Functional Operators (AM-21), Volume 1: Measures and Integrals. (AM-21) 9781400881895

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A N N A L S OF M A T H E M A T IC S STUDIES Number 21

ANNALS O F M ATHEM ATICS STUDIES E dited by Marston Morse and Emil Artin 7. F inite Dimensional Vector Spaces, by

Paul

R.

H a lm o s

11.

Introduction to N onlinear Mechanics, by N.

14.

Lectures on Differential Equations, by

15.

Topological M ethods in the Theory of Functions of a Complex Variable,

by

K r y lo ff

and N.

B o g o liu b o ff

S o lo m o n L e f s c h e t z

M a r sto n M o rse

16. Transcendental Numbers, by

C a r l L u d w ig S ie g e l

17.

Probleme General de la Stabilite du Mouvement, by M. A.

18.

A Unified Theory of Special Functions, by C. A.

19.

Fourier Transforms, by S.

20.

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

21.

Functional Operators, Vol. I, by

22.

Functional Operators, Vol. II, by

23.

Existence

Theorem s in

B ochner

Partial

and

L ia p o u n o ff

T r u e s d e ll

K. C h a n d r a se k h a r a n

John von

edited

by

N eum ann

John von N eu m ann

Differential Equations,

by D o r o t h y

B e r n s t e in 24.

Contributions to the Theory of Games, edited by A. W .

25. Contributions to Fourier Analysis, by S. M. M o r s e , W . T r a n s u e , and A. Z y g m u n d

B ochner,

T ucker

A. P.

C a ld e r o n ,

FUNCTIONAL OPERATORS BY J O H N V O N

NEUMANN

Volume I: Measures and Integrals

PRINCETON PRINCETON

U N I V E R S I T Y PRESS

1950

C O P Y R I G H T , 1 9 5 0 , B Y P R I N C E T O N U N I V E R S I T Y PR ESS LONDON!

GEOFFREY

CUMBERLEGE,

O X F O R D U N I V E R S I T Y PRESS

P R I N T E D IN T H E U N I T E D S T A T E S O F A M E R I C A

F 0 REffORD

The lectures on "Operator Theory", of which the present volume constitutes the first part, were given in the academic years 1933-34 and 1934-35, at the Institute for Advanced Study,

The notes were prepared in these years

by Dr, Robert S. Martin and Dr. Charles C. Torrance, respectively.

They

were multigraphed and distributed by the Institute for Advanced Study short­ ly thereafter, but the original edition has been completely exhausted for several years.

The interest in these lecture notes appears to have been

continuing, and therefore a new edition is now being brought out0

The pre­

sent volume comprises Chapters I - XI, dealing with preliminaries, namely, with the theory of Measures and Integrals.

The second volume, on Operator

Theory proper, will be'published subsequently.

The present edition is

identical with the original one, except that typographical errors have been corrected and some notations and references have been elaborated.

I would

like to express my warmest thanks to Dr. H. H. Goldstine, for his advice on this edition, and also for having most obligingly undertaken the exacting task of proof-reading the typescript.

JOHN VON NEUMANN

The Institute for Advanced Study Princeton, New Jersey November 1949.

FUtJCTIOHAL OPERATORS

CHAPTER I .

POINT SET THEORY The p o i n t s P o f t h e sp a c e u n d e r c o n s i d e r a t i o n a r e o r d e r e d s e t s o f n r e a l num bers (x_ , • . . 3x ) : x . i s c a l l e d t h e i 1 n i

th

c o o rd in a te of th e p o in t .

r

D e f i n i t i o n 1 . 1 : The d i s t a n c e b e tw e e n tw o p o i n t s ( x 1 , . . . ,x ) a n d -n r ^ - — 1 n — ( y if - .y .)

is

i/

£ i

N

.

D e f i n i t i o n 1 . 2 : An open i n t e r v a l I i s d e te r m in e d b y tw o s e t s o f num­ b e rs X

an d Yv ( v =s l , . . . , n ;

( x ^ ,...,x ^ )

Xy < Y^ f o r e a c h y ) an d c o n s i s t s o f a l l p o i n t s

s a t i s f y i n g t h e c o n d i t i o n Xy
• • • ^

a n y c o n d e n s a tio n p o i n t o f t h e c lo s e d i n t e r v a l I :

X ^ = x_y = Yv ( ”V = l , . . . , n ) , € Xv an d Y^ t h e l a r g e s t i n t e g e r < Yv . number o f i n t e g e r s i n t h e i n t e r v a l Xv = x v = Yv i s (Yv - Xv + 1 ) .

The

H ence

th e num ber o f i n t e g r a l p o i n t s i n I i s n

(2)

H = TT 1

But x v < x Y x v + 1,

(Y - Xy + 1).

V=1

Tv - 1 5 Yv < Yv , Yv- x v - 1 5 Yv -

+ 1 < Yv - x v + 1.

H ence, b y (2 ) n

n

1) .

(3) TT (Yv- XV“ 1) = NT< TT (Y - xv+ V =1

V=1

S im ila rly ,

(4) By ( 1 ) ,

TT (4 i} - xv ')- 1) 5 11 (i) < TT

V=1

+ —

+

p -* (S g ),

K even

pi*(Sg~

K odd

“ P-*(sr+2^ “ ^*0T) < °o«

Hence t h e tw o s e r i e s w i t h p o s i t i v e te r m s

B ut

’ 0r

III.

MEASURE

17

p * (S 3- S1 ) + Ji*(S5- S g ) + . . .

,

F * ( S 2 ) + p*(s4- S2) + . . . , t o g e t h e r w i t h t h e i r sum

(s)

p K i y r - ly s r) + jll* ( m3 r - n y O + . . *

,

8j*e c o n v e r g e n t , s i n c e , b y ( 2 ) , t h e p a r t i a l sums o f e a c h o f t h e s e r i e s a r e By t a k i n g K s u f f i c i e n t l y l a r g e t h e re m a in d e r R_ 1 o f ( 3 ) c a n be — K—i

bounded.

made l e s s t h a n € , w h ere

i s t h e s e r i e s ( 3 ) w i t h t h e f i r s t i te rm s o m itte d *

B ut

0S - M p = ( l l ^ p - M p ) ♦ ( i p p - M ^ p ) + . . . . By T heorem 2 * 6 ,

(4 )

- M p) =

S in c e jtfcl =

- M p) +

+ [jzfo - Mg(jzfR)]* ^

- l ^ +p )

=

< € *

f o l lo w s b y T heorem 2*6 t h a t

H* ( $ 0 5 n+tM p/Zfe)] +

= ju^l/LgR) +

M p /fa )] =

- MgJ)
°»

5y Theo-

rem 2*7,

(l)

+ ( rr - jzfo) ] = ^ O y i )

Since H D [M^H + (H - JZ&0], H

+ ja*(if - jzfc).

follows by (l) that

ju*(ll) = p *(11^11) + ju*(h - jZih)•

The theorem follows from the preceding lemma when K. becomes infinite. THEOREM 3.3 i Proof;

If M is measurable, then -M is also.

M is measurable if ju*(i:i) = p*(MU) + p*[(-M.)H] for every H.

But this relation remains unchanged if M is replaced by -M, since ~(~M) = M. THEOREM 3.4: Any closed set is measurably. Proof:

This follows directly from Theorems 3.2, 1.2, and 3.3.

THEOREM 5.5: If Pr oof ;

S inc e

(1)

are measurable, then

+ M 0 is also.

is mo s.sur ab 1e ,

|U*(K) -

where N is any set.

(2.)

and

Since

^ ( M p : ) +^*[(-1^)1)],

is measurable,

^[(-M-gU] = fa*[M2 (- M ^ N ] + >i*[ (-M2 )(-M1 )N],

But

(S)

(-I/p (-Mg) = -(M1+ Mg).

By (1), (2), and (5),

(4)

Again, since

p* (K ) =

is measurable,

+ fi*[M g(-M pN] + p.*[ I - ( 1 ^ + Mg) } N] .

III.

(5 )

H (1 V

MEASURE

=

V

19

H] + F * [ ( - V ( M1 + V

K] =

= ^ ( l ^ N ) + p.*[ ( -M1 )MgN ] .

The th e o r e m f o l lo w s b y s u b s t i t u t i o n o f ( 5 ) i n ( 4 ) . THEOREM 3 .6 g I f P ro o f: a b le .

an d Mg a r e m e a s u ra b l e , t h e n M^Mg i s a l s o .

By Theorem s 3 .3 an d 3 . 5 , “M^,

B ut -M j+ (-M g) i s -(M ^M g).

“^ i + ( “^ g ) a r e m e a s u r­

Hence M^Mg i s m e a s u r a b le .

Theorem s 3 .5 a n d 3 .6 may be e x te n d e d im m e d ia te ly t o a n y f i n i t e num­ b e r o f m e a s u ra b le s e t s . THEOREM 3 . 7 : I f P ro o f: Lemma 1 :

an d Mg a r e m e a s u r a b le , t h e n

M-^Mg i s s im p ly M ^(-M g).

a Mg c

If

p o i n t s e t s w i t h sum M, th e n

1 P ro o f:

M^Mg in a l s o .

...

i s a n i n c r e a s i n g se q u e n c e _of m e a s u ra b le

li m p * (M y i) = ju*(MH), w h e re H i s a n y s e t . K-^oo

I f p*(MgN) i s i n f i n i t e f o r an y k , t h e n i t i s i n f i n i t e f o r a l l

l a r g e r k an d t h e lemma i s t r i v i a l .

H ence i t i s n e c e s s a r y t o p ro v e o n ly t h e

c a s e w h ere p*(M^!T) i s f i n i t e f o r a l l K , MIT i s t h e sum o f a l l th e s e t s M^N, i t

S in c e

c MgN c

• • • c MIT a n d s i n c e

is p o s s ib le to w rite

m = FLjH + (MgIT - M^IT) + (MgN - MgIT) + . . .

.

By T heorem 2 . 6 ,

(1) Sdnce

|U*(MH) 5

- J^N) + . . . .

^ i s m e a s u r a b le ,

= ^ ( u^

n)

+ |i* ( iy r -

20

III.

MEASURE

so t h a t

(2.)

p * ( Mg.1T) - p ^ M ^ H ) = p*(MgN - M ^ N ) ,

p ^ M ^ H ) b e in g f i n i t e .

By ( 2 ) t h e sum o f K te r m s o f ( l ) i s p ^M ^H ) « p*(MM). s e r i e s ( 1 ) i s t h e l i m i t o f t h e sum o f K te r m s , i . e . , ~ B u t, b y ( l ) , S = p*(M N). C o ro lla ry ;

Hence S =

The sum S o f t h e e n t i r e S =

li m p*(M_JT) = p*(M \f). K -* qo

li m p*(M^H) = p*(M N). K-xoo

I f i n Lemma 1 t h e s e t N be t a k e n a s M, t h e n lim p*(M _) -

« p * (M ). Lemma 2 : I f

3 Mg D

s e t s w i t h p r o d u c t M, t h e n

.

-

...

i s a d e c r e a s in g se q u e n c e o f m e a s u ra b le p o i n t

lim p*(M -N) = p*(M H), w h e re H i s a n y s e t w i t h f i K -

n i t e o u t e r m e a s u re . P ro o f:

(1 )

S in c e

I t I s e v i d e n t t h a t -M^ c

lim K- sKX)

- M gC . . .

• By Lemma 1 ,

= ju*[ (-M )N ].

i s m e a s u r a b le ,

]i* (E ) = p . * ( M + |li* [(-1 ^.)W ], so t h a t p* (H ) =

(&)

a

lim p*(M_H) + lim p * [(-M )H] K -x » K-xoo lim K-iw»

=

by (1 )

+ ju*[(-M )H ] -

+ p * [( -M )R ],

s in c e

3M U .

Hence M i s m e a s u ra b le an d

(3 )

p * (N )

“

ju*(MW)

+

p .* [

(-M )N ].

The le m m a .fo llo w s b y c o m p a riso n o f ( 2 ) a n d ( 3 ) , a n d t h e f a c t t h a t p * (N ) i s f i n i t e .

III.

MEASURE

21

C o r o l l a r y : I f i n Lemma 2 some p*(M ^) i s f i n i t e , t h e n N c a n b e t a k e n as th i s

---------

and

lim p*(M ^) = m (M ).

The f o l lo w in g th e o r e m was p ro v e d i n c i d e n t a l l y i n t h e p r o o f o f Lemma 2 : THEOREM 5 . 8 ;

• • * . ! £ . & d e c r e a s in g se q u e n c e o f m e a s u ra b le

p o in t s e t s w i t h p r o d u c t M, t h e n M i s m e a s u r a b l e .

!EL i n c r e a s i n g se q u e n c e o f m e a s u ra b le

THEOREM 3 . 9 : I f l l ^ c M ^c

p o i n t s e t s w i t h sum M, t h e n M ^is_ m e a s u r a b le . P ro o fs

S in c e

i s m e a s u r a b le ,

« jjL*(MgN) + ja*[(~M g.)N ].

The

th e o re m f o l lo w s im m e d ia te ly fro m t h e p r e c e d i n g lemmas w hen jK becom es i n f i n i t e . THEOREM 5 . 1 0 s I f m e a s u ra b le s e t s , t h e n P ro o f:

Ng, • • • i £ . f i n i t e £ £ i n f i n i t e

se q u e n c e o f

t h e sum o f th o s e s e t s i s m e a s u r a b le .

Form t h e p a r t i a l sums 3^= M^+ M^+ . . .

+ M^..

Then S ^ c S ^C S ^ r . . . ,

t h e sum o f t h e S ^ i s t h e sum o f t h e M^, a n d T heorem 3 .9 a p p l i e s t o t h e S^.. THEOREM 5 . 1 1 : I f M^, M g,. . .

i s any f i n i t e

or i n f i n i t e

se q u e n c e o f

m e a s u ra b le s e t s , t h e n t h e p r o d u c t o f t h e s e s e t s i s m e a s u r a b l e . P ro o f;

T h is f o l lo w s fro m T heorem 3 .8 when a p p l i e d t o t h e p a r t i a l

p r o d u c t s o f t h e M^.. Theorem s 3 . 2 ,

3 . 4 , 3 .1 0 an d 3 .1 1 show t h a t a n y B o re l s e t i s m e a s u r a b le .

THEOREM 3 . 1 2 :

If

P ro o f:

jji*(M)

= 0 , t h e n M dn m e a s u ra b le w i t h m e a su re z e r o .

F o r a n y s e t H, p*(MH) = jjl*(M) = 0 a n d p * [(-M )H ] = . p * (N ), h e n c e

jn*(MH) + ju*[(-M )N ] = jx* (H ), a n d t h i s i s a l l t h a t r e q u i r e s p r o o f . THEOREM 3 . 1 3 : I f M o r N is_ m e a s u ra b le an d p.* (M l) _is f i n i t e , t h e n p*(M + N) = ju*(M) + ji*(N ) - ji*(MN). P ro o f:

S uppose M m e a s u r a b le .

Then j i *(N ) - ji*(MN) +

p*(MH) i s f i n i t e t h i s c a n b e w r i t t e n

(1 )

ji*(JST — MB) = >i*(N) - jz * ( M ) .

MR).

S in c e

III.

22

MEASURE

A g ain sin e© M i s m e a s u r a b le ,

( 2.)

}i*(M + N) = p*[(M + N) M] + p*[(M + U) -

(M + N)M] «

= p*(M ) + p*(N - MN).

S u b s titu tio n of

( 1 ) i n ( 2 ) g iv e s t h e r e s u l t s t a t e d .

THEOREM 3 . 1 4 : I f M^, Mg, • • • in _a f i n i t e o r i n f i n i t e se q u e n c e o f m e a s u ra b le p o i n t s e t s su c h t h a t no two o f th e m hav e a common p o i n t , th e n Mg+ . . . ) P ro o f: i f th e

=

+

+ ...

.

T h is i s a g e n e r a l i z a t i o n o f T heorem

se q u e n c e i s f i n i t e .

3 .1 a n d o b v io u s ly h o ld s

I f t h e se q u e n c e i s i n f i n i t e i t i s n e c e s s a r y

t o n o te t h a t b o th p*(M^+ M^+ . . . )

and

+ p*(M g) + . . .

m e r e ly

a re re p re s e n ta b le

THEOREM 3 . 1 5 : The o u te r m e a su re p * (U ) o f an y s e t M may be d e f i n e d a s t h e g r e a t e s t lo w e r b o u n d o f t h e m e a s u re s ju(N) o f t h o s e e le m e n ts I o f a c e r ­ t a i n s e t S o f m e a s u ra b le s e t s w h ic h c o n t a i n M.

H e re in S may be a n y s e t o f

m e a s u ra b le s e t s , p r o v id e d o n ly t h a t i t i n c l u d e s a l l op en s e t s ;

in p a r t i c u l a r ,

S may be t h e s e t o f B o re l s e t s . P ro o f:

S in c e S c o n t a i n s t h e s e t o f o pen s e t s

c o n t a i n i n g M, i t c o n -

t a i n s t h e sums o f se q u e n c e s o f open i n t e r v a l s c o v e r in g M. m e a su re i s n o t i n c r e a s e d . one o f t h e s e t s N i s i t s

Hence t h e o u te r

N e i t h e r i s i t d e c r e a s e d , f o r t h e m e a s u re o f a n y o u t e r m e a s u re , an d i t s

o u t e r m e a su re may be a p p r o x i­

m ated t o an y d e s i r e d a c c u r a c y b y t h e sum o f t h e v o lu m es o f a se q u e n c e o f o pen i n t e r v a l s w h ic h c o v e r N. THEOREM 3 .1 .6 : s e t o f ty p e P ro o f:

Any m e a s u ra b le s e t M c a n be r e p r e s e n t e d a s a B o re l

m inus a s e t o f m e a su re z e ro * C o n s id e r t h e c a s e w h ere M i s a b o u n d ed m e a s u ra b le s e t .

T h ere

III.

e x i s t s a n open s e t

MEASURE

23

3 M s u c h t h a t p . ( ^ ) < p*(M ) + ~ - }i(M) + i

.

L et

oo m - t t j&.m i= l 1

M i s a B o re l s e t o f ty p e /zf

ju(M) = p (M ). But M =

an d i s m e a s u r a b l e .

B ut M c: jzf an d p.(M) 5

M + (M- M ).

< ja(M) + ~ .

M 3 M.

T h e r e f o r e ja(M) = jlx(M )•

By T heorem 3 . 1 , jji(M) = p(M ) + ji(M ~ M ).

p.(M - M) b OoThe th e o r e m f o l lo w s fro m t h e v id in g t h a t M i s bounded.

Hence

Hence

f a c t t h a t M = M ~ (M - M ), p r o ­

The c a s e w h ere M i s n o t b o u n d ed w i l l b e t a k e n up

im m e d ia te ly a f t e r t h e p r o o f o f THEOREM 3 . 1 7 : Any m e a s u ra b le s e t M c a n be r e p r e s e n t e d a s t h e sum o f a B o re l s e t o f ty p e P ro o f:

an d a s e t o f m e a su re z e r o .

L e t M be a

b o u n d ed m e a s u ra b le s e t a n d l e t I b e a f i n i t e

i n t e r v a l c o n t a i n i n g M.I~M i s m e a s u ra b le a n d , b y bound ed

s e t s ) , I-M «

m e asu re

z e r o . Hence M = I - ( M - Z )

T heorem 3 .1 6 (p r o v e d f o r

M-Z, w here M i s a B o re l s e t o f ty p e

-M i s a B o re l s e t o f ty p e

c lo s e d

= ( l ) ( - M ) + IZ*

an d Z a s e t o f

S in c e I i s c l o s e d

( l ) ( - M ) i s a l s o a B o re l s e t o f ty p e C *

and S in c e

IZ i s a s e t o f m e a su re z e r o , t h e th e o re m i s p ro v e d f o r t h e c a s e w h ere M i s bounded.

I f M i s n o t b o u n d ed , l e t 1^ be t h e c lo s e d i n t e r v a l -N 5 x v = N.

M = MI^+

+ •••

m easu re.b le so t h a t , b y i s a B o re l s e t o f t y p e M = (B^+ C

cr

and (Z n+ Zn+ . . . ) ' I B

E ach o f t h e s e summands i s b o u n d ed a n d

th e c a s e j u s t p ro v e d , M( I - l rr _ ) = B„+ Z _ , w h e re B__ Jl K -l K. ii K an d Z i s a s e t o f m e a su re z e ro * ^ 2+ • • • ) •

o ..) +

•

Hence

B ut (B^+ Bg+ • • • ) i s a B o re l s e t o f ty p e

i s a s e t o f m ea su re z e ro *

Thus t h e th e o r e m h o ld s f o r

an y m e a s u ra b le s e t M. I t re m a in s t o c o m p le te t h e p r o o f o f Theorem 3 .1 6 . th e o r e m i s n o t b o u n d e d .

By T heorem 3 .1 7 , i t

Then M = ( - C ^ ( - Z ) = ( mm^ (T) "* (-C )Z i s a s e t o f m e a su re z e r o .

S u p p o se M o f t h a t

i s p o s s i b l e t o w r i t e -M = a B o re l s e t o f ty p e

Z. and

IV .

24

in n er

MEASURE

CHAPTER IV . INNER MEASURE D e f i n i t i o n 4 . 1 ; The in n e r m e a s u re , yi (M ), of_ a_ p o i n t s e t M is_ t h 8_ l e a s t u p p e r b ound o f t h e m e a s u re s o f a l l m e a s u ra b le s e t s N c o n ta i n e d i n M. THEOREM 4 . 1 :

F o r an y s e t M, p +(M) = p * (M ), a n d i f M i £ m e a s u r a b le ,

p*(M ) « jx*(M) = r (M )» P ro o f:

The f i r s t p a r t o f t h e th e o r e m i s o b v io u s a n d t h e se c o n d p a r t

f o l lo w s fro m t h e f a c t t h a t i n D e f i n i t i o n 4 .1 one o f t h e s e t s N may be M i t s e l f . THEOREM 4 . 2 : ^ ( M ) in t h e l e a s t u p p e r b o u n d o f t h e m e a s u re s o f a l l t h e c lo s e d s e t s c o n ta i n e d i n M. P ro o f :

T h e re e x i s t s a m e a s u ra b le s e t N c o n ta i n e d i n M s u c h t h a t

M) - t .

ju(N) >

I t i s shown i n th e p r o o f o f Theorem 3 .1 7 t h a t t h e r e i s a

c lo s e d s e t N 1 c o n ta in e d i n N su c h t h a t bound o f ^ ( N 1 ) i s n o t l e s s t h a n ^ ( M ) .

) > ji(N ) - £ •

Hence t h e l e a s t u p p e r

The th e o re m f o l lo w s fro m D e f i n i t i o n 4 . 1 .

In T heorem 4 .2 t h e s e t o f a l l c lo s e d s e t s may be r e p l a c e d by a n y s e t S o f m e a s u ra b le s e t s , p r o v i d in g S in c l u d e s t h e s e t o f a l l c l o s e d s e t s 0

In

p a r t i c u l a r , S m ig h t be t h e s e t o f B o re l s e t s . THEOREM 4 . 3 : I f M is_ an y p o i n t

s e t f o r w h ic h p*(M ) i s f i n i t e

and i f

p.^(M) = p * (M ), t h e n M i s m e a s u r a b le . P ro o f:

T h e re e x i s t an open s e t jzf 3 M an d a c lo s e d s e t C C M s u c h

1

1

oo

that p(/zfK) < p*(M) + - and ^(C^) > p+(M) - j.

If ^ = TT

oo

and if Cff= 5 I Cg,

K=1 t h e n jzf 3 M 3 C • TT

O'

3 C^.,

V-itf-p)

H ence u ( ^ _ ) = p*(M ) a n d u .(M ) = u(C ) . Tt

< P*UO + ^ an(i

“ P * (M) 821(1 P ( C

P * (Iv0 ~

0"

K—1 S in c e $

TT

C

f!

K

and

S in c e k may becom e i n f i n i t e ,

B ut P * (M) = P * W , so t h a t )x(f6r ) =

A l l t h e s e num bers a r e f i n i t e b e c a u s e p*(M ) i s f i n i t e .

S in c e

IV .

t h e n , b y T heorem 3 , 1 , p ( ^ T ) = p ( ^ )

IM ER MEASURE

+ F-O^r" cff)s so t h a t

cff) = 0 .

(M - CCT) c ( j ^ - C ^J, p * ( H - C^) 5 m e a s u ra b le w i t h m e a su re z e r o .

25

~ °*

A g a in ,

3y Theorem 3 . 1 2 , (M - C^) i s

S in c e M = C + (M - C ^ ), M i s t h e sum o f tw o

m e a s u ra b le s e t s an d h e n c e i s m e a s u r a b le . THEOREM 4 . 4 ; P ro o f:

I f MK = 0 , p ^ M ) + p ^ N ) 5 ^ ( m + J l).

I f P a n d Q a r e a n y tw o m e a s u ra b le s e t s c o n ta i n e d i n M a n d N

r e s p e c t i v e l y , t h e n , b y T heorem 3 . 1 , p.(P) + ju(Q) = ji (P + Q) « /^ (M + N ).

The

th e o r e m f o l lo w s fr o m t h e f a c t t h a t /^ ( M ) a n d ^ ( N ) a r e t h e l e a s t u p p e r b o u n d s o f p ( p ) an d ju (Q ). THEOREM 4 . 5 : I f M^, M^,, . . . h av e a common p o i n t , t h e n

is_ a n y se q u e n c e o f s e t s s u c h t h a t no tw o

M^+ M^+ • • • ) =

+ ...

.

The p r o o f i s o m i tt e d . P ro p e r tie s I ,

I I , an d IV o f o u t e r m e asu re o b v io u s ly h o ld f o r in n e r

m e a su re an d P r o p e r ty I I I i s t h e a n a lo g u e o f T heorem 4 . 5 .

(N o te , h o w e v e r, t h a t

t h e s e p r o p e r t i e s c o u ld n o t b e u s e d a s a s t a r t i n g p o i n t f o r t h e t h e o r y o f mea­ s u r e i n w h ic h one b e g a n w i t h

i n s t e a d o f p * m The 5 o f

i s le s s a p p ro p ri­

a t e f o r s u c h a p u rp o s e th a n t h e = o f ji* . THEOREM 4 . 6 ; I f MtJ = 0 , t h e n ( I ) p +(M + U) = p + (M) + p * (N ) = p*(M + N ), a n d ( 2 ) p ^ M + H) = p * (K ) + p ^ K ) = p*(M + II ). P ro o f:

I t i s s u f f i c i e n t t o p ro v e p a r t ( l ) .

s e t c o n ta i n e d i n M.

L e t P be a n y m e a s u ra b le

By Theorem 3 . 1 , ^.(P ) + ju*(N) = jx*(P + N) = p*(M + N ).

S in c e p ^ M ) i s t h e l e a s t u p p e r bou n d o f p ( P ) , ^ ( M ) + p * (N ) = yu*(M + N ). A g a in , l e t P be an y m e a s u ra b le s e t c o n ta i n e d i n M + N, l e t B be a n y m e a s u ra b le s e t c o n t a i n i n g N, a n d l e t A = P - PB. T heorem 3 . 7 , A i s m e a s u r a b le .

H ence p (A ) = p ^ (M ).

S in c e AB = 0 , A C M.

By

S in c e P C (A + B ), t h e n ,

by T heorem 3 . 1 , p ( P ) 5 p (A + B) = p (A ) + p ( E ) 5 p*(M ) + p ( B ) .

S in c e p ^ M + II)

26

IV .

IBRER MEASURE

i s t h e l e a s t u p p e r bound o f p ( P ) an d p*(l'T) t h e g r e a t e s t lo w e r b o u n d o f p ( B ) , + N) = fi^ M ) + p * (U ). THEOREM 4 . 7 ; I f M i s a sejfc o f f i n i t e

o u t e r m easu re, a n d i f N _is an y s e t

c o n ta i n in g M a n d h a v in g a f i n i t e m e a s u re , t h e n p + (M) = p (H ) - ^i*(N - M ). P ro o f;

I f t h e s e t (M + II) o f T heorem 4 .6 i s m e a s u ra b le w i t h f i n i t e

m e a s u re , t h e n p ^ M ) + p * (ll) = p(M + h ) , w h e re MD = 0* p (M)

= p(M + 11) - p * ( l l ) .

S in c e pi*(ll) i s f i n i t e ,

M + 11 may be r e g a r d e d a s t h e s e t h o f t h e th e o re m *

D e f i n i t i o n 4 . 2 ; I f M _is a n y p o i n t s e t f o r w h ic h p*(M ) _is f i n i t e , yi*(M) - p.+(M) i s a m e asu re o f t h e n o n - m e a s u r a b i l i t y o f M an d i s

th e n

c a lle d th e non­

m e a s u re , v ( M ) , o f M. I t f o l lo w s fro m T heorem 4 .1 t h a t V(M) = 0 an d fr o m T heorem 4 .3 t h a t , i f *v(M) = 0 , M i s m e a s u r a b le . THEOREM 4 . 8 : I f MN = 0 , t h e n (1 )

-v(M) + V(N) = -v(M + U ),

( 2 ) v(M + TIT) + v(M ) = v ( K) , and ( 3 ) v(M + N) + v ( t l) = P ro o f s

P a r t ( l ) f o l lo w s im m e d ia te ly fro m Theorem s 2 .6 a n d 4 . 4 .

T heorem 4 . 6 , p ^ M ) + p * (ll) = p*(M + N) a n d p*(M) + (2)

= F * (M + N ).

By P a rt

f o l lo w s fro m t h e s e r e l a t i o n s a n d p a r t ( 3 ) i s a n a lo g o u s t o p a r t ( 2 ) . The r e l a t i o n s i n T heorem 4*8 a r e a n a lo g o u s t o t h e t r i a n g u l a r la w o f

d is ta n c e s .

V.

INVARIANCE OF MEASURE UNDER TRANSFORMATIONS

CHAPTER

27

V.

INVARIANCE OF MEASURE UNDER.TRANSFORMATIONS O nly o n e - t o - o n e t r a n s f o r m a t i o n s w i t h a n i n v e r s e a r e c o n s i d e r e d h e re * D e f i n i t i o n , 5 # 1 : A t r a n s f o r m a t i o n i s c a l l e d m easur e - p r e s e r v i n g i f i t le a v e s o u t e r an d i n n e r m e a su re i n v a r i a n t a n d p r e s e r v e s m e a s u r a b i l i t y . I t i s o b v io u s t h a t m e a su re i t s e l f i s a l s o i n v a r i a n t u n d e r a m ea su re p r e s e r v i n g tr a n s f o r m a t io n * THEOREM 5 * 1 : I f a t r a n f o r m a t i o n T le a v e s o u t e r m e a su re i n v a r i a n t , t h e n T i s m e asu re p r e s e r v in g * P ro o f:

I f M i s a m e a s u ra b le s e t an d N an a r b i t r a r y s e t , th e n

p* (N ) = p*(MN) + p*(N - MN)# o f N, t h e n ju*(N f ) = u n d e r T#

I f Mf i s t h e t r a n s f o r m o f M u n d e r T an d N! t h a t

) + ju*(N ?- M*Nf ) s i n c e o u t e r m e a su re i s i n v a r i a n t

B ut NT may b e r e g a r d e d a s a r b i t r a r y .

]u(M) = p(M ! ) .

H ence M1 i s m e a s u ra b le a n d

I t f o l lo w s fro m D e f i n i t i o n 4*1 t h a t i n n e r m e a su re i s a l s o i n ­

v a r i a n t u n d e r T* I t is a ls o *

o b v io u s t h a t i f T i s m e a su re p r e s e r v i n g , t h e i n v e r s e o f T i s

L ik e w is e t h e p r o d u c t o f tw o m e a su re p r e s e r v i n g t r a n s f o r m a t i o n s i s mea­

su re p re s e rv in g . THEOREM 5 * 2 : I f T i s a t r a n s f o r m a t i o n s u c h t h a t p ( l ) = p * ( l ! ) a n d p . ( j T) = p * ( J ) j w h ere I i s an y i n t e r v a l w i t h t r a n s f o r m I 1 a n d J 1 a n y i n t e r v a l w i t h i n v e r s e - t r a n s f o r m J , t h e n T i s m e a su re p r e s e r v in g * P ro o f: o pen i n t e r v a l s j

S u p p o se M t o b e an y p o i n t s e t .

T h e re e x i s t s a se q u e n c e o f

i ^ , * . , c o v e r in g M s u c h t h a t p ( l ^ ) + p ( l g ) + y u ( l3 ) + . . . = p * (M )+ £ *

Mf i s c o v e re d b y I£ + I£ + I£+ *•* an d ji*(M f ) = ] u * ( l|+ I £ + . * * ) = p * ( l £ ) + + ^ (Ig )

+ ••• " ^ (Iq ) +

+

+ •••

“ JU*(M) + 6 #

I t may b e shovm i n

28

V.

INVARIANCE OF MEASURE UNDER TRANSFORMATIONS

a s i m i l a r way t h a t ji* ( M) = p*(M T) + £ •

lie n e e ji*(M) =

)•

The th e o r e m

f o l lo w s fro m Theorem 5 * 1 . I t i s o b v io u s ly p o s s i b l e t o r e s t r i c t t h e i n t e r v a l s I an d J 1 o f T heorem 5*2 t o r a t i o n a l c u b e s w ith o u t l o s s o f g e n e r a l i t y * The re m a in d e r o f t h i s c h a p te r i s d e v o te d t o sh o w in g t h a t t h e g e n e r a l l i n e a r u n im o d u la r t r a n s f o r m a t i o n i s m e a su re p r e s e r v in g *

T h is w i l l be e f f e c t e d

by r e s o l v i n g s u c h a t r a n s f o r m a t i o n i n t o a p r o d u c t o f m e a s u re p r e s e r v i n g t r a n s ­ f e r n a t i o n s o f t h e ty p e s o c c u r r in g i n t h e f o l l o w i n g

11

The tr a n s f o r m a t i o n s M

Lemma? I

X v

+

/ X .

I I

x

=

1

/ .=

1 ,

n ) ,

iy

1

x r

/

/ e x . ,

I I I

I V

( V =

b v

x i

=

/ X .

=

1

1

X .

x

.=

± X

. ,

X XV

=

X V

( v

a

+

1

c x

.,,

x ^

-

x v

( v

/

i )

y

a r e m e a s u re p r e s e r v in g * P ro o f ?

The p r o o f s o f p a r t s

I - I I I a re t r i v i a l .

IV , l e t I b e t h e i n t e r v a l X ^
0 f o r e v e r y K

s e ts

su ch t h a t

is a

p o i n t c o n ta in e d i n

an y i n f i n i t e

se q u e n c e S o f m easur a b l e

an d su c h t h a t y*

Mk ) < OO, i f

F

s u b s e t o f S , and i f N i s t h e s e t o f a l l

p o i n t s P , t h e n N is_ m e a s u ra b le an d ju(N) = G •

(T h is i s known a s t h e A r z e la -

-Y oung th e o r e m .) P ro o f:

L et

R

N_+ N R

R+i

_+ . . .

an d 3 .1 1 , Sg an d N a r e m e a s u r a b le .

•

oo Then N =* T T S~. _ _ R —J.

B ut

...

By t h e c o r o l l a r y o f Lemma 2 o f C h a p te r I I I , ^j.(N) =

R

By T heorem 3 .1 0

an d ^ ( S g ) =

® ^ •

lim |i(S ^ .) = t . K—x x)

THEOREM 6 . 2 : in g M,if_ Cg.(P) is_ (V = 1 , . , . , n )

I f M i s a m e a s u ra b le s e t , i f jzf i s a n o p en s e t c o n t a i n Sg sK a n i n f i n i t e se q u e n c e o f c u b e s Xv — g - < x ^ < Xv +

w i t h c e n t e r P : ( X ^ ,. . . ,X ^ ) su c h t h a t

i s a d e c r e a s in g s e q u en c e

w ith l i m i t z e r o , i f su c h ia se q u e n c e o f c u b e s i s a s s o c i a t e d w ith e a c h p o i n t P 6 M,

a n d i f w ith e a c h cube C _ (P ), P £ M, t h e r e i s a s s o c i a t e d a c lo s e d 1—

N g(P) c: C ^(P) su c h t h a t

- - ———-

------ — ------------------

R

p y p)]

— — ----------

set

------—

> = G > 0 f o r e v e ry k and P , t h e n t h e r e e x i s t s

> [ ck ( p )] a s e q u e n c e S : N^ ( P ^ ) , N^ ( P ^ ) ,

...

t h e s e t s o f S h av e a common p o i n t ,

o £ th e s e t s N ^(P ) su c h t h a t ( l ) no_ tw o o f ( 2 ) e a c h s e t o f S i s c o n ta i n e d i n p $ an d ( 3 )

cc h i= l

(q ) i

c o v e rs M e x c e p t f o r

s e t o f m e a s u re z e r o . (T h is i s known a s t h e

1

V i t a l i c o v e r in g th e o r e m . ) P ro o f: triv ia l. an d

I t may be assum ed t h a t ^i(M) > 0 , f o r o th e r w is e t h e th e o re m i s

Suppose t h a t M i s b o u n d e d .

L e t J2f* be a n op en s e t su c h t h a t M e j r f 'c j6

V I.

(1 )

COVERING THEOREMS

35

jiOrf*- M) = -=-=■ 4*3 F or each P 6 M th e r e i s a s m a lle s t su b ­

(S u c h a s e t ft' o b v io u s ly e x i s t s . )

s c r i p t K ,^v s u c h t h a t CTr (P ) c / ' . (P ) (P )

T “~ CL (P ) c o v e rs M. K(P )

By T heorem 1 .1 0 ,

a se q u e n c e T o f t h e c u b e s 0

(P ) c o v e rs M. S u p p o se T t o be s e l e c t e d so t h a t K( P ) no two cu b e s i n i t h av e t h e same c e n t e r . L e t T be o r d e r e d so t h a t t h e ed g e s of i t s

e le m e n ts fo rm a m o n o t o n ic a l ly d e c r e a s in g se q u e n c e a n d t h e n l e t t h e

c u b e s i n T b e re n u m b e re d G / - o \ ( P ^ ) * w h ere C r

o f CL

/

L et J

(P ^ ).

}

C / \ ( P ^ ^ ) b e d e n o te d b y H

M~mZI i s a d eHence t h e r e

o f p s u f f i c i e n t l y l a r g e so t h a t ji(M -M $Z , J < 6 , w h ere 6

e x i s t s a v a lu e

i s an a r b i t r a r y p o s i t i v e n u m b er.

- i < fl( M S ^ , ) S

Some o f t h e cu b e s C , , \ ( P ^ ) , K V

w e e d in g -o u t p r o c e s s . as fo llo w s :

.

00 j ] (M-mZT ) ~ 0, an d lim u (M -m F ) - 0. p=l P K-»oo r

c re a s in g seq u en ce,

ji(M )

is w ritte n in s te a d

k

’

Then

p [c ^ ( 1 ) ;p ( 1 ) )]

...,

+ ...

C , ,\( P ^ ^ ) r

f

i

+ p [ y f t t ) (p ( P p ) ] .

a re to be d is c a rd e d by a

1

The cu b es r e t a i n e d , c a l l e d t h e s u r v i v o r s , a r e d e te r m in e d

C

i s a s u r v iv o r and C ^ j ( P ^ )

i s a s u r v i v o r i f an d

o n ly i f i t h a s no p o i n t i n common w i t h a n y p r e c e d i n g s u r v i v o r . v i v o r s b e re n u m b e re d CL ( P . ) , • JIt 1

1

CL R

(P

L et th e s u r­

) , w h ere t h e lo w e re d i n d i c e s a r e

Pi

t o be d i s t i n g u i s h e d fr o m t h e r a i s e d i n d i c e s i n t h e p r e c e d i n g e n u m e r a tio n . ( a ) Ho tw o s u r v i v o r s h a v e a common p o i n t ,

( b ) E ach s u r v i v o r i s c o n ta i n e d

.j . $ 3n f o r i f t h e e d g e s o f t h e s u r v i v o r s b e t r i p l e d ( w i t h a m u l t i p l i c a t i o n o f volum e in ^ .

by ^

( c ) By ( 2 ) , t h e t o t a l volum e o f t h e s u r v i v o r s i s

g r e a te r th a n

) w ith o u t d is p la c e m e n t o f t h e c e n t e r s , t h e e x p a n d e d s u r v i v o r s c o v e r t h e

34

V I.

COVERING THEOREMS

o r i g i n a l s e t o.C cu b es C K

v=

^ i s t a k e n t o be

t h e n t h e t o t a l volum e o f t h e s u r v i v o r s i s g r e a t e r t h a n — -n- J*(M). 2*3 N

K1

(P _ ),

ji ( m ) ,

L et

N (P ) be t h e s e t s N (P ) a s s o c i a t e d w i t h t h e s u r v i v o r s . KP l P i K(P )

1

T h ese s e t s a r e t h e f i r s t

s e t s o f t h e se q u e n c e S o f t h e th e o r e m .

v i o u s l y s a t i s f y c o n d i t i o n s ( 1 ) an d ( 2 ) .

A lth o u g h t h e y do n o t s a t i s f y co n ­

d i t i o n ( 3 ) , y e t t h e i r t o t a l m e a su re i s g r e a t e r t h a n L et H

= If

(P ) + . . .

+ Nk

1

They ob­

(P

).

—

‘

By ( 1 ) , ( 3 ) , and ( 4 ) ,

p(M -M E jj) = JJ-(M) - Ja( E j j ) ♦ p ( / ' ~ M)

< p(M ) - -£--=■ 2*3

p(M) +

• p(M) 4*3

= (1 - - £ — ) p(M ) 4*3 = 9 p (M ).

L et

0 be a n y num ber

se q u e n c e S f :

it,

in th e

(P n) , • -L

i n t e r v a l 0 < 0 < 1 su c h t h a t t h e r e e x i s t s a f i n i t e 1SL. (P ) o f t h e s e t s NTr(P ) s u c h t h a t ( 1 ) no tw o Jl p Ji

of

P t h e s e t s o f £>* h av e a common p o i n t , (S')

= 0 jx( M), w h ere

( 2 ) e a c h s e t o f S 1 i s c o n ta i n e d i n ft9 and (P ^ ) + •

i e x i s t s , f o r 0 may be 0 .

(p )•

S u ch a num ber 0

f p

i s m e a s u ra b le a n d c o n ta i n s no p o i n t o f

V I.

COVERING THEOREMS

0Pe n arLC* c o n ta i n s no p o i n t o f

ft

a b o v e , t h e r e e x i s t s a s e t T: N^

By a n a rg u m e n t s i m i l a r t o t h a t

(P +^ ) , ♦•, N^. (P ^ J o f t h e s e t s N^.(P) su c h

p+1

a ] an d S p [ f ( P ) = a ] a r e m e a s u ra b le f o r

For le t a ^ a ^ ,

b e a s e q u e n c e o f t h e num bers a. o f t h e t h e o ­

rem a p p ro a c h in g a g iv e n num ber a a s a l i m i t fro m a b o v e 0

Then a l l t h e s e t s

S p [ f ( P ) > a ] a r e m e a s u r a b le , a s i s a l s o t h e i r sum w h ic h i s sim p ly S p [ f ( P ) > a ] ; t h e c o m p le m e n ta ry s e t , S p [ f ( P ) = a ] i s l i k e w i s e m e a s u r a b le . L e t a a n d b be tw o num bers s u c h t h a t a < b ,

By T heorem s 5 ,3 an d 3 ,6 t h e s e t

S p [ a < f ( P ) = b ] i s m e a s u ra b le s i n c e i t i s t h e common p a r t o f Sp [ f ( P ) > a ] an d S p [f(P ) = b ] .

L et

( x .., , • » , x ) t v 1* 9 n' of p o in ts (x 1,
oo

^ ^ n ( ^ ) ^v p 3

th e n

^Tp P r o v i d e d t h a t a t l e a s t

t h e f u n c t i o n s f Q(P ) is_ sum mable. The p r o o f i s a n a l o g o u s t o t h a t o f

th e p re c e d in g theorem .

THEOREM 8 . 7 : I f ' f ^ ( P ) , f g ( P ) ,

i s any sequence of m easurable n on-

...

- n e g q t i v e f u n c t i o n s w i t h a. l i m i t f u n c t i o n f ( P ) a n d i f t h e r e e x i s t s a summable f u n c t i o n g ( P ) s u c h t h a t ** ( P ) * g ( P ) ? or a l l n , t h e n f ( P ) is_ m e a s u r a b l e and lim f f n -xx)

( P ) dvp -

P roof:

^ f ( P ) dv .

Let

F

^(P) m^kv J

= minimum o f f ( P ) , f

.-.(P ), mv /9 m+1 J9

$ f m+k( P )1,

F ( P ) “ g r e a t e s t lo w e r bound o f f ( P ) , f m+1 ( P ) , . . .

50

V III.

Tt f o l l o w s t h a t F_ ( P ) = F x m ,lx 7

LEBESGUE INTEGRAL

0( P ) = . . . an d t h a t F ( P ) a p p r o a c h e s F ( P ) fr o m m , 2x 7 m ,k 7 m 7

above a s k - > o o .

S im il a r l y , F^(P ) = F^(P) = . . .

b elow a s m-xoo *

A ll th e s e fu n c tio n s a re o b v io u sly m easu rab le.

la tio n s f and a l l =

=

n-m . H ence, by Theorem 8 . 6 , i f

lim [ F ( P ) dv - C l i m F ( P ) dv d m ,k P J, m ,k 7 P k~>oo 7 k-xo ^

lim in f n -xo

F o r any n r e ­

(P ) = F V( P ) and C f ( P ) dv_. = f p V( P ) dv_ h o l d f o r a l l m = n nv 7 m ,k' 7 i n P 1 m ,kv 7 P

k

lim in f n~>oo

an d F ^ P ) a p p r o a c h e s f ( p ) fr o m

^v p ”

y n (P ) dvp ?

^

n = m,

= f F ( P ) dv _ . 0 in 7 P

^v p ^ o r e a c -^ m*

l i m £ Fm( P ) dvp = J l i m m-^oo m-xoo

I t f o l l o w s by a s i m i l a r a rg u m e n t t h a t l i m i n f n-xoo = ^ (g (P ) - f ( P ) ) ^v p*

l i m i n f [ $ g ( P ) dvp n~>oo

Hence

H ence, by Theorem 8 . 5 ,

F j P ) dvp -

^ (g(?) ~

£ f ( P ) dT p .

( P ) ) dVp *

®ut the l e f t side of t h is in e q u a lity is

£ f*n ( P ) dvp ] =

Hence l i m s u p ^ f ^ ( P ) dvp ^ ^ f ( P ) dv p0 n-xoo

= l i m i n f ^ t n ( P ) dvp*

£ t n (P ) dvp =

Hence

n-xoo

£ g ( P ) dvp - l i m su p n-xoo

£ ^(P )

dvp .

T h e r e f o r e l i m su p ^ t n ( P ) dvp = n -x io

l i m £ f (P ) dvp e x i s t s an d e q u a l s § f ( P ) dvp . n -x o o n

D e f i n i t i o n 8 . 5 ; I f f ( P ) i s a n o n - n e g a t i v e f u n c t i o n w h ic h assu m es o n l y th e v a lu e s 0, v ^ , . . . ,

v ^ , i f S^. i s t h e s e t o f p o i n t s P a t w h ic h f ( P ) =* v^., K and i f jd(S^ ) e x i s t s and i s f i n i t e f o r e a c h v ^ , t h e n f ( P ) i s c a l l e d a f i n i t e l y E v alu ed f u n c tio n . THECREM 8 . 8 : I f f ( P ) i s a f i n i t e l y v a l u e d f u n c t i o n , t h e n i n t h e n o -

t a t io n of D e f i n i t i o n 8 .5 ,

^ f ( P ) dvp =

v ) . The sum of tw o f i n i t e l y K=1 ^ K v a l u e d f u n c t i o n s i s f i n i t e l y v a l u e d an d t h e i n t e g r a l o f t h e i r sum e x i s t s and e q u a l s t h e sum o f t h e i r i n t e g r a l s . The p r o o f o f t h i s t h e o r e m i s a p p a r e n t .

V III.

LEBESGUE INTEGRAL

51

THEOREM 8 . 9 : E v e ry r e a l n o n - n e g a t i v e m e a s u r a b l e f u n c t i o n f ( P ) i_s_ t h e l i m i t o f a_ s e q u e n c e f ^ ( P ) , f ^ ( P ) ,

• • • £ £ . f i f ti ^ e ly v alued f u n c t i o n s .

q u en ce c a n be c h o s e n i n s u c h a, manner t h a t 0 = f P )

T h is s e ­

= f^(P ) = • •. = f(P )

everyw here. P roof:

Let

be t h e c l o s e d cube i n R^ w i t h edge N and c e n t e r a t t h e

o r i g i n , w h ere N i s a p o s i t i v e i n t e g e r . then th e re e x is ts a p o sitiv e in te g e r

I f P i s a p o in t such t h a t f ( P ) < 2^, V = "V ( p ) s u c h t h a t

5!— 2

5 f(p ) < —

Z

Let

f (P ) = J

j £ L . 3* 0

if P £ v

an d f ( P ) < 2N ,

* f o r a l l o th e r P.

f ^ ( P ) i s f i n i t e l y v a l u e d an d i s m e a s u r a b l e , ^ ( P ) i n N, a n d

m o n o to n ically in c r e a s in g

lim f (P ) - f ( P ) . N->oo

THEOREM 8 . 1 0 : I f f ( P ) and g ( P ) a r e r e a l n o n - n e g a t i v e m e a s u r a b l e f u n c ­ t i o n s , t h e n f (P ) + g ( P ) is_ a l s o me a s u r a b l e , an d £ [ f ( P ) + g ( P ) ] dvp = =

J f ( P ) dvp + $ g ( P ) dvp . Proof:

T h is i s o b v i o u s l y t r u e f o r f i n i t e l y v a l u e d f u n c t i o n s , an d i t

f o l l o w s f r o m Theorem 8 . 9 t h a t t h i s i s t r u e f o r a l l f u n c t i o n s . THEOREM 8 . 1 1 ;

I f f ^ P ) - f g ( P ) » g ^ P ) - g 2 (P )» t h e n

,[ ^ ( P ) dvp- J f 2 ( P ) dYp = - £ g l ( P ) dvp - ^ g 2 ( P ) dvp , p r o v i d i n g t h a t f ^ ( P ) , f 2 (P )» g - ^ P ) . an d g g ( P ) a r e n o n - n e g a t i v e . P roof:

The th e o r e m f o l l o w s i m m e d i a t e l y f r o m Theorem 8 . 1 0 upon t r a n s ­

posing te rm s. T heorem 8 . 1 0 s t a t e s t h a t P o s t u l a t e 2 i n D e f i n i t i o n 8 . 1 i s s a t i s f i e d by t h e i n t e g r a l o f D e f i n i t i o n 8 . 2 .

I t is app aren t t h a t th e o th er p o s tu la te s

a r e a l s o s a t i s f i e d by t h i s i n t e g r a l when f ( P ) i s r e a l a n d = 0.

.

52

V IIIo

LEBESGUE INTEGRAL

I f two s u c h f u n c t i o n s a r e i d e n t i c a l e x c e p t o v e r a s e t o f m e a s u r e ' z e r o , t h e n i t i s o b v io u s t h a t t h e i r i n t e g r a l s a r e e q u a l*

Hence i t i s p o s s i b l e t o

g e n e r a l i z e Theorem 8*7 as f o l l o w s ; THEOREM 8 *1 2 s I f f ^ ( P ) , f ^ ( P ) ,

• • * _is si s e q u e n c e o f m e a s u r a b l e n o n ­

n e g a t i v e f u n c t i o n s w h ic h a p p r o a c h a l i m i t f u n c t i o n f ( P ) e x c e p t o v e r £i s e t o f m e a s u re z e r o and i f t h e r e e x i s t s a_ summable f u n c t i o n g ( P ) s u c h t h a t e a c h f n ( p ) 5 s ( p ) e x c e p t f o r a s e t o f m e a s u re z e r o , t h e n f ( P ) i s m e a s u r a b l e a n d lim n->oo

5 f n(P) dTP “ I f ^ Proofs

and Theorem 8 . 7

dTP*

The t h e o r e m f o l l o w s i m m e d i a t e l y fr o m t h e when t h e v a l u e s o f e a c h f n ( P ) , f ( P ) , and

p r e c e d i n g comment g(P ) a re changed t o

z e r o o v e r a l l t h e s e t s o f m e a s u re z e r o m e n t i o n e d i n ifche th e o r e m , D e f i n i t i o n 8 . 4-s L e t f ( P ) _be any r e a l - v a l u e d f u n c t i o n an d l e t f ( P ) = = f ^ ( P ) - f ^ ( P ) , w h ere f ^ ( P ) a nd f ^ ( P ) a r e r e a l an d n o n - n e g a t i v e *

I f such

a p a i r o f f u n c t i o n s f ^ ( p ) a n d f ^ ( P ) e x i s t w h ic h a r e sum mable, t h e n f ( P ) i s c a l l e d sum mable| i f s u c h a p a i r o f f u n c t i o n s e x i s t w h ic h a r e m e a s u r a b l e , t h e n f ( P ) i s c a l l e d m e a s u r a b l e a n d ^ f ( P ) dVp i s t a k e n t o be ^ f ^ ( P ) dVp- J f ,,(p )d V p . I t f o l l o w s fr o m Theorem 8 . 1 0 t h a t t h e v a l u e o f £ f ( P ) dVp i s i n d e p e n ­

80 ^ onS as

d e n t o f t h e manner i n w h ic h f ( P ) i s r e s o l v e d i n t o P-^(P) andth e d iffe re n c e

^ f ^ ( p ) dVp -

j £ g ( P ) dvp ^ a s s e n s e .

L e t f ( P ) be an y r e a l - v a l u e d f u n c t i o n an d l e t

f f ( p ) i f f (p) 5 o, f?(P) =
a ] i s o b v i o u s l y Rn i f a < 0 , a n d

I f a = 0 , t h e c o n d i t i o n m a x [ f ^ ( P ) - f ^ ( P ) , Q] > a

i s e q u i v a l e n t t o t h e c o n d i t i o n f ^ ( P ) - f ^ ( P ) > a , t h a t i s , t o .th e c o n d i t i o n f

( P ) > f 2( P ) + a .

But S p t f - ^ P ) > f 2 (P ) + a ] = YL Sp [ f 1 ( P ) > p > f g (P )+ a ] , p ra tio n a l

an d Sp [ f 1 (P ) > p > f 2 ( p ) + a ] = Sp [ f ;L(P ) > p ] .

Sp [ f 2 ( P ) < p - a ] .

two s e t s i n t h i s p r o d u c t a r e m e a s u r a b l e , f ° ( P ) i s m e a s u r a b l e ; f shown t o be m e a s u r a b l e i n a s i m i l a r way.

S in ce th e may b e

That t h e c o n d i t i o n o f su m m ab ility

i s n e c e s s a r y f o l l o w s f r o m t h e f a c t t h a t f ° ( P ) = f ^ ( P ) a n d f 2(P ) = f ^ ( P ) . THEOREM 8 . 1 4 ; A n e c e s s a r y an d s u f f i c i e n t c o n d i t i o n t h a t a r e a l - v a l u e d f u n c t i o n f (P ) be^ m e a s u r a b l e i s t h a t t h e s e t s S p [ f ( P ) > a ] be_ m e a s u r a b l e f o r an e v e r y w h e r e d e n se s e t o f v a l u e s o f a . P roof:

( C f . Theorem 8 . 3 ) .

The f a c t t h a t S p [ f ( P ) > a ] i s m e a s u r a b l e f o r a l l r e a l a i f

m e a s u r a b l e f o r an e v e ry w h e re d e n s e s e t o f v a l u e s o f a f o l l o w s a s i n t h e p r o o f o f Theorem 8 . 3 .

S in c e Sp [ f ° ( P ) > a ] i s

i f -a < 0 an d Sp [ f ° ( P ) > a ] =

~ S p [ f ( P ) > a ] i f a « o , a nd s i n c e s i m i l a r r e l a t i o n s h o l d f o r f g ( P ) , i t f o l l o w s by Theorems 8 C3 and 8 .1 2 t h a t t h e c o n d i t i o n i s s u f f i c i e n t .

S i n c e Sp [ f ( P ) > a ] =

= Sp [ f ° ( P ) > a ] i f a « 0 a n d Sp [ f ( P ) > a ] = Sp [ f ° ( P ) < - a ] i f a < 0, i t f o l l o w s b y Theorem 8 . 1 2 and 8 . 2 t h a t t h e c o n d i t i o n i s n e c e s s a r y . C o r o l l a r y : I n Theorem 8 . 1 3 t h e c o n d i t i o n f ( P ) > a may be r e p l a c e d by any of th e c o n d itio n s = a , < a , = a . Proof: 8 .3 .

T h is may be p r o v e d i n t h e same way a s t h e C o r o l l a r y o f Theorem

54

LEBESGUE INTEGRAL

VIII.

THEOREM 8 . 1 5 : A n e c e s s a r y an d s u f f i c i e n t c o n d i t i o n t h a t t h e m e a s u r a b l e f u n c t i o n f ( P ) be summable i s t h a t | f ( P ) | b £ sum mable. P roof:

T h is f o l l o w s i m m e d i a t e l y f r o m t h e r e s o l u t i o n ! f ( P ) | "

= f°(P ) + f°(P ). I t i s r e a d i l y s e e n t h a t P o s t u l a t e s 1-5 a r e s a t i s f i e d by t h e i n t e g r a l o f D e f i n i t i o n 8 . 4 and t h a t Theorem 8 .1 2 g e n e r a l i z e s t o s e q u e n c e s o f r e a l - v a ­ lu e d f u n c t i o n s when t h e c o n d i t i o n u n (p)i

“ g ( P ) i s r e p l a c e d by t h e c o n d i t i o n

= g (p). D e f i n i t i o n 8 . 5 : I f f ( P ) = g ( P ) + i h ( P ) , w h e re g ( P ) an d h ( P ) a r e r e a l ­

v alu ed f u n c t io n s , th e n

^ f ( P ) dvp i s t a k e n t o be

^ g ( P ) dvp + i ^ h ( P ) dvp .

Thus f ( P ) i s m e a s u r a b l e (sum m able) i f g ( P ) an d h ( P ) a r e b o t h m e a s u r a b l e ( summ able). L e t f^ C P ) -

C t (P ) f o r P I M, “S Lo f o r P e. -M,

w here f ( P ) i s an y complex f u n c t i o n

an d M i s a n y m e a s u r a b l e p o i n t s e t . D e f i n i t i o n 8. 6 :

^ f ( P ) dvp i s t a k e n t o "where M i s a s d e f i n e d i n t h e p r e c e d i n g p a r a g r a p h an d £ f ^ ( P ) dvp i s t h e i n t e g r a l o f D e fin itio n 8 .5 . THEOREM 8 . 1 6 : ^ f ( P ) dv

h a s t h e f o l l o w i n g p r o p e r t i e s , w h e re M i s a

m e a s u r a b l e p o i n t s e t , f ( P ) i s a m e a s u r a b l e complex f u n c t i o n , and j[mf ( P ) d v p^ :-----------------------------------^He i n t e g r a l o f D e f i n i t i o n 8 . 6 *

1) [ c f ( P ) dv M

= c f f ( P ) dv , w h e re c i s an y c o n s t a n t . M

2 ) £ W P ) + g (P )] dv M S) y 1 dv M 4 ) [ f (P ) M

= ^ f ( P ) dv + $ g (P ) dv M r M

.

= p(M ).

= 0 i f f ( P ) i s r e a l and = o f o r a l l F i n M.

V III.

5)

$ t-

LEBESGUE INTEGRAL

f (TP) dvp = j g j

55

J f ( P ) dvp , w h e re T i n an y l i n e a r

R m)

m

t r a n s f o r m a t i o n o f d e t e r m i n a n t D.

6)

I f f ( P ) = g ( P ) + i h ( P ) , f ( P ) i £ m e a s u r a b l e when a n d o n l y when S p [ g ( P ) > a ] and S p [ h ( P ) > a ] a r e m e a s u r a b l e f o r a l l r e a l a , and f ( P ) i s summable when and o n l y w h e n |f ( P ) |

7)

J f ( P ) dtr M

+

f ( P ) dY K

8)

I f f ( P ) i s m e a s u r a b l e and i f M.M. i j = 0 for i / in g statem en ts

=

J f ( P ) dY M+N

i n t h i s c a s e we haYe

j , th en th e fo llo w -

...

when and o n l y

1 ^(P ) I

f ( p ) dvp= i

If f^ (P ), f^(P ),

i f MN = 0 .

h o l d ; f ( P ) is_ summable o v e r

when i t is_ summable o v e r e a c h ML a n d

9)

i s sum mable.

< 00 » an a . or < b . , b . ) = 0a c c o r d i n g a s x . J = a. or = b . , r e s p e c t i v e l y , th u s / < a . or > b . . L. i i

V III.

LEBESGUE INTEGRAL

57

f+°°l f1]

l i m N (a . - x . ) ( x . - b . ) « 4 0 f i n t h e s e c a s e s r e s p e c t i v e l y , an d l i m f ( u ) - 4 0 L N«*oo i i i i | q\ C+oo"\ a c c o rd in g as lim u = 0 V. ) Each g ^ ( x ^ , . • . , x^) i s a n a l y t i c a l i n x^,. . . , x^ l ~ o o j

n

n

and t h e r e f o r e t h e l i m i t o f a se q u e n c e o f p o l y n o m i a l s 0 o f Cg o f 8 . 7 c (o n e c o u l d e v e n r e p l a c e F i n a l l y i t i s o b v io u s t h a t

Thus

of 8.7b i s p a r t

by C^, b u t t h i s i s u n i m p o r t a n t t o u s ) . of 8.7c i s p a r t of

I t o u g h t t o be m e n t i o n e d t h a t t h e c l a s s e s c o n t i n u e d b eyo nd t h e f i n i t e num bers m = 1 , 2 ,

of 8 .7 a .

o f B a i r e f u n c t i o n s c a n be

. 0 t o a l l elem ents of th e

so -

c a l l e d **Cantor*s s e c o n d c l a s s o f o r d i n a l numbers*1, b u t i t i s n o t d e s i r a b l e t o go i n t o t h e d e t a i l s o f t h i s p r o b l e m h e r e . THEOREM 8 . 1 7 : - — —----------- — g (x_ , °m 1

f ( x 15 • • • , x ) and v J. nr

g (x , • • . , x ) , 1 * rr

x ) a r e B a i r e f u n c t i o n s , t h e n h(x_. , 1 n / -— • — — —------------- - -------

- f(g ^(x ^,

•..,

P roof: i.e ., J

If —

x ^),

...

, g^(x^,

t h a t i t is c o n tin u o u s.

,

9

x ) = n

i s a ls o a B aire f u n c t i o n 0

. • • , xn ) )

C onsider D e f i n i t i o n 8 .7 a .

•..,

...

Assume f i r s t t h a t f b e l o n g s t o C^,

Then t h e

t h e o r e m i s o b v io u s f o r , • • • , gi n

°1

io O o, f o r c o n t i n u o u s f u n c t i o n s , and i t f o l l o w s by

in d u c tio n fo r g^,

i n an y o t h e r c l a s s e s C .

i n C^ an d a r b i t r a r y

g^,

...,

g^.

ry S l,

Thus i t i s p r o v e d f o r f

An o b v io u s i n d u c t i o n e x t e n d s i t t o f i n an y c l a s s

and a r b i t r a -

g^. Hence i t i s p o s s i b l e t o c h o o s e any c o n t i n u o u s f u n c t i o n

f(x^,

1

• • . , g^

...,

xn ) , ? or i n s t a n c e , max ( x ^ ,

ly n o m ia l; or a g a in th e B aire f u n c t io n f ^ ( x ) fr o m a B a i r e f u n c t i o n g ( x n , . . . . . f H( g ( x . ,

. . . . x ))

0

fo r

• . . , x ^ ) , min ( x ^ ,

fx

~ |q

i f J K ! == N o th erw ise *

x ) a n o th e r B aire f u n c t io n i f l g ( x 1 , . . . , Xn) | ^ H o th erw ise.

• . . , x ^ ) or anypo"thus o b t a i n i n g

g,T(x_ ,

x ) ~

,

THEOREM 8 . 1 8 ; I f f o r t h e r e a l B a i r e f u n c t i o n s f ^ P ) , f J P ) ,

• ••,

V III.

58

LEBESGUE INTEGRAL

f*m( P ) ex3-s 'bs e ^ Q ry w ^ e re , t h e n t h i s l i m i t i s a B a i r e f u n c t i o n 'above'' T h is l i m i t c e r t a i n l y e x i s t s i f f - ^ ( p ) , f ^ ( p ) ,

...

a r e u n i f o r m l y bounded b elo w V

Proof;

I t i s s u f f i c i e n t t o c o n s i d e r l i m i n f f ( P ) • T h is l i m i t c a n mv J m-^-oo be e x p r e s s e d by u s i n g o n l y t h e o p e r a t i o n s "m in" ( f o r a f i n i t e number o f f u n c ­ t i o n s ) an d “ l i m ” ( f o r e v e ry w h e re c o n v e r g e n t s e q u e n c e s ) a s c a n be s e e n i n t h e b e g i n n i n g o f t h e p r o o f o f Theorem 8 . 7 .

T h i s , t o g e t h e r w i t h t h e r e m a rk s p r e ­

c e d i n g t h e t h e o r e m , c o m p l e te s t h e p r o o f e THEOREM 8 . 1 9 : I f f ^ ( p ) , f ^ ( P ) ,

...

i s a sequence o f B aire f u n c t i o n s ,

t h e r e e x i s t s a n o t h e r B a i r e f u n c t i o n f ( P ) s u c h t h a t , w h e n e v e r l i m f w(P ) e x i s t s , ~ m-^oo th is lim it is f(P ). P roof;

I t may be assum ed t h a t f ^ ( P ) , f ^ ( P ) ,

. o. a r e a l l r e a l , a s

o t h e r w i s e t h e r e a l an d i m a g i n a r y p a r t s c o u l d be c o n s i d e r e d s e p a r a t e l y .

o th erw ise. =

Let

m -sK X)

l i m f ( P ) , t h e n f ( P ) m e e ts t h e r e q u i r e m e n t s o f t h e t h e o r e m 0 N-^oo A f t e r t h e s e g e n e r a l th e o r e m s on B a i r e f u n c t i o n s , i t i s d e s i r a b l e t o i n ­

v e s t i g a t e t h e c o n n e c t i o n b e tw e e n m e a s u r a b l e an d B a i r e f u n c t i o n s . THEOREM 8 . 2 0 ; E v e ry B a i r e f u n c t i o n f ( P ) jls m e a s u r a b l e . P roof;

T h is i s o b v io u s f o r t h e c l a s s

( u s i n g D e f i n i t i o n 8 . 7 b ) , an d

i t f o l l o w s b y i n d u c t i o n (b y Theorem 8 . 7 ) f o r a l l c l a s s e s C^. THEOREM 8 . 2 1 ; A f u n c t i o n f ( P ) i s m e a s u r a b l e when and o n l y when i t i s e v e ry w h e re e q u a l t o a B a i r e f u n c t i o n e x c e p t o v e r a s e t o f p o i n t s P o f m e a s u re zero. Proof;

That th e c o n d it io n i s s u f f i c i e n t f o r t h e m e a s u r a b i l i t y of

f*(p) f o l l o w s f r o m Theorem 8 . 2 0 .

T h e r e f o r e o n ly t h e n e c e s s i t y o f t h e c o n d i t i o n

V III.

m u st be proved©

LEBESGUE INTEGRAL

Suppose t h a t f ( P ) i s m easu ra b le©

59

f ( P ) may be assum ed r e a l ,

a s o t h e r w i s e t h e r e a l and i m a g i n a r y p a r t s c o u l d be c o n s i d e r e d s e p a r a t e l y © I t may e v e n be ass u m e d n o n - n e g a t i v e , a s o t h e r w i s e i t i s t h e d i f f e r e n c e o f tw o s u c h f u n c t i o n s ( f ° ( P ) and f ° ( p ) , D e f i n i t i o n 8 . 4 ) .

By D e f i n i t i o n 8 . 3 a n d Theo­

rem 8 . 9 , f ( P ) i s t h e J i m i t o f a s e q u e n c e o f f i n i t e l y v a l u e d f u n c t i o n s , so t h a t , by Theorem 8 . 1 9 , i t i s s u f f i c i e n t t o assum e t h a t f ( P ) i s o f t h e f o r m f 1 i f P t M, v 1 (P) + . . . + v 1 ( P ) , w h e re 1M( P ) 58 *< an d w h ere e a c h s e t M. 1M 1 n M (O ifP i-U , i s m e a s u r a b l e an d o f f i n i t e m e a s u r e .

But s u c h a f u n c t i o n i s a B a i r e f u n c t i o n

i f an y f u n c t i o n f ( P ) - 1,_(P ), w h ere M i s o f f i n i t e m e a s u r e , i s a B a i r e f u n c tion ©

By Theorem 3 . 1 7 , M i s a B o r e l s e t o f t y p e

e x c e p t f o r a s e t o f mea­

s u r e z e r o , so t h a t i t i s s u f f i c i e n t t o c o n s i d e r t h e c a s e w h ere M i s a s e t o f th e ty p e

© B u t, i n t h i s e v e n t , f ( P ) i s o b v i o u s l y a B a i r e f u n c t i o n . D e f i n i t i o n 8. 8 : A s e q u e n c e f ^ ( P ) , f g ( P ) ,

approach a l i m i t f u n c tio n f ( P ) u n ifo rm ly i f , number

5,

...

of fu n c tio n s is s a id to

c o r r e s p o n d i n g t o an y p o s i t i v e

t h e r e e x i s t s an i n t e g e r N such t h a t I f ^ ( P ) - f ( P ) | = 6 f o r a l l P

a n d f o r a l l n = N.

A sequence of f u n c t io n s i s s a i d t o a p p ro ach a l i m i t f u n c ­

t i o n e s s e n t i a l l y u n i f o r m l y i f , f o r e v e r y £ > 0, t h e r e e x i s t s a s e t o f m e a s u re < t

such t h a t over i t s

co m p le m e n ta ry s e t t h e a p p r o a c h i s u n i f o r m .

THEOREM 8 . 2 2 ; I f f ^ ( p ) , f ^ ( p ) , • . • is^ a s e q u e n c e S o f f u n c t i o n s sumraable o v e r a p o i n t s e t M o f f i n i t e m e a s u r e , an d i f S h a s si l i m i t f u n c t i o n f ( P ) , t h e n t h e a p p r o a c h o f S t o f ( P ) is_ e s s e n t i a l l y u n i f o r m. Proof:

C o r r e s p o n d i n g t o an y 6 > 0 l e t N^ g be t h e s e t o f p o i n t s P oo

such t h a t l f n (P ) - f ( P ) | > 6. n

I t fo llo w s t h a t

n=n

f o r a ny f i x e d p o i n t P, I f ( P ) - f ( P ) | < 6 n

oo

TT o

£=0

MNn n g = 0 sin ce, no

fo r a l l s u f f ic ie n tly larg e n.

oo But X T ML *(. A o A=o n0

Y 1 MN o

c => . . .

n 0+J-+^ » u

.

E ach N

r is m easurable sin c e

60

V III.

each f . ( P ) i s m easurable*

H ence, by t h e C o r o l l a r y o f Lemma 2 o f C h a p te r I I I ,

1

f o r 6 > 0 an d

LEBESGUE INTEGRAL

> 0 t h e r e e x i s t s a v a l u e n 1 o f n s u c h t h a t ju[ V~~

L e t 6 hav e t h e s e q u e n c e o f v a l u e s 1, l / 2 , l / 3 , —, JL.,

...

| if

• • • and l e t

00

< ~

have t h e v a lu e s

i s t h e i n t e g e r n T c o r r e s p o n d i n g t o 8 = ~ an d

^ ^ ^°° < th en u [ > _ m da] = r i= Q V ^ k 2

VC3C> r 00 a-nd u [ ^ Z zZ T MN . /? J ' k = l JL~ 0 V ^ k

< = u (E ) = £ r

,

^ Hence t h e

.

s e q u e n c e f ^ ( P ) a p p r o a c h e s f ( P ) u n i f o r m l y o v e r t h e p o i n t s e t M - E. THEOREM 8 * 2 5 ? I f f ( P ) i s d e f i n e d a n d m e a s u r a b l e over_ a s e t M, t h e n i t i s p o s s ib le t o e x c lu d e from M a s e t ^

o f a r b i t r a r i l y s m a l l p o s i t i v e m e a s u re

so t h a t , when f ( P ) i s d e f i n e d o v e r o n l y

ous^ o v e r

, f ( P ) i s everyw here c o n tin u ­

.

P ro o f ?

Suppose t h a t M i s o f f i n i t e m e a s u r e .

By Theorem 8 . 2 1 i t i s

s u f f i c i e n t t o c o n s i d e r t h e c a s e w h ere f ( P ) i s a B a i r e f u n c t i o n o f some c l a s s Cmo

I f f ( P ) i s o f c l a s s C^, t h e n t h e t h e o r e m i s a p p a r e n t .

rem p r o v e d f o r B a i r e f u n c t i o n s o f c l a s s C^. f(?)

If f(P )

S u p po se t h e t h e o ­

i s of c la s s

th en

i s t h e l i m i t of a s e q u e n c e S o f f u n c t i o n s f ^ ( P ) o f c l a s s C^.

s i s , f ^ ( P ) i s c o n t i n u o u s o v e r M-IL, w h e re

.

By h y p o t h e ­

By Theorem 8 . 2 2 ,

t h e f u n c t i o n s f ^ ( P ) a p p r o a c h f ( P ) u n i f o r m l y o v e r M-E, w h e re ja(E ) = ~ .

Hence

oo th e fu n c tio n s

f . ( P ) a re c o n tin u o u s i

o v e r t h e s e t M-T, w h ere T = E + / 5 7~T

1=1

a n d w here j i( T) = £. , an d S a p p r o a c h e s f ( P ) u n i f o r m l y o v e r M-T. c o n t i n u o u s o v e r M-To

I f M i s of i n f i n i t e m easure, th e

c o n s i d e r i n g t h e p a r t 'M^ o f M l y i n g i n t h e cube -N = c o n s tru c tin g th e e x c e p tio n a l s e t p u t t i n g M.= m} + M? + • . . , r i i I t Is th e preceding

f o r N = 1 , 2., . . . ,

an d r e p l a c i n g E by YU N=1

d e sira b le to ex h ib it

M. i

Thus f ( P ) i s

o o f i s o b t a i n e d by = N, i = 1 ,

...,

n,

*

w h ere w h ere u ( E ^ ) =

2n + i

a m easurable f u n c t io n f ( P ) such t h a t £

t h e o r e m c a n n o t be z e r o 0

. in

L e t f ( P ) be I^ (p ), w h e re M i s a m e a s u r -

VIII.

able set defined as follows:

LEBESGUE INTEGRAL

let all rational open intervals a < x < b be

ordered into a sequence 1^, 1^, ... • Jg, ... ^2V-1

in such a manner that

Define a sequence of intervals 5 ~ Ji(j^ ^ ) for every A and such that

XV ^ or every v °

J2V are

Consider those points x which belong to

only a finite positive number of the intervals there is

a last interval

tion of x.

61

•

For each such point x

which contains x and its index

A = A(x) is a func­

Let M be the set of all such points x for which Ais even, and let QO

N be the

set of all such points x for which

A is odd.

CO

Let K = J A A

CO

TheV

(KA ) ^ ( J A ) - p j A ( I l J A + i )] ? y j A ) - 2 Z ^ ( J A + i ) =

?

)(1 - I - 1 - ...) = i)a(JA ).

ju (J

J. (V ~ J .) i=l A +1

Since M = Kg+ K4+ ... and N = Kx+ K g+ ...

it follows that M and N are measurable.

Furthermore 1 , 3 V

, and I,, 3 2V v

n. 2V-1

so that in every interval there is a part of M and a part of N and each such part is measurable and of positive measure.

Hence in every

interval f(P)

assumes both of the values 0 and 1 over sets of positive measure, and £

must

be positive. Definition 8.9: If for each £ > 0 and each 9, 0 < 9 < 1, there exists a 5 > 0 such that, for every cube C: x^°~ < x_^ < x^°^+ ~ with edge a < 6 ,

(i = 1, ..., n)

the set of points P in C for which |f(P°) - f(P)l = 6

is

of measure = 0an , then f(P) is called approximately continuous at P°. THEOREM 8.24. If f(P) is defined and measurable over a set M, then f(P) is approximately continuous over M except for at most a set of measure zero. Proof:

Let

be the complex plane so that, for any point P, f(P) £ R^

Let all rational intervals in R^ be ordered in a sequence 1^, I , .. . .

In

the notation of Theorem 8.3, each set

By

= Sp[f(P) 2. 1^] is measurable.

Theorem 6.5, for every point P, except for a set Z of measure zero, it follows that all SV containing P have unit density at P.

Let P be a fixed point in

62

V III.

R b u t n o t i n Z. n

LEBESGUE INTEGRAL

L e t C be t h e c i r c l e i n R0 w i t h c e n t e r f ( P ) an d r a d i u s £ . Z

In sid e C th e r e e x is ts a square 1^ c o n ta in in g f ( P ) . set S

S in ce th e c o rre sp o n d in g

i s of u n i t d e n s i ty , t h e c o n d itio n f o r approxim ate c o n t i n u i t y a t P is

sa tisfie d .

IX .

MONOTONIC FUNCTIONS

63

CHAPTER IX. MONOTONIC FUNCTIONS D e f i n i t i o n 9 . 1 ; In th e space d e f i n e d o v e r an y f i n i t e

o £ r e a l n u m b e r s , l e t f ( x ) be_ r e a l an d

or i n f i n i t e i n t e r v a l 0 f ( x ) is c a lle d

i n c r e a s i n g i f f ( b ) > f ( a ) whenever b > a , d e c r e a s in g i f f ( b ) < f ( a ) w henever b > a , m o n o to n ic a lly i n c r e a s i n g i f f ( b ) » f ( a ) whenever b > a , m o n o to n ic a lly d e c re a s in g i f f ( b ) 5 f ( a ) w henever b > a . I t i s a p p a r e n t t h a t t h e t h i r d c l a s s i n c l u d e s t h e f i r s t an d t h a t t h e f o u r t h in c lu d e a th e second; i f f ( x ) i s in th e f o u r t h c l a s s , - f ( x ) is in th e th ird .

Hence i t i s s u f f i c i e n t t o c o n s i d e r o n l y m o n o t o n i c a l l y i n c r e a s i n g ( m . i . )

f u n c t io n s in d ev elo p in g t h e p r o p e r t i e s of th e s e v a rio u s ty p e s of f u n c t io n s . D e f i n i t i o n 9 . 2 : I f f ( x ) i s m . i . , t h e n t h e l e a s t u p p e r bound o f f ( x )

fo r x < x q i s denoted by f (x q) and th e g r e a te s t lower bound o f f ( x ) fo r x > x

i s d e n o te d by f , ( x ) . + o'

o ____________ 1

THEOREM 9 . 1 : I f f ( x ) is_ m . i . , t h e n

lim f ( x ) = x~*x < 0

and

lim f ( x ) = f (x ) . ' 7 +v o x~>x > o P r o o f : T h e re e x i s t s a number x n < x 1

o

such t h a t f (x n ) > f ( x ) ~ F ' 1' o' Q

Hence, f o r a l l x s u c h t h a t x^ < x < x ^ , f ( x q ) - £ c f ( x ) = f

•

( x ^ ) , and t h e

f i r s t p a r t o f t h e t h e o r e m i s im m e d i a te ; t h e s e c o n d p a r t may be t r e a t e d a n a ­ lo g o u sly . THEOREM 9 . £ i I f f ( x ) is_ m . ^ . an d _if x q < x ^ , t h e n f + ( x Q) = f Proof:

L e t x^ be s u c h t h a t x q< x^< x^.

f +(xo) S f ( x 2) 5 f _ ( Xl).

(x^).

Then, by D e f i n i t i o n 9 . 2 ,

64

IX .

MOHOTOHIC FUNCTIONS

9 . 3 ; ___ I f f (' x )J ____ i s __ m . i . ,5 ______ t h e n f —(x o ) = f ( x o ) 5 f + ( x o ) . _THEOREM _ _ _ _ _____ The p r o o f i s t r i v i a l . D e f i n i t i o n 9 . 3 : I f f ( x ) i s m. i „, t h e d i f f e r e n c e f +( xq ) - f ( XQ) i s c a lle d th e o s c i ll a ti o n ,

osc f ( x q ) , of f ( x ) a t x ^ .

I t f o l l o w s fro m Theorem 9 .3 t h a t o sc f ( x q ) = 0 .

I t is apparent th a t

o sc f ( x ^ ) ~ 0 when an d o n l y when f ( x ) i s c o n t i n u o u s a t Xq . THEOREM 9 . 4 : I f f ( x ) is_ m .i_ ., t h e n t h e number o f p o i n t s a t w h ic h i t i s d i s c o n t i n u o u s i s a t m ost c o u n t a b l e . P roof:

!y Theorem 9 . 2 , f o r x ^ / x^ t h e i n t e r v a l s f ( XQ) = y = f +( x q )

a n d f (x ) = y S f + ( x ^ ) a r e n o n - o v e r l a p p i n g e x c e p t p o s s i b l y f o r a common end p o in t.

The i n t e r v a l -oo < y < +oo can be d i v i d e d up i n t o o n l y a c o u n t a b l e

number of n o n - o v e r l a p p i n g i n t e r v a l s of l e n g t h = fc. > 0 .

Hence t h e number o f

p o i n t s a t w h ic h o sc f ( x ) = £ > 0

I f £ is g iven th e v a ­

lu es l / 2 ,

l/ b , l/4 ,

...

i s a t most c o u n t a b l e .

, th e o s c i l l a t i o n of f ( x ) a t each p o in t of d is c o n ­

t i n u i t y is g r e a t e r th a n a lm o st a l l of th e s e v a lu e s of £ •

Hence t h e number

o f p o i n t s o f d i s c o n t i n u i t y i s a t m ost c o u h t a b l e . THEOREM 9 . 5 s

I f f ( x ) i s m . i . , t h e s e t of p o i n t s o f d i s c o n t i n u i t y of

f ( x ) may be e v e ry w h e re d e n se o v e r R ^• P roof:

By Theorems 1 .8 and 1 . 9 t h e s e t o f r a t i o n a l p o i n t s i n R^ i s

e v e ry w h e re d e n se i n R^ and may be o r d e r e d i n a s e q u e n c e S : x ^ , x ^ , x be a p o i n t o f R_. and l e t x , x , r 1 n n *

1

1

Then f ( x ) = ——- + — + . . . 2n1 2n 2p o in t o f R^.

±

be t h e e l e m e n t s o f S l e s s t h a n x .

6

i s m . i . an d i s d i s c o n t i n u o u s a t e a c h r a t i o n a l

D e f i n i t i o n 9 . 4 : I f f ( x ) and g ( x ) a r e m . i , and i f t h e c o n d i t i o n s f ( x q ) - g ( x q ) an d f + ( XQ) = g+( x Q) h o l d a t e v e r y p o i n t x q , t h e n f ( x ) an d g(x) are c a lle d e q u iv a le n t.

Let

IX .

MONOTONIC FUNCTIONS

65

THEOREM 9 . 6 : I f f ( x ) an d g ( x ) a r e m. i_., & n e c e s s a r y and s u f f i c i e n t c o n -

(f(xb) < (f (x2^)

d i t i o n t h a t f _an_d g be e q u i v a l e n t i s t h a t J| gand ( x ^ jI = j/ and ^ j V w h e n e v e r x 1< x 9 .

Proof:

The c o n d i t i o n i s n e c e s s a r y , f o r f ( x ^ ) == f ( x ^ ) - g ( x ^ ) = g ( x ^ ) .

S i m i l a r l y , g ( x 1 ) 5 g „ ( x 2 ) = f_^(x2 ) = f ( x 2 ) . and g ( x ^ ) 5 g ( x g ) . ber. f

For a l l x
g ( x Q) , w here g ( x ) = f + ( x ) - Cx.

---> c,

Theorems 9 .1 6 and 9 .1 7 a p p l y 0 x q i s i n tbs

i n t e r v a l s a^


x < b .

s e t jzf,

th a t is ,

i n one o f t h e

By Theorem 9 . 1 7 , p a r t 2 ) , t h e sum o f a l l s u c h i n t e r v a l s

may be made a r b i t r a r i l y s m a l l by t a k i n g C s u f f i c i e n t l y l a r g e .

T h is c o m p l e te s

th e p roof of 1 ). P roof of 3 ):

S uppose D ^ f ( x Q) < C < D < D * f ( x Q) .

Then t h e r e e x i s t s

< C; t h e r e f o r e £i~jLx, ) (,.x_q.) < q x* - x o o ( x q b e i n g a p o i n t o f c o n t i n u i t y ) so t h a t f +( x ^ ) - Cx q< f + ( x * ) - Cx’ . L e t

p o in t

x ’ < x f o r w h ic h

o

x1 - x

a

IX .

Y q- - x o

and l e t y* « - x » .

MONOTONIC FUNCTIONS

73

Then y* > y Q and. f +( - y Q) + CyQ > f +( - y f ) + Cyr . f(x” )- fix ) f o r w h ic h — — r------- i-JU- > D, so t h a t o

A gain, t h e r e e x i s t s a p o in t x n > x

c u ssio n

" > D an(* ) ” Dx c f + (x ” ) “ Dx” . ( i n t h e p r e c e d i n g d i s o x f an d x ” c a n be a r b i t r a r i l y c l o s e t o x q *) By Theorem 9 . 3 7 , p a r t 4 ) ,

w h e re A

- C, i f y^ = ~x^ i s r e g a r d e d a s l y i n g i n some i n t e r v a l

I s a f < y < b 1,

then y

l i e s i n one o f t h e i n t e r v a l s a < y < b an d *5 [ f (~ a ) ~ f , ( - b )] = n J n z— n n/ n = C ( b f - a * ) 0 S uppose y ^ l i e s i n t h e i n t e r v a l a ^ f < y < b ^ , • Then

°o

By Theorem 9 . 1 7 , p a r t 2 ) . w h ere A = D, i t f o l l o w s t h a t x

~b .< x < - a , . ni o n’

J

* *

l i e s i n one of t h e i n t e r v a l s “ i

[ f - ( _ a n ' > “ f +( - b n- 3*

**

9

-b t < x oo

>

,

9

+ w .(x ) +

y

n-.

(x ),

*9

1

(x), n0

3

...

such t h a t ...

•

co

w (x) = pV+1 H p i

w ( x )° ^

S i n c e t h e r i g h t member

e q u a t i o n i s d i f f e r e n t i a b l e o v e r a = x = b ex -

i s bounded, each term i n t h i s

c e p t f o r a s e t o f m e a s u re z e r o c w (x ) - w (y) no tL— = H I x - y p !

S in c e

2

L e t t h i s s u b s e q u e n c e be d e n o t e d b y w ^ ( x ) , w ^ ( x ) ,

oo Then w ( x ) + . . . 1

4^ ( x ) ,

x - y

l o s s o f g e n e r a l i t y i n a s s u m in g t h a t v+'n ( a ) = 0 f o r a l l n .

p vp ( b ) i s f i n i t e - . i= l i

...

>

- 0 and 4^n ( y ) ~ 4 ^ ^

ip ( b ) = 0, t h e r e e x i s t s a s u b s e q u e n c e In

oo + H p i

,

>

x - y

T h ere i s no

n y

f (*f)

WqCx) - w1 ( y ) Then ----------------------- + . . . x - y

w (x) - w (y) E ll x - y

wv (x ) " + ------------------------ + x - y

an d th e sum o f th e f i r s t V terms

i s n o t g r e a t e r t h a n t h e r i g h t member o f t h i s e q u a t i o n .

L e t y b e c o n s t a n t an d

l e t x approach y .

The l i m i t of t h e r i g h t member e x i s t s ( e x c e p t o v e r a s e t Gf oo m e a s u re z e r o ) and i s i n d e p e n d e n t o f V • Hence H w ' ( y ) i s bo u nd ed an d c o n p=i p v e r g e n t ( e x c e p t ov er a s e t o f m e a s u re z e r o ) a n d l i m w T( y ) = 0. u-^oo ^ THEOREM 9.].Ss

_

I f f ( x ) i s r e a l and summable o v e r a = x. = b an d i f

T ( x ) - ^ ^ f ( y ) d y , t h e n 1) 9 ( x ) i s c o n t i n u o u s and o f b . _v 0, 2)Cp ( x ) i s d i f f e r e n tia b le

o v e r a = x = b e x c e p t f o r a s e t o f m e a s u re z e r o . , an d 3 ) ' f ’ ( x ) = f ( x )

e x c e p t o v e r a s e t o f m e a s u re z e r o . P roof:

I t i s s u f f i c i e n t t o c o n s i d e r t h e c a s e w here f ( x ) 5 0, f o r any

IX .

r e a l fu n c tio n f ( x ) -

MONOTONIC FUNCTIONS

~1

— 1----- . e a c h o f t h e s e f r a c t i o n s

2

i s a lw a y s

= o , an d i t

75

2.

i s e v id e n tt h a t t h e th eo rem

b e tw e e n tw o n o n - n e g a t i v e f u n c t i o n s .

Since f ( x )

ex ten d s t o t h e d if f e r e n c e (x) is m .i. ;

= 0,

* f( x ) i s o f b . v . a n d , by Theorem 9 . 1 8 , p a r t Z) o b t a i n s . a sequence of v a lu e s of x w ith l i m i t x*. x

. . . be

L e t f^_(y) = ^ o ^ e l s e w h e r e "

~ x ~

S i n c e | f x ( y ) | = | f ( y ) | f o r a l l x , an d s i n c e

t ( y ) , Theorem 8 .1 6 i s a p p l i c a b l e and i t f o l l o w s t h a t

l i m *f(x ) =* ^ (x * ) , so t h a t p a r t 1 ) o b t a i n s . n~sKX) I f f ( x ) i s n o t bounded, l e t f (x ) = i K J 9 N y (0 x ‘V * )

Let x^, x ^ ,

b

Then f ( x ) = ^ f ( y ) d y = £ ** ( y ) d y . a a li m f (y) = f n-H)o n

th e re fo re

“ S f N^ y ) dy i s a

I t r e m a in s t o p ro v e p a r t 3 ) .

, f ( x ) > N,

^ N + l ^ X^ “ ^ N ^

N = 1, 2,

9

i s m#io> an d

...

.

Then

l i m " V X^ = ^ N-xx> .

x )*

The p r e c e d i n g lemma shows t h a t i t i s s u f f i c i e n t t o c o n s i d e r f ^ ( x ) , t h a t i s , bounded f u n c t i o n s . a = 2 Ih | •

In D e f i n i t i o n 8 .9 l e t 6 = 1 - 8 .

, make | h |
0 and h < 0 ) , and l e t N^ be t h e

complement of

¥ ( x +h)~ f ( x ) x +h -------------- — - f ( x 0 ) = r j f(x )d x -f(x h

in th is in te r v a l.

) = -($

xo

2

Then

f(x )d x + $ M1

f(x )d x )-f(x N1

= — [C [ f ( x ) - f ( x )J d x + f [ f ( x ) - f ( x ) ] d x ] . & x/t O O M1 N1

B ecause o f t h e a p p r o x i m a t e c o n -

t i n u i t y of f ( x ) , th e f i r s t i n t e g r a l is =

•

= 2 A ( a h ) , w here A i s a

bound o f f ( x ) . Hence

The s e c o n d i n t e g r a l i s th e e n t i r e e x p re ssio n

is

= (4A+l)£. , and p a r t 3) i s im m e d i a te . , If

f(x )

i s m . i . and i f f ( x ) = f ^ x )

(w here i t e x i s t s ) , t h e a s s e r t i o n

)=

76

IX .

MONOTONIC FUNCTIONS

x t h a t J f ( y ) d y = 'f(x ) i s o b v io u s ly f a l s e b ecause a * f( x ) n e e d n o t be c o n t i n u o u s . I t w i l l be shown t i o n i s g e n e r a l ly f a l s e even i f

x j* f ( y ) d y i s c o n t i n u o u s , w h i l e a in th e sequel th a t th i s a s s e r­

Y (x ) i s m .i . and c o n tin u o u s .

L e t f ( x ) be summable, un bo u n d ed , an d n o n - n e g a t i v e o v e r a = x = b ; l e t lim { f ( x ) d x = f f ( x ) d x 0 Hence t h e r e ,T J Nv ' J N—3«oo a a b b e x i s t s a n Nq s u f f i c i e n t l y l a r g e su c h t h a t § f (x)dx = £ f ( x ) d x - t • , a o a

f ._ ( x ) « N 10 ^

^ ~ w* i f f ( x ) > N. v '

l^ en

b

t

Thus

= J [ f ( x ) - f ^ ( x ) ] d x = £ f ( x ) d x ? o , w here S = Sx [ f ( x ) > Nq ] . a o S

T h is

proves THEOREM 9 . 2 0 ;

I f f ( x ) i s summable, u n b o u n d e d , an d n o n - n e g a t i v e o v e r

a 5 x = b , th e n i t i s p o s s i b l e t o i n t e g r a t e i t over such a s e t S i n t h i s i n ­ te rv a l th a t it s

i n t e g r a l i s a r b i t r a r i l y s m a l l w h i l e f ( x ) i s b ou n d ed i n t h e s e t

co m p le m e n ta ry t o S. D e fin itio n 9 .9 ? A fu n c tio n f ( x ) said

t

to

d e f i n e d o v e r an i n t e r v a l a = x = b is^

be a b s o l u t e l y c o n tin u o u s o v er a= x = b i f , c o r r e s p o n d in g t o

each

> 0, t h e r e e x i s t s a 6 > 0 s u c h t h a t w h e n e v e r y ( b . - a . ) < 0 , w h e re

N ], w h ere a = x = b .

Then, by

Theorem 9 . 2 0 t h e r e e x i s t s a s u f f i c i e n t l y l a r g e N s u c h t h a t £ f ( x ) d x < -■ 0 M be an y s u b s e t o f t h e i n t e r v a l a = x = b o f m e a s u re < Then

f f (x )d x = C f (x )d x + f f ( x ) d x < n6 + % » M Mx MS

th en J f(x )d x < £ . M

If

60 6 is

Let

Le

M - MS.

t a k e n t o be -~rr ,

Now l e t M be t h e sum o f a l l i n t e r v a l s a . S' x = b . , w h ere

n = a = b = b , and w h ere u(M) = X I ( b . - a . ) < 6 . Then n n. 9 rv ' r “: i i i= l n b. a. n I f ( x ) d x = X I [ [ Xf ( x ) d x - § Xf ( x ) d x ] = X I [ ' f ( b . ) - 4*( a . ) ] . T h ere e x i s t s a M i= l a a i= l 1 1

a * a .= b . » 1 1

...

6 > 0 such t h a t t h i s l a s t sum i s alw ay s < £ . S i n c e * f(x ) i s m . i . , t h i s l a s t n sum = H l ' f ( b J - Y ( a . ) | and f ( x ) i s a b s o l u t e l y c o n t i n u o u s . i= l 1 1 THEOREM 9 . 2 2 ? I f

* f ( x ) is_ m . i . a n d a b s o l u t e l y c o n t i n u o u s , t h e n t h e

t r a n s f o r m a t i o n x ’ = ' f ( x ) maps s e t s of m e a s u re z e r o on s e t s o f m e a s u re z e r o . P roof:

L e t M be a s e t o f m e a s u re z e r o i n R^ and l e t N be i t s image

under th e tr a n s f o r m a tio n x 1 = ^f(x).

L e t M be c o v e r e d by a s e q u e n c e o f n o n oo o v e r l a p p i n g i n t e r v a l s I 1 , I 9 , . . . su c h t h a t y If I. is th e in 1 Cn , t u ( l 1. ) < 1 1=1 t e r v a l a^= x = b ^ , t h e n can b e t a k e n s u f f i c i e n t l y s m a l l so t h a t

6

/ ~fyfb ) ) ] < £ b e c a u s e o f t h e a b s o l u t e c o n t i n u i t y o f vf ( x )° (The a b i= l n s o l u t e c o n t i n u i t y g iv e s t h i s d i r e c t l y o n ly f o r V , b u t n may be a l l o w e d t o i= l become i n f i n i t e . )

But t h e s e t o f i n t e r v a l s

LP (a ^ ) ® 1 *

^ ^ i^

GOV0rs

Hence ja*(N) < t $ t h a t i s , ^i(N) = 0 . I t i s d e s i r a b l e t o show t h a t t h e c o n d i t i o n o f c o n t i n u i t y a l o n e i s i n -

IX .

78

MONOTONIC FUNCTIONS

s u f f i c i e n t in t h e p re c e d in g theorem .

T h i s w i l l b e done by e x h i b i t i n g a m . i .

c o n t i n u o u s f u n c t i o n w h i c h d o e s n o t map a s e t o f m e a s u r e z e r o on a s e t o f mea­ sure zero.

Let

( x ) be d e f i n e d o v e r t h e i n t e r v a l 0 = x = 1 a s f o l l o w s :

be t h e p o i n t x = ^ and l e t

be t h e p o i n t

i n t e r v a l of 0 = x 5 1 w ith c e n te r be a s u b i n t e r v a l o f 0 =

f ( x ) 5 1 w ith ce n te r

L e t P-j.^1 ^ e a s u b ­

( x ) = -g-.

0

A set M 6 and ( 2 )

MN.

D e f i n i t i o n s 1 0 .1 .1 and 1 0 .1 .3 t h e c o n d it io n (P-^), (p ^ )

r e s p e c t i v e l y a re s u p e r f lu o u s , = I

th e n so does M -

i s s a id

N- M£

c

S ) ) be any c o l l e c t i o n o f s e t s .

t o b e ” im m e d ia te ly b elo w ” a s e t N € '(T* i f :

. T h is r e l a t i o n i s

i t i s d en o ted by M