.4), and (ii) from (1 5 .1 ). It is not difficult to show that a function is a U-function If and only If it is bounded below and lower semi-continuous. In particular, the characteristic function of an open set is a U-function, so by (17*5) is a measurable function. Thus all open sets are measurable. In par­ ticular, with e = 1 , we obtain mT = I (1 ) « u. Since differences, countable unions and countable intersections of measurable sets are measurable, all Borel sets are measurable. Consequently all Borel-measur­ able functions are measurable, which implies (iii). Conclusion (v) follows at once from (19-5) • In particular, if 2 f is the characteristic function of a measurable set M, then f = f , so by (v) 1(f) = [1(f)]2, that Is, mM = (mM)2 ]> 0. If f1 and f2 are the characteristic functions of measurable sets M 1 and Mg respec­ tively, f^ f2 is the characteristic function of M 1 fl Mg, so (mMr)(inM2) - m(M1 fi M2), and (vii) is established. Suppo'se that f Is summable (hence measurable) and that 1(f) y 9, but m(t: f(t) < 0 ) ^ 0. Since m{t: f(t)< 0} = o-lim m(t: f(t) < -1/n), there exists a positive integer n such that n —1i 00 ~ the set M = (t: f(t) ^ -1/n) has measure mM ^ 9. Then on the one hand I(f • XM ) = I(f)I(xM ) = I(f)mM > 9. so

I(f •

On the other hand,

i -n ’1l(X M ) =

f • X M < -Xj/n,

.

Combining these estimates yields -n-1mM > 9, whence mM < 9. But by (vii) mM y_ 9 , so mM = 9, which is a contradiction. This establishes the first sentence In (viii). If f is summable and has a lower bound -b, and the set N * (t: f(t) < 0) has mN = 9 , then the function f + bxjj is ^ 0 on T, so 9 < I(f + byN ) = Kf) + bI(xN) = 1(f) + bmN * 1(f), establishing the second sentence. If f1 and f2 are bounded and equivalent, and f1 is summable, so Is f2 by (2 2 .1 5 ). The sets (t: f2(t) - f1(t) < 0} and Ct: f1(t) - f2(t) < 0 } both have measure 9, so by the proof just completed we have I(f2 - f 1) ^ 0 and I(f1 - fg) > 9, establishing the third sentence of (viii).

130

VI. APPLICATIONS Suppose that f 1

and

f2

are summable and fInite-valued, and

I(f1) =I(f2) . By the first sentence of (viii), both - f2 (t) < 0 }

m(t: f^t)

m(t: f2 (t) - f ^ t ) < 0 }

and

Hence the union of these sets, which is 0,

are equal to

(t: f ^ t ) / f2 (t)},

0.

has measure

establishing the last sentence of (viii).

Returning to (iv), we let K be the subset of A consisting of those elements x to which there corresponds a bounded function of Baire f such that 1 (f) = x. All elements of the and having an upper bound; it then has a supremum V S in A. By (2 .1+), we may suppose that S is an isotone sequence x 1 < x2 < ... converging to VS. To each xR corresponds a bounded function of Baire f^ such that = xn* ^e^ a a num1:)er such that V s < au. Since xn £ cu for each n, by (viii) we have f^ < c except on a set of measure 0 . Hence f^ Is equivalent to A c, which Is a Baire func­ tion; and by (viii) K f ^ A c) = xR . That is, we may as well suppose that all the f^ satisfy f^ < c to begin with. Next we define inductively f 1 = f j, ..., fR = f^ V fn_-, • There are clearly Baire functions. The relation 1 (f) = xR holds for n = 1 by definition. If it holds for n = k -1 , then > xk-1 = K f k_i), so by (viii) f£(t) >fk-i(t) except on a set of measure 0 , and f£ isequivalent to V f^, which is f^. Then by (viii) = = xk* anc^ 30 by induction we have Kf*n) = *n for all n. Since f 1 < f2 ... j( c, they have a limit f which is also a Baire function and is < c. By (ii),

= °_lirn

= 0_liin xn =

In a like manner, if

S

having an infimum in

A,

cr-complete.

Hence

is a countable subset of

AS

Is in

K.

Hence

K

K

Vs

is also in

directed by