WHICH CAME FIBSTs THE MEASURE OB THE BJTBOHAL?

AVPB0VB)«

• I £ . OJO-CCX.' • v5Le> wmrmmv

1

/ - ^ • / .

m & n <®f th« Graduate sahool

m i d came wtmfi tm mkmrn on ms mrgmj ,?

THESIS

Prwentad to th« Graduate Counoll of tho

north Texas 3tat« tfelvortlt? to Partial

PulfllliMnt of th« Bmulraaanta

Fbr the D#®r®« of

M&STER OF SCXEHCE

IT

John Barms Ghapsan, 1* 8.

Denton, faxas

JtUM, 1966

< m m op Q w m m ®

Chapter Page

X. C G S P U O T U AJSOIfTO SSf fractions

ASD HSASUBABX3 FOHCTIOirS . . . . . . . . . . . 1

I X * XSrSgaSATXQN VIA 3BB HUBUHE. 1 5

i n * msmm v i a m mmm&x* • * • • * • • » • • • 3 8

X?. COHPASXSOB of rat 36

BXBLXOCEBAPHT* 6k

ill

< m f « » i

commmm Mmvsvm SET FOMOTIOMS

AND wBAsmmw wmcsmm

This thesis provides a developswnt of integration fioi

t m different paints of Ttw. la Chapter !t a neasure sad

• measurable funetion are defined* A theory of integration

is then developed in Chapter 11 based on tho measure* la

Chapter 111, the integral is introduced direotly without

first going thxm^h. tho proeess of defining a measure» and

a measure is developed from the integral* The eonoluding

ohftpter shows tho equivalenoe of the two integrals under

rather general oonditions*

Throughout the thesist the letter 1 will be used to

denote the spaoe under disoussion. Lower oase letters suoh

as x and y will denote elements or points of tho spaoef sets

of these points will he denoted toy capitals suoh as A and B|

and olasses of these sets will be denoted by soript oapitals

suoh as (L and ® « Hie letter 1 will bo used to designate

the set of real numbers, and S® will denote the set of

extended real ambers*

If £fnj is a sequenoe in B« and if there exists f£ B

suoh that given any £ > 0, there exists a positive integer V

suoh that |tn~fI< € whenever n> ie » then f is said to be the

limit of {fa} • this definition of limit will be denoted by

X U tn * f • the notation 11* fa « oo will sonetlnes be need

to indloate no suoh llalt exists.

Chapter I provide® m m & m m r y baekground for the

develepaent of the integral. A measure funotlon 1® defined

en a 0- -algebra, t hus fentlng * atoasunt iptM. t he Integral

@f Chapter I I 1® based ©a this neasure spaoe. the theorjr of

the integral le oonoerned with a partloular ©las® of funotlons,

t h e olass of Measurable funotlons. A measurable funotlon is

defined* and several of the properties of neasurable funotlons

a r e established in Chapter I . fhe ehapter oonoludes with an

Important convergence theoren.

Definition* An algebra of sots is a noneaptj olass

G of sets sueh that

i) l f A £ < 2 a n d i e ( g 9 then AUB z 0 « and

11) if A € <8 • then Xe (3 where X denotes the ooa-

pleaent of A with respeot to the spaoe X.

*•*•1 IMfinltlon. A -algebra of sets is an algebra

of sets (8 such that if lt£ (3 » 1 m lt 2* 3# • ••» then

U % £ <B • i=i *

2ft may spaoe X, the olass of all subsets of X and the

olass consisting of the two sets X and 0 are both <r-algebras.

Iheorea. If (8 is a cr —algebra* then X 6 (B and

#£ (8 . Proof, fhe olass <$ is nonempty by definition, Let

A £ <0 • then A £ (3 # fhus, AU A€ © and A U 1 * x. Heneet

X£ (8 . Since X € (8 » X£ (8 and 1 • If.

*•*•3 fflSBSSE* Xf (X ia a r -algebra, then Oi is olosod

under the formation of differences wad countable iattniotioni,

Let & and 3 be elements ©f Ob, Shon

A^B « Af|§ » (X B) £ #. •

Lot £%} be « sequence froa # . Shan mm oo _

An £ Q* * B • i | 2) •«*! U AM €• uL i tlz 1 •*

aad

n. i ** Hence,

riAa- U % e d . *si ^

Bssssi' I f £ 1* any class of subsets of X, there

is • saallost r -algsbia d whioh contain* £ j that ls# there

Is a o- -algebra & containing C suoh that if (B Is any <r -algebra

containing f » then ftC0,

2$sa£e I * t P be the collection of a l l <r -algebra® #

of X suoh that C c 0 • P Is not eapty since the olass of

a l l subsets of X Is aa olsasat of P. t»t & « QO3 . siaoo

Ccz 6 for miy (0 £ F» C CZ Q. * If A e & t than A e (0

for every (8 € F. Hence, X € $ for every (Be? and X must

ba ia& • I f Q, , n » 1, 2t «•«, than Ajj£ (8 for every oo ^ <50

$ £ W m that yAjjC (H for ovory <B £ p§ and JJAn aust

bo in Q • Thue, (2 is a o* -algebra containing C and CL C. (3

fo r ovary (Sep.

*•**5 PftffftlfrflU* ffl» sot I is a lorol sot If 1 can be

obtained by a oo«atable number of operations* starting froa

open sots, each operation oonsistlng of forcing unions#

intersection®, or ooapleaents.

1.1.6 ®i® oXass ffl ©f Bor@X set® is Mi# ®»Xlest

V -algebra which contains all of the open sets.

Proof* 4 snaXXest 0" -algebra exists by 1.2. ® is

oXosed under the tarnation of oonpXenents and countable unions

by the definition of Borel sets and thus is a 0"-algebra.

Let Q. lie any (T-aXgebra containing aXX of the open sets,

ftien Q. aaust contain all oonpXenents, countable unions, and

countable intersections by the definition of a <r--algebra and

i«i«3« Heno«# (2 must contain (B «

1.2. Definition* & set fimetion a is a napping from a

oXass (8 of sets into the extended real number system by which

to eaoh B e ® there corresponds a unique element u(B)£ # •

1.2.1 Definition. A set funetion u is said to be com-

pletely additive if it satisfies the folXovlng oonditionst

1} the domain of u la a (T-aXgebra (8 |

ii) if [»B] is a sequence of disjoint sets fmm <8 » Co

then XL <*<Sg|) is defined in the extended real lumber system nsl eo CO

«nd u( LJ Bn) *

ili) «Clf) « 0.

If % • 0 for n 2 in the definition of a oompXoteXy

additive set function* then u C % U % ) • «(B^ )*u(B2> sinoe

n(#> * §• Tmn* every coapletely additive set function sat-

isfies a finitely additive oondltlon.

1*2.2 fheorem. If a la a completely additive set

funetion defined on Oi » if A and 3 are in (2 suoh that BC

aad if u(B) Is infinite, then m|&) » u(B)«

Immt* Bo* A « B U (A-B) whor© BO (A-B) « gf. Slnoo u in oosplotoly uddittTo on & » u(B)«Hi(A-B) moat fco doflnod. SinOO ti(B) is laflalto, u(B) m u(B)+ti(A-»B)« Honoo, • u(A) - u(BU(A-B)) » »(B)*l(*.B) « *(B).

1.2.3 SSSSlMtt* If « i« • oddlttro set function defined on Q. , If A and 1 in (X mot that SC A* and if u(A) lc flnlto, thon u(B) la flnlto.

ftwf* Hie proof follows inroadlately ff®a 1.2.2. i.2.fc fhoogois. If u Is a oonplotoly additive Bet

fnnotlon doflnod on (I» if AtB 6d, and if u(A) » oo t thon u(3) ff • «o »

?>oof. Lot a(A) « 00 and aaauno u(B) • -»<*>» thon A « UnB)U<iUl} whoro (AOB)n (A»B) « *, and

B • (BnA)U<B-A) whoro (Bf| A)n (B~A) » Thtta, u(A) • u(aHb)-mi(A~B) »<*> sad u(B) » ti(BnA>4«(B*A) m moo » if h(aDB) is infinite* #m of th® two otwtioita U impostibis. Ttma, u(A^B) «oo and b(B*A) • - °o . ®t*fc (A~B)r\{B«A) « 0 and u(A-B)*u(B-A) aaa* I90 doflnod. , HioTofovot a oontxadlotion la shown. Honoo* tho assumption lo f*loo9 ond u(B) • oO »

1*2.5 fhooaom. If a la * nomto&atiTr®* ©osaplefcoly additlro aot function on d wad if ia any aoqnonoo of aota f*o« d » thon 11 is oountably subadditive, 4*0.

ml y Ag) £ m(Aul« /)?l

fgoof* 1m% [jJ to a sofiionoo of 00ta fmm d * Lot

A h | § J i g m * * A y C A j j ^ L J A g | § t » « t «* ^ « * *

n - i

1 1 1 ^ n * * C

u < r u l

• % ~ C J J % J # • • • •

9 m k 9 L i % « , j * l % « i t i # ® t I j f l B j • 0$ i # i # S h u * #

u * p , V • Q V j ^ w ( B a ^

% C i ^ f o r • • • i t B t s o t h a t l o a o o »

iO o o *)" t

® ( J J ^ ) ~ H r n C % | #

1 . 2 . 6 ! h — m « X f u l o o o o a p l o t o l y o d & l t l v o n t

f u a o t l o a © a ( 2 » « a d I f C I ( 2 « a d a • 1 8 2 t . . » ,

t h o a 1 1 a u ( B a ) » « 1 1 a S ^ .

o o

fttof* U S j j s l a o o a • 1 , 2 , . . . .

A l s o * u % « % U ( l / l % | U ' * * U ( S ^ i H ^ ) U • • • w h i c h i * a

u n i o n o f d i s j o i n t s o t s . H i a s t t s y t h e o o a p l o t o s d d l t l r l t y o f u 9

« ( l l a S ^ ) • a | 1 ^ % )

« «(% )• % )

• l ^ a j a C % > 4

• 1 1 m

1 * 2 . 7 B i o o r o a . I f t t I s ft o o a p l o t o l y a d d I t I T S s o t

f u a o t l o a © a fl 9 a a d I f C Q . s a o h t h a t

» • l t 2 »

• a d s u o h t h a t u f S g ) I s f l a l t o f o r s o a o a f t h o a

l l a m ( % ) * a ( l l a DO

g r o o f . L o t u { ^ | ) b o f l a l t o . f f e o a 1 1 a % • Q % C % t

o a t a ( l l a 8 n > I s ' f l a l t o I f 1 * 2 . 3 . S o w

v j - ( Q % > u < v „ C ) * » >

B O t h a t » ( % , ) - « { n V - tt(V $ b B > > * o 4

uUia ( V - V * " • *UW>-m(lta % K K®w SnDS i+l so that [%-%} ie expanding f « n> m. fhus,

m{lta C%»Sj|>) • lift »*%»%) br i«2%6* Also, %CEa| for

n>nmA «<%} is finito for a>a. Thoiwforo,

uOy-uUla « u(ilm (%•%*)

- Xlm

» 11a (ttCSgV-uCBg))

• mCS^-lia «(%)•

Hoaoo* u(li* Sn) • lis «{%)*

i*3« D£flnltlon. 4 MnMgfttlvo, extended roal-mluod

sot function u is * aoftsuvo prorldod it Mtlafiti tho

following conditionss

1) tho dMiia ef a ia a t -olgobxct (8 $

li) if fljj] is » sequence of disjoiat sets tmm (3 • t h , n u< u v - £><•.>»

ill) u(0) « 0.

*»r a noxmogotlre i«t function, tho conditions ©f 1.2.1

®*® ill® m m as thoso of 1.3. Henoe, & measure is a sol**

negative, eo®pl#t«ly additive s®t fuaotion.

i«3*i Oefliiltloa. A wmmxm%%* «pm« <X#c8) is * sot X

togothor with a (r-ol«©b» (8 of subsots ©f X. A subsot B

@f X is Mid to bo Mftswrablo with i<ospoot t® (B if »€ (8 .

l»3.2 Definition. A meaaura space <Xf(H» tt) is a

measurable spooo (X, (8) together with ® neasure u dofiMd

@» <8 .

8

to eloaositaxy oxaaplo A aoasuro «pa@# la (P« (S» u)»

where F is the sot Of p*altlY« intofsrs, (3 is tho olas® of

all subsets ef 9, and *(») is tfco nabsr of of 1*

ffe© theory of the integral in Chapter XI Is ©onoemed

with a particular ilui ef funetlona fro* X onto this

clase 1® called the class of a®asumbl« functions. It is

the purpose of th© reaaitider of tills ohapter to define a

measurable funotlon and to ostablish some of the properties

of aeasumble tmrntima*

Bgflaition, Let (X, (B ) be a Jtoasasafcl® space. An

extea&si funatioa f defined m i l s sold to be

measurable If [x|f(x)>a] £ (B for m i y roal mrnber a.

i.fc.i If f Is an extended real-valued function

defined on X» then th® following statement® ar® o tlvalontt

1) [x|f(x)> a] e (B for eTery ae 1§

11) £x|f(x)> aj £ (0 for m i y a elf

ill) [x|f(x)< a} e (8 for W I T ae i|

IT) fx|f(x)£ a J *• (8 for ovo*jr a6 B*

Proof. Hi# atatoaont (1) 1® true if and only If (IT)

Is troo since (x|f(x)> a] «• [x|ffx)£ a}# The statement (11)

is true If aad only If 1111) Is troo siaoo

(x|f(x)~ ft] « [x|f{x)< a] •

Ww$ {xjfCx)^ a} m Q [x|f(x)> $ is closed under

oountable latorsootioxu Heaee, (1) lasplloe (U)* Also*

{xlf{x)> a] m LI {x|f(x)> *•*£} » and (B is olosod iwdor oouat*

able union. Henoe« (11) implies (1), and the proof is ooaplsto,

1he following theorems show some of the properties of

aeaSUTftble fun® t ton*.

*^- 2 B s s m * If f is Measurablet then |f\ 10

aeesurable.

lE2S£* 1st a< 0. most [xllfCacjl > a} » {x} £ ® . Let

a * 9. fhen

f*| lf(*)l > a} » {[*|f(*)> •} U [*|f(*x -a}} £ (3 .

*•*•3 Bmmwmu If f li measurable, then

[*|f<*> • °°} e (8

and fx|f(x) » - °o] £ $ .

£!S2£- - Q n] e (8 , a m

[x|f(x) • - ooj • fi fx|r(x) < -n] 6 $S .

SSSESI* If f is measurable and If oe Rt then

f*® is a M i m b U and of is seasitrable provided of does not

assume the fon (£>)(£°o ).

fm&t* Lot f be measurable. Bio*

fx|f(x)4<e> a} • [x|f(x)> »-o} £ d? »

and f+o is measurable. If o « 0, then

fX) £ fi if A < 0

io] e (8 if *2 o

If •< 0, then {x\ef(x)> a} • (xlf(x) * -§»} £ <0 .

If ®>0, then [xjof(x)> a] « fx|f(x)> -J.} € (0 .

*•*•5 Sieorem. Xf f and g ara measurable, then

£x|f(x)> g(x)} # fxif(x) >g(X>} t

[x|f(x) • g(x)3 » {x|f(x)f g(x)] #

and {x|f(x)< six)}

are in (B •

j^x|of(x) > aj

to

P w f . Hit proof is given for the set £x|f(x)> g(x)} .

4 sinilar proof hold* for the regaining sets*

Let f sad g las measurable. Aon

fx|f(x) > g(x)j m ^{[xJfCx)* tjn[x)g(x)< t}] € <8

where T is the set of rational ambers*

1*4*6 Theorem. If f is neasurable, then f2 is aeasurable*

Proof. Let f be aeasurable* If a* ©*

[x|f2(x)>a} • f[x|f(x» Va]U{x|f(x)< - Va}}* (8*

If a < 0, fx|f2(x)> a} • fx]€ (0 .

1.4*7 Ihooroa. If f and g are measurable, then ffg Is

measurable provided f*g does aot assuste the undefined form

Proof* If f (x) > a~g(x), then by the Jrohisedean property

there exists a rational mraber t such that f(x) > t > a-g(x)*

Thus,

[x|f(x)+g(x) > a} » [x|f(x) > a*»g(x)J

-<U[fT|f(«)>^nfx|g(x)> a-t}} € 6

where T is the set of rational musters*

1*4*8 theorem* If f and g are aeasurable, then fg is

measurable provided fg does not assume the for* (0)(t <# ) or

it«°) CO).

Proof* The set

{x|f(x)g(x)> a} « [x|i[(f(x)+g(x))2-f2(x)-g2<x)]} £ (B

bjr 1.4*4, 1*4.6* and 1*4.7*

lote. It is also interesting to exanine seise of the

properties of setueaoes of measurable funotions*

a

ghooroau If [faj la a sequence of aeasturable

f«netionst then sup fa, iaf fn» I S fn» and f a a n

measurable.

Proof* The set

[*|«? fa(*) > *J • jj [*lfn(*)> •} £ & §

•a* [*! is»f fB(x) < * » JJ [*| ftt(x) < a] e (8. thus, sup fft

and inf fn are measurable. It follows that I S fft and

1 H fn are measurable sinee I S fn • £nf(ira| fB) and

It* fn " | S ( H f *»>•

l.**X© Hieoreau If f and g are measurable, then aax(f,g)

and mln(f#g) are m*mm%!#•

Proof. If at least one of f and g is infinite* the

result follows fro* 1.^.3*

Suppose both f aad g ax# finite. Ttimn

£x|aax(f(z),g(x))> •} . .j c <S .

by l.tfr.2, and 1.4.7* Henee, nax(f,g) is neasorable.

Likewise,

r , i r , -|f (*>-«<*)! •*<*)•«<*) T |x|ala(f(z).«(>))> •) « £*| i '• ' a' ""*>*)

and the proof is oomplete.

1«&.U Definition. The functions f* and f" axe defined

to be »ax(fvO) and -*in(f»0) respeotlrely.

HS££31« ®*# function f is measurable if and

only if f* and f~ are measurable.

12

I, It f is aeasarable, then it folloae immediately

txm 1*%.1® that t* and f* are naasaxable. ®otr f •

Benoe* If f* and f" or® measurable, then f 1s measurable by

1.4.7.

1.5* Definition* A property iff mid to bold alaost

everywhere if the sot of points where it fails to hold is a

set of aeasore sero. la particular, f > g alxost everywhere

if f and g have tho same doaain and a (x|f(x) ft g(x)"£ • 0.

® * tiBSl ^ wltten a*«*

1.5.1 fheorem. 'if f is a measurable funotion, if

f • g a.e.» and if g is eonstant on sots of neasare sero

where f |f g, then g is measurable*

Proof. Z#t B • |z|f(x) g(x)^. Than E li a sat of

measure zero.

{zjg(x) ?a$ rn |fx|f<x) ?a}«fx tBIgfcr) < a ^ U fxeB|«(x)> a5 .

How fxjf(x) > a) is neasurable slnoo f is a measurable funotion.

Also, {xt S|g(x) £a^ and {x tBjg(x) > aj are aessurable sinoa

thai' ax® subsets of E and B is a sot of measure zero. - fhus,

$x(g(x) > §k\ is measurable, and g is measurable.

1.5*2 Definition. Let I CX. ®*e funotion lg{x) defined

by (1, Its

* ( x ) - [o. , M

Is sailed the oharaeterlstlo funotion of B.

1.5.3 Definition, tot s be & real-valued funotion on X.

If the range of s is a finite set of real numbers, then s is

said to bo a staple funotion.

If b is m @inpl# function with a2# aa] tha

aoasava v&lms ©f a, then

8 " t " A i

% « fx|a(x) » St] » i « 1| 2# ...» a*

*%im9 a is aaMastfbla if mtA only if aat® 1| ars

»«*eurabl©. Also* twy ai*pla fvaatlaa la * liaaar aaabl~

ntlMi ®f ahaxaatartatia fuaatlaaa.

fcti. A OMMtnaftly *aa**riaff thaaa in analyaia la tha

approximation of ttelag* laar alaplar thlaga« It la mm paaalbla

t# appsoslaftta measurable functions 1sy measurable functions

af axt alaaaataxy tjrpa. tha alapla fuaatlaaa dafinad 1a 1*5*3*

1«5«& SSSSH* Jf t 1# a funotlon on B» thora axlata a

aaquaaoa [faj af alaple ftnatlaaa aaah that far mqr xcS#

lis fa<*) » f(x). Xf f la aanaegatlira and maasurablo, thra

{" fm] ®»7 be aheaaa ta ba a a&adaaaaasl&c sequanoa af non-

negative, simp%Ȥ aaaimrabla functions.

Sappaaa f > 0« fur aaah poaitlva latagar a and

aaah latagar 1 a&ah that Of 1 f n^-l, lat

r 1 141-1

®la m j

'a * [*1» - f<x)}

If x t * 1| 2| •••t #2®»1

and lat

%{xl If x Fn.

1*

Thu«, f n Is el*pl« sm& mmmmttTtf [ f n ] i s fiftad«oxea«i&gt

and l i s f n * f • If f Is *#a«ur*bl»,, thtn f«r •aota n, t% and

th« mm mftmaabl* W BtaM* tn It M**ux«bl«*

Suppose f l i a s **bltzftvy fmn©ti©ii« U t f * By

%h« ftbm MMtnfltloa, th«r* « i i t «nu&M* [%]

©f *K»m«g&fciTr© «impl« functions aueh that Ilia % • f* «®t

l i s ^ « f « Let £f t t] « [%-^aj • ®*®» [ *»} *• sliaplet and

l i s f B « f t

mmm xx

mmwmmm m tm w&mm

«!• of Integration, aa presented in this ohapter,

1® eoaeaxaad finst with the olasa of aeasmmble functions as

described la Oiaptar I# next with the definition and atwatajra

the elaae of integxable ftmotlena, and than with the

properties of tha Integral. ©ie integral aan be looked upon

aa a function whoae range Is In R and who®# domain U all

m i m (f»K) where f e «£(u) and K le in the olaae af aaaatumbla

•ata«

M tha fl*at eeotlon af thla ohapter, tha Integral af a

alalia funotlen la defined. Thla definition la toxa la aaad

la defining the Integral of a aeairaimfele foaatlea* 2,i Definition. Let <x,(B ,u) be a measure space* a a

noanagatlTe* meaeurable, elaple fmnetion on X with values

["#!» «gg OJJ] a and Be $ • Biaa

L® * IT •i*(*Ab1)

r , 1 k1 mrnm % » [x|a(x) » ,1*1, 2, .... a.

2*i«l Definition* Let f be a nonnegative, extended f

real-mlued, aeaanisable funotlen on Mia aaaawe space (X# $ fu)

and lat » £ <8 • «ta»

f f - imb fa Jm Jm

*5

u where the 1ml is taken onr *11 measurable staple fwmttmm a

mmfa that 0 £ « £ f •

2.1.2 definition. 4 aoanegative, ezteMed, real-*»lmed

function f i® mid to to® tategmttle ($8 S with lespeot to is)

if f t« aeAMiMi and if f f is flo&t*.

2«1.5 Piflaitlaiu Im% t toe seasmratole* If at least ene

«f f*» r* is integiable, than

f t m f !•. f r . •'l 'I 1

2.1.* Definition* 4 function f is said to toe integvatole

if toeth f* and f" are lntegrabl®.

®» nation \ t U „U.d int.** Of f *•

Both the rang® and domain of the integral depend upon m» and

the notation

lEf4tli» 4fd*2"

at oetera. will too used ia any discussion which t p » l m more

than ens measure. ®*e integral aay be infinite* 4«&*

[ t m t 'n is defined* However, f is not integrable unless its integral

is finite*

Let £ Cm) denote ma sat of functions which are integrable

am a sat with reapeot to m* Zf f is integrable on s« then

f e. £ (m) on 1*

Several of ilia nost important eleaentary properties of

the integral are now proved.

' ' 2 # 2 Shaorem* The funotion f ia in (u) on E if and only

if f is measurable and bounded on B and m|S) is finite.

1?

Proof* If f £ <£(«) on S9 thoa f is aoosuvoblo oad f f i i flalto* Thus, f auot bo bouadod on B oad u(B)<°° •

L*t f bo aoasnmblo and bounded on I and lo t m(B)<°®.

Thoa f+ oad f* or« aoaaogotlvo by definition oad mm

®oosu»blo by 1.4.12* Mow f o lo olwoyo flalto olaoo u lo It flalto. Tints, f f+ oad f f " oso flalto oad f £ *£(u) oa S J I JM

by 2ii«4«

2.2.1 Thooroa. If f lo oajr aooouxoblo fuaotloa oad If tt(X) * 0# thoa ft mo.

Wmof* If f lo aoaaoootl'ro the vomit follows lan* dlotoljr f*M tho oddltlrlt? of u oad froa 2.1.1. fte tho gen#al oaso# J f + • 0 oad f f • 0« Hoaoo

JM JS

f t rn f f * . ft- . o, ; !

2*2.2 Thoogoa. If f € X <«) on S rM If a£ tB thoa

®f (u) oa 1 oad I of » o f f. JS JM

Proof* Tho fnaotloa f Is botiadod, oad u(8) lo flalto olaoo te£(u) on B. Thas, of is bouadod, of Is aoosaxoblo br 1.4.*, oad of ei.Cu) oa B bgr 2.2.

If f lo olaplo, thoa f t * f ' E«o 1«t*nE 1) > a Bj) » * J ^ f .

It t Is aoaao0otlTo» f of * lab f os • o(lub f .) . • f f JM JS JS ' B

who*o tho l«b Is toteoa oror oil aoooaxablo, slaplo fuaotloas • 0 l i f . Ooasldor tho goaoxml oooo. If a > 0, (of)4, * of+

18

*»& (*f )• « af*. 8ttM««

F a f - f (af )*- f («f T JE yB

f a f * . f a r -m J*

•Vf- tri # i f f ,

Jig If a< Of <af)+ • «af~ wad <af)~ * -af*» m that

f af « f •af* «* f "-af4,

JE JE

• -a f f - ( -a) f f* Je J i

[• j/* [Bf+J m &

• i f f . J* E 2 . 2 . 3 Theorem. If f and g mm mmmmtol* funotlone

saoh that f (x ) f g(x) for x€S» and If J f and f g art r f

JS defined, then ( f i g .

'B K Proof. X*t f £ g an I b» atasurable. If f and g are

nonnegatl-re* f t * lub f 8 for every measurable, simple

function t wham tha l*b la taken oTar a l l •aaaus«bla»

simple functions a suoh that O-B^g* Henoe*

f f • lub f t - lab f a * P g. ' I

Bar the general #stae* f+£ g+ a»d g"^ f . Urns, I f4 , £ I g* r r

JM JM and J g~ f I f"» W&mm9

U " / / * - f / ~ - 1 / • J / • J , -

19

v y 2**«* &SS235* If fest(tt) on 8, A e <8 » and AC 8* then

f e iLM on A# Farther, if f Is aennssatlYe, then f f > f f. ;S A

Proof* 2he funotion f Is beiiisdsd on 1 ant is measurable

by 2.2. SImm AC B, f aast also be bounded cm A* Sew u(A)

is finite sinoe % is additive and *(S) is finite. thus,

f e Cm) on A t)/ 2.2.

X*t f b» nonnegaUv©* fbr ©very asMUStibl** single

function a sueh that © * s * f f

"To&«(ln%)2 ^ •lu(An®l)

where S t • fxjs(x) « ej simm a is additive wad ACS. Bras,

f f « l«b f.> lllll f H m f f. J% JM JA A

Siere ar® matrons definitions of an integral in the literature of iaat§temtl®g* Stftitl ©f these definitions will

be investigated and will be shown to be equivalent to f f. •'t

I«t f be a wmnegativ®, measurable funotionp aM let 1

be any msaammbl# set* Let [fftJ be a nondeereasing sequenoe

of noimegatlve, staple fusaetiens auoh that for every x S,

11m fa(x) * ffx). Monroe (i) defines the integral ef f ever

1 as followsi

J ? " " * J"/"*

*•3. £at f and g be mnnegative, measurable

funotions auoh that f £ gj Mien J* r < f v y 8 J 1

l*roof« % 1.5*^ there exist iwmnegafctv® sequence* £faJ

•a* {«»] ®f nondeoreasing, aeasuxmble, staple functions suoh

that li» fn » f and 11® «® g. flier® exist subsetuenees

20

[" »? f'n] ®aAf%] 3P»spe®tlv#ly# suoh that

fSnrZ %fc *#r *V **' * I M I t t n U f t» st**Xo Amotion S,

[*.- f / * J*

s. 1 'B

9mm » for ovoiy Mjj

!/«%- " • fl*% • 1 / B %f 2*2*3. Ihoroforo*

i t a I N - [,• sni.

{ * ' - ' { *«• J I J I

2*3*1 Hieorea. Lot f 1m t a©iuisgatl"rOi aoaewmfcX#

function and Xot 1 *• measumble. ®ien

f f • r#f* JS Jf

Proof. 3r 1*5*^ thoro oxlsts a noaAooroaslag sovtonoo

j" faj of soanegatlvef aoasnrablo9 slaplo functions such that

for oyoiy x€ «# 11a fft(x> • f(z). ffcr m * y n» fa Is a

•laplo function, tni f, and

f fa*li|l> [ « JE JM

whar® the lub Is taken ovor all aoasuxablc, slaplo function®

• 3 Of f. Thcroforo, H a f fni XmH f s « J f*

B ' B B But s f 11* fft for ovoxy slaplo funotlon s 3 <M s* f* sad for

a aoaaurablo, slaplo funotlon s9

f . - r.. '« ; i

Thus* for oToxy a9 0 *s^ f»

1r 2*3* Hoaoo9 J,"" J / 1 £ l u ' «

21

I, f » lub f a » J *

lub J*s £ | # 11a fm

f 8 # f E

" l l a [Bfn»

where the l*b is taken w v all «e*Sttmble» staple funetiens

8) Oi 8 it* HlUS*

f f • Ito f *« • f **• _7 <m J w ** J ®t

£ £ ^ E 2,3*2 Definition. Let f be * maasurabla function and

let E IN ft MMvitbli «•!• then f f f • «lb *

S U f JM where t Is sinple.

f

2*3*3 Thoorea* If f ia a seasureble function and 1 is

« aeasurable set* then If

f • f. 1 s

Proof. If f isi unbounded en B or if f is defined on 1

sad u(S) Is infinite, then both integrals are infinite suet

ths equality holds. Thus, it is suffiolent to oonslder only

the ess# where f is bounded and u(E) is finite* &st 1! b e an

upper bound for f • Let

k •n<*> - —

n for k k+i

— — £ f(x) < «—», k • ©» 1# 2f si n

k+l t (3E) • —

a

22

t9T , k *' — f ( i ) § k * Ot 1| 2# • •

a a

Bath «n and t f t mm mt daflaad far k > Hi* Baaoat »m and t n

@mm% tate @m mora thaa kQ • ^a^] valmaat and 8^ ami ar®

eiapla twmttms* Bow, sib r > ®ife r t ^ i«n r ® i x«b r ®wt

J l t>f Jt J 8 J l ®

r r Jt. k+i r k M . n i * x * L * a - J *n " 2_ — « W > { * | — i t(x) < — } J.

JM J % fio n L a a J

and ^

l a b / , • » - IB"» " ^ T ° ( b A ^ 1 T i f < " ' T ^ -

89 that

0 4 sw. js«» 4 [ K v f a " n

A . k+l-k ^ r k kti i • r — * i —»- r(*> 4. -»— ] )

m L a a J

t tf lc+l « — U(«nfx | — * f ( * ^ — ] 3.

n JTZ L a a

Sat u(B) is finite, «a& since a oaa ba mad« arbitrarily large,

^ ( , *» - W Jg»B

and

) f » «lb f t • lwb f a « f f , J 1 t i f J l " f Jb

Saks (2) deflaaa the iatagml of a nomegatlve aaaauxabla

funotlon f as

I

g n f - Ink /_«(Ei-)lnf f ( i )

* fT?

23

w h s r o t h o l * b is t a k o n ova? t h o M l l M t l o a of a l l f i n i t o h

elasaos o f d i s j o i n t * » a « i n u r a b l o s o t s n o b t h a t I m L}x%*

2 » 3 « f c f h — t — » U t f tet a o a s w r a b l t f n n o t i o n o n X s a d 1 s t S b o t h s i m i s n o f a f i n i t o o l a s s o f disjoint» n o a s u x a b l o f i m o t i o a s . 2 h o n

f 8 ' - f < -J i >n

f » o f . Bar t h s x o o x i s t s a n o n & f t o r o a s i n g s s q o o a o o [ f B J o f a o a s a s a b l e * s i n p l e f t u i s t i o n s s n s h t h a t l i s f R • f « ®8wr

f » " t l 9 » S

f *\ ) * i

x | f a ( x ) » O j J $ s n d f . S s n o o * o « £ i n f f ( x ) * *e E; s o t h a t f o r o s o h n L f n " I T * i a ( B i )

ill |« j[ - * < * >

i I n b 5 2 u < * 1 ) i n f f ( x ) jsl

« & i i i the l « b is taHeia o * o r t h o o o l l s o t i o n of a l l infiait# h

o l a s s o s o f d i s j o i n t * a o a s n s n b l o s s t s s a s h t h a t B 1 ^ . flwaotartt

J / - M / - '

B a t f(x)

i s t h o i n t o g z a l o f a s i a p l o f t m o t i o n * s a y g, u h o v o g £ f . Hondo*

h f ~ « { ! * ) i a f f ( x ) £ l t t b f s •TI 1 *«*/ J l

2k

utiar® the Xab is taken ©*ar aXX slapla faaatloas ® 3 § i • £ f.

Bierefora* ^

) f • X«b ZL afK^iaf t(x) i Xub f a » f f. J % n-l *£ & Jv JK

I . / ' • J m

Thus, " 8 f S 'I

m m M i m t@ tfunro© (1, p. 178), labascpta has defined

fclta integral of a bounded, nonaegative, measmmbXe funotion

©a a Hft§« »f finite measure as follow f L & & w,i J f - 1I» -Lif(x>< ), J 1 *rs n L a a J

whara « la an upper bouad for f. 2«3*5 I**aorea. If f is t bouadad* noaasgatlva* measurable

funotion on a space of finite measure, than

y M J i k Proof. Let X ba aa upper bound for f. Let f (x) * —

14 a for

fe M — £f(x)< _ » k • 0, it 2, ... . a »

8&w tn la not defined far *>a*. Saaaa, fR aaaaat take oa

aora than kQ m [an] values, and fn la a aiapXa ftmatioa. Also,

lis fa • f* Hence e

"vf /-- I / - 1/ by fitaoraa 2.3.1. Let

•a " if-§-»(*n[*| SSI] ).

Par i > ax*

k k+i

c a a J

Tben,

25

% " j ^ a *

j I n t u te#*t

r « i i « T — — - i f ( x ) < — - } > J M " I 1 a n

m ism ®s

- 11* f s f B

m f f« ) ft

temthex important pyopsrfey of ths Is its

aMlfcivtfcy over fch# OI&BB of m u o i k b l # ssfca. I m t h i s

pmm%W* which i t j o w s d to th« fo l lowing tt»evt»» 1%

follmrt that ths iatsgml Is a. ssaplstsly lAdl t l t i ««t

f\*n©fcio*u

2«fc. Tkimmm* I f f Is assswsbls sad aoaasgstiv* *a

Z| snA i f te («) s * S» thsa J f I s ssssplstslx sAd i t t r s

m (B t tbs slsss « f assstixfcbl* ssts«

y»»Sf. &St bS S SS OSBOS Of d l s j s i t t t ssts tmm

(8 » « M 1st A » U % * 2b* »x»*f I s sssplsts i f i:1 r Vs- f

/ a * " f r L / "

I f f Is a as&snxftblst simple function* then

- I l ' i g x v " 1 " ! '

* 2Z ®^C aO )

/. f . 4

26

U t t 1st measurable ana aonmgatlve. far n u m l i l t ,

•tapl* twmttm 8 * 0 s f > oo „ c*3

Shtn the

f . - l f «f i f f. JA hi >K ^ 'A*.

u b f • £ Z f f •U jM J A

•o that

J > * £ V -

If f»r «qr k»

r *• *<*>,

the& th® pnwf i® ©#mpl«t« slaos A^CA and

f f > f f •

./ A jAjg

8®smm

r ? < o<s

fcr Meh k« Chooa* 6 > ©• % i«5*4» atMttXftttU* slapl*

fflM&tMM st and ®2 o«a b« found suoh t&at t9

0*8 2f f.

L 8 1 - L . ' - f '

J 8A f flHMWi

A 2 2

M 8<*) • ms;(8i<x)9 82(*)) Vh8S« X€A 1U Ag. th«»

f t > f 8 • f 8 + f 8 > f f f f f t . € #

J A | U A g JA U Ag J&i J Ag JAg j A g

27

so that

r r > r f* r r. J*lUAg JAu JAJJ

Iimii for «Mh k>

J V - J/* ••• * f *- e J/-U % ^ *k >:i ' *1

4 SIbm AD(J% f»* oaoi* ki

f " f / • J A Jfy&-

/«j * OlOytfOfOi

( ,

f*v «aoh Je. Letfetag 3c lao»s»«® without brniM,

1

r go r*

/»' ' S f^'

J/ - i t

J&S232S* 3Df f £^{«) ©a X» then jf Is an o*»iy~

«tM*o flBltei oo*pl«t«iy oddttlTo so* fttootfcm on {8 .

SBEBt* domin of J f is the <r «&Xg©t>f® $ of

measurable sots, «nd J f » 0 by 2.2.i. fllaoo f£^.(«) on X,

f f Is everywhere finite, out the countable additlrity of / f

follows from the application of 2.fc to f « f+«f. fhe oon-

ditioas of i«2*i &a?e satisfied* sad the integral is * ©oapletely

additive sot fmnotiosu

2»*«2 SffiSSB* U ACS e ICAi aai u(B) * 0, then

f t - f J A J A

f a 'A ^ A-B

28

Smt' »nw A = BUU-B) «M4 jf Is adaniTB,

[•tm f f* f f, U J9 ;A-B

But

J f - 0 B

% 2*2.1 «o that

f f • f f • JK JA-B

tt la seen fmm th® preceding th«or®m that aata of

nteaaur© ax* nagligibU in l&tagx*tlaa» The proof ®f

tti* naxt thMzn follow dtiMtljr from 2.4.2 and 2.2.3. 2,^3 JSsffiSSS* 3tf f m& g e £<t*> oaStnl iff »g a.a.

©a 1« then

f / " l« w

Ml tf f-g a.a. ob Sf than

J > ' J, , .

1 2*5 If f and g aar# ssasi«iUv«t aia«u«U««

clapla fuiatla&a, mi if B 1® aajr aaaauxmbla eat* than •

r (f g) ; 8

•xlsta, aid

f <*•§) « [ f» f g.

ftMftfr Ste®« f and g ar« nonnegatlre and alapla, f+g

la nanaagatlv* and atapla. Bf f*g la aaaaiuabla.

Hence, / <*•#) «*leta Isy 2.1. Let

29

X > i K i - - f c V i' i *• /r»

« " J ***•1 *

iMiJI ni

\ - f i • ^ H B r ®i«a « 0 «»t hmt m q t maA

C 2±L tr>*1 r -.

h4l »- rn+i H4i r rmi UB^DAa]

J2tx mI -t

^ ^ [ ( t A 1 ) n Bj ) n s ]

ft+i r _ mn r -

J v l V * ] m L / * L®*

^ R • S ^ 5 4 ftfiHi* I f * S nomegatl-re, measurable

functions, than

f (*«#) JM

n l i t i aa&

1 (f+«) » f r *• j I ^8 J l

isssi.* ®*® fuaotlda f*« i s nonnegativee sad lay *.£*?»

1« WMMttxml»X«« Buim, f (f+g) t*y 2.1.1. l r b

I*5»^t «!•*• exis t uraAMmitBg miw&mi and ©f

alapl*, aoaiieg&tive functions suoh that l i s f„ m f

and l i s " S* Bum 11a {*»**»] *" «nd

" » L ( f*+«n> - ] ( « * )

3©

2»3«i» 7t»» each n,

/,<'»•*,» - [/»* 1/a

If 2*5 eo

J s

m Ji*®* Jg*a^

"» 11a J f * • )% n Ji»

HttMi lay 2»3«1»

f C**®)

• lis J W )

. xi*

• f t+ r •• )% J s

2*5*2 SSHSl* X* fc £ («) on I ant if f * fj-fg wh«x«

f| and f2 are aftiSMgativ» and in <£ (u) ©n I, then

f * m f r ^2» j m j m s M

SJLno© f • •»!""» *• fj»fg and *»

H«moe,

r ( r ^ f 2 ) « r (f%ft), •/ E

and

r f r f 2 * r t~+ r t* J m •'w JM 1

bjr 2*5*1* fhttfi#

f f « f r+- [ t~ » f f t - r f2 J* J r Js *

#

3% 2,5,3 mmmm* If f «a& g 1b ^ (ti) m S» tfa«&

(f+g)& &(*) «& l | MA

V f + a > " ^ 4 -

2m£, S i m f«g € £(u) <m 8, f «ai g **• bounded and

•Mumzabl* «ad n(S) is finite. ®ww# f+g U MMunueMble W

i.*.7» anA f +g is bomtad. H&m, f*g e £ (n) ©n B bjr 2.1.

Since f « t+^t* and g • g^g-# f+g - (f^g+M ?*%•). »«®t

r <r««) - r <?••«•>- r <?-•*->

s y i J i

by 2«5*2V and

J (f++8+)- J < r + r ) • [ f++ f t*- f I V

E J% ;B

J* 'W Jn w 2*5.1.

- j B f , j v . j v - ; B r - j » j v

2.5«fc ^teorsa* ' j®i« iatagsal J id a liBta* ftmoti@aal

#m 3C <u) ©a E.

Proof. L»t f and gbs la st («) on. f and lat * and b fee

ra&l ixmbara. ©MI proof is ooaplete if

f <Af+*g) « i f f*b f «.

Bf 2«5»3»

Br 2*2.2,

ami

f (*t+bg)» r »f + r i»g j* •/i ; i

V " * k' J tgmbj g. J 1 •>*

32

itius,

f tftf+bg) * & r f+b r g* H Jm Jw

tt ham boost shown in 2*5*4 that tho integral Is a linear

fttMttftnal* M gen® ml, tho• i a t o g a a l is n ts b i l i n o a r * for if

8| and. % m m i n (8 anft m C % D Sg) ?* 0V than

r f < r f* r f. J % U % >H

H r th« p a r t i c u l a r oaao i n whioh t t f%n B 2 ) » 0 , tha i n t e g r a l

in b i l i n o a r » a® Bh®w3ft in th« following theorem.

2*5*5 lieorera. If % and Mm art la (0 , utaHfe) . o,

* t t d % » i « l A B , « n A f « n A g i 2 « l B ^ ( a ) « a X f than

r («f<fbc> • ft f f4« r m* r r g* J%u»2 •>% y it J %

froof. Uto proof is tar i i i t t t a p p l i c a t i o n o f 2 . 4 , 2 . 4 . 2 ,

sisd. 2 .5*4*

2*5 .$ B i i o w « » I f f and g a m i n / (u) on 1 ant if

J / • J /

f o r ovox? noaaarabla ACS# than f » g a . o . on 8*

Froof* Imt A m [x| f f * ) > g(x)9 X € b] » a r t l o t

B * [x|f(x)< g(x)» x£lj*

If A and B *m wmW* tho thooron ie satisfiod.

Sappom A 4 0* H05i- | f « J" g a© that J (f-g) w O by

2*5*4* Alao* f«*g> © m A# Hoaoof ts(A) « 0 , f o r i f othoxwiao,

0 * 2.1.1.

3ttppo»« 3^0* Btmm f f • f I, f (c»f) » o. But •/B J B >B r

g»f > 0 o» B. Bonoo, tt(B} • 0» f o r if othoxwiao* I ( g « f ) > © B

bjT

33

Ttm fallowing two thooxwas axv

etatod without pmefSm A proof of oaoh it gives la fejrXor <3).

2.6. Maneton* gffffyTOfVH IttES- » {'„} *» »• "»»-

Aaovoaaiag soqaoaoo of notmogatlvo fanetl®2ia9 oaeh latogzafclo

m tho mmmmtelm oot St tfaoa

lis j^fB • j lla fa.

2*6.1 Dominated Convergence fhoagoa. Let [fa] be a

iiiwas* of aoaoasablo fuaotloaa with that 11a fa » f a.o. oa

2» Xf there oxloto gci (a) oa 1 moh that for oaoh a |fa( g

•a 1« thoa

2#7» Jfettfitt? A aoaoavablo faaotloa f «b X 1« la («) •&

1 If aa& ©air If If I £ £ <a) oa 8.

Proof. Let f e £(a) oa B. 2*oa f* «aA f aro la £. (a) oa

8* 91aoo |fI • f++r, If !e£(u> « 1 w 2.5.3.

Lot |f|£ £ (u) oa B. 9toa a(B) lo finite aad |f| Is

bounded m WW 2*2* fhmt f is tou&Aod oa t« S*« ooaAitloas

of 2*2 are oatloflod m that f (u) «& B.

flaoo if t £ £ (a) oa 1* then If I £^<u) oa B« f is Mid to

M alwolmtolr Intsgrobl© oro* E, sal f f is said to lit aMo* JB

lately oowwgoat*

2.7.1 fitiwrea* Zf f £ i (a) oa B» thoa I f f I f f \f\. » JX

It la o rloai that f £ Jf/aa& that -f * /fl • Br

2.7, I f I £ («} oa E. 2hue,

Jr

3%

fl

by 2.2.3. Honoot • r . ' • r

I f . ' U J , " 1 -2.7*2 Theorem. If f is bounded and measurable, and if

i(u) on E» then fg e ;£ (u) on E.

Proof. Since g€t£(u) on E, u(S) Is fialto by 2*2*

©ms# f e (m) o n E b y 2.2. Sinoe f and g are bounded and

aoasurablo, fg It teui&«4 a M measurable by 1.4.8. Bence* fg£ -£(u) on 1 Isgr 2.2.

Stooroas 2.7 and 2.7.2 prorlde tho proof for ftwna 2.7.3» the first mean mine theorem for Integrals,

2.7*3 ftniwi* Xf *£ f(x)± K for *11 *6 1» and if f Is •oaswrablo and ge/<«) on f» thoa there Is a Ml number * suoh that mi K a M suoh that f^lsl » • J )sl •

Proof. Bar 2.7 ami 2*7.2, f |g| € («) on 1* ants*

I,-1*1 £ L " * 1 4 J,"1*1

so that

• J « l , u "Jm"1-X* JKi«| • Of tho vomit is trivial. If Jg|«i * thoa

i - f l«( Jj J « | e |

f |g| * x. 1

Let

m m i r — figl.

n

Than mi a £ If» ft8d

[ f li( • * f lei • J g )%

2.8. Definition. Im% « to • measure i m ® t l o a and f any

SOt fUB*tiO& Wh©B# 4Mft|B is & SUbOlaSS Of tllO OlOSS Of

aoasttimfclo sots. twmtim r is absolvtoljr Miltiumi with fospoot to * SHAHS that glYon 6 > 0# tbovo islatf a

<f > @ sush that | •(!)! < € nfcoit®*®* s is to tbo doaain of r

ms& u(E)s <f •

2*8.1 flfroogsa. tt t £ L W ©a X# thoa J f is atosolatoly

mmtlmmm m (B » th# class of ooaimrabl® mIi, Pi—f. I*t v(k) » 1 |f)« Sim t Is a aosmoffctlto*

J S

ottssyshoxo fialto» ooaplotoljr additlro oot foaotloa 0m fcfes

olass of a««ni»tt« sots lagr 2.*.l. Suppose bjr way of ooatzm*

ft lotion that there exists a real nmabsr 6 > 0 aM a sot B of

atsasumfels sots smell thai fet «aeh a*

tt(V < -J;

art j j t J iC . X*t

*" ^i- M « k and lot 0 0

a » n a, />=x B

so that AC for oaoh a. Bsw 60 « r4»

iiwiiiigi! SSI 1 *=i a*

n 1 I « 2L ixl 2® 2

36

m that

*2. t 1

>U) ± « ( V - | l «(*k> < 'jjL * p • ——{

f#J* aaali »• BlflMi *<A} * 0 « B i T U ) « 0 *jr a.2.1. Sw

AgpAft+x and. tUh) la flxilta «« that

11a r U , ) - T i n . Ag) - t ( Q V - r(4)

*T 1.2.7. >«* T(s^) > | J f | t r 2.7.1. k m , * V i T « V * I J / | - > ' > 8

f»v aaah BU ttraa* •(*) > 0 and a oaatzadlatlaa la aatabllahadt

•ha aeauaptlon la falaa, and J f la atoaolutaly aantlBuoixa.

i w i t t i 2.fc.l ami 8*8*1 aHaw that If f £ ^(u) am x$ m m

J t la an erarywhere f in i te , a a^ l a t a l r addlttra, absalmtalr

oa&tlxmoua funotion ©a the aaaaazabla aata. Xt la interesting

t# not# that tha a&a©ii-Hlfc§&3m Shaoxasi atataa tsmt thaee ttisaa

pvapartlaa give a ooaplata afcawaataatettlasi af tha IntagmX aa

a ftaatlaa an & • A pxoaf af thla thaaraa la g l n a tqr Nuazaa CD.

CHAPTER BIBLIOOBAPHT

1. ttuwyn*- •• a.. mtrgdaatlaa to MHB» Baft P»t.CT»n<n>, Steading* Mdison-wasiey, 1953.

•• Tor W7*

3' ^ 8 ^ Bi^88*886

3?

CHAFCEB H I

MEASURE VIA THE XHfEGHAL

fhe classioal theory of integration la based on the

theory of Measure. Bowfltsr, the purpose of this chapter Is

to develop a theory of Integration without first defining a

neasure. This approaoh focuses attention on funotlons and

their Integrals at the outset, as well as alleviating the

pxoblen of obtaining a suitable Measure. Ifce final section

of the chapter shews that a aeasure apice mn be obtaiBed

from the integral•

The type of aathematloal structure which serves as a

base for development of the integral will be described first

in this chapter*

3*0. Definition* Let f and g be real-valued functions

on some space 1. Define f Vg to be nax(f»g} and fAg to be

»l»{f,g).

3«i. Definition* Let L be a aemeipty class of real*

valued functions on a space X. L is said to be a vector

space if af+bg is in L whenever f and g are in L and a,b are

real numbers. A vector space L is said to be & vector

lattice if f Vg and fAg are in t> whenever f#g are in L.

3A.I Theorem. If L is a vector spice of functions such

that for each he L» h V 0£ L, then L is a veotor lattice.

38

£{£&£• 8 w f*ge L « a i f V g « ( f - g ) v o + g i iaoi if f > g t

then ( f - g ) V ( t a g » (f-g)+g * f # mA if t< g t H u m

(f*g)V 04-g • Otg ** is*

SHms« < f - g ) V 0 + g I s i n L s o t h a t f V g l a i n L . Bow

f A g * ( f + g ) - ( f V g )

where f V g hag alraaity b a a * shown to bs» In L. Hanaa» fAg

must ba In L and L la a faator lattiaa.

O i a o r a * 3 , 1 * 1 s l u m s t h a t a v a a t o r s p a o a o f f t t a a t i a a s i s

a T a a t a r l a t t i a a i f i t i s a l o a a d a a d a r t h a a a p p i a g a f f o n t o

f * s l a a a f * - f V ® ,

3«i»2 fhaornu A v a a t o r s p a o a L i s a v a a t o r l a t t i a a i f

a a d o a l r i f f o r a a a h f e L , | f | e !#•

toSt* S o p p s s i t h a t J f | 6 I» f a r a a a h f € L , t h a a 1

f * » — ( f ^ i f i >

2

• a t h a t f * n u t Da l a L a n d t p 3 « i * i » I* i s a r a a t o r l a t t i a a *

Suppose t h a t L ia. a vaator lattie®. L«t fh«»t

I f I * 2 f * * ( - f ) a a A fcoth f + a a d « f a r e i n !»• 9 m m 9 | f | i s i a L

and tha proof ie ©amplet«.

F o r t h a r a a a l a d a r a f t h i s o h a p t a r , L w i l l d a a o t a a -raator

la t t iaa of r e a l - v a l u e d fttaatioas*

3 * i » 3 P a f l n l t l a n * A f u n c t i o n J on L I n t o S i s s a i d t o Da

a l i a a a r f n a a t l o a a l i f J ( a f + f e g ) » a J ( f ) + b J T ( g ) f o r f aaA g i a I .

•ad a and b in B. I t a mid to b® aoaaagati-ra i f J(f) > 0 for

aaah noaaagatlva faaotloa f € l»«

40

3*1.4 Theorem* I f 1 i s * nsaasgatlvo l inear funotlonal

®xi L« and I f f £ g where f»ge L» thta J ( f ) f J(*)#

Proof. How fff(-f) Is noimosetl1*®* m m 9

!•&«•« J(g)> J<f)«

3*2. Definition* A nonnegatlve l inear funotlonal I on L

Is said to bs an intsgral I f the following: condition Is

satisfied* I f J f J Is * decreasing sequenoe of fttaotien* l a

It such that Ilia f B » @9 then 11a X(f t t) m 0.

For the smaiador of this ohapter, 2 w i l l denote * fixed

integral m &•

3.2*1 fteorea, I f [ f a | and f a r t I n L sueh that [ f n ] I s

nondeereaslng and 11a f a • f» then I I® Z(f t t ) * X ( f ) .

Proof. Slnoe l l a ( f - f n ) m Q and I® n@nln«reaslng»

W 3.2 ,

0 •» 11a I ( f » f a )

• xm utbutmn • l l n I ( f f t )

• X ( f ) - l l » I ( f B ) .

H.no., 1 ( f ) - 11. X ( f n ) .

H* the extended m l m t e f sjrstea, a l l nondeereaslng

seqisenoe® of fitnotlons are convergent. this faot loads to

the following extension of

3*2.2 Definit ion. Let L be a veotor la t t loo of funotloas

on X. M oxtended real-mimed ftinotlom f Is called an o*or»

function I f there exists a nondeereaslng sefttoae# [ f a j I n L

*1

mmh ttiafc 11m tn • f. ®ia slaa® @f such fuaatlons w i l l %t

denoted If U»

3*2.3 j ^ n u * The vector lattloa L la e ntataed InU *

Fwafi &at f e &« Bta aaqttaaea of fuaotloaa fn « f for

• w y a Is aoadooraaalag la L and 11a fn • f. Ihua, f e U •

3«2.% Thearea. If f»ge U and * «al b tx« asmaagatlTa

raal mwbera, then af*$»g la in U and U la a lattloa.

fm®t» Oiora mmt exlat aoaioovoaalag aaquaaaoa [fn]

[%] *t & 11* tn m t and 11a gB » g. Then

{aaaeC%t%)] la la L and la aoadaeroaalag and

1.1a aax(fats&) •

ittt aax(f,g) » f vg. Thua, f v g la la U • Slallarly*

fAg e U «

and U la a ?wt«f lattice where aultlplloatloa tsy nonnegatlve

*aal numbers only la coneldered.

3*2.5 fhooroa. If (fn] and fg«] art nondecreaaing L mJ I J

aatuaaaaa of fuaotloaa f*oa L auoh that 11a fn • 11a g^, then

1 U I(fB) « 11a K^).

Proof. Choose k£ L euoh that % f 11a fn. Since

fojp every a and lla(faA k)« k, 11a I(ftt) > I(k}« far avoir a»

Kg,) 6 11a X(tn). Hence, 11a K g ^ l l a I(fn).

Sepeatlng the argument with the voloa of j"ftt] and [g ]

latarohaagod, the inequality 11a I(fn) £ 11a 1(9^) la obtalaod.

Hence, 11a I(fft) « 1U IC%)«

krZ

3,2*6 Definition. If f is l a U , then Z(f) » lis I(fn)

where }fB^ is a nond®creasing sequence in & such that

lis fjj, * f •

Theorom 3 , 2 . 5 show* t h a t the value o f X(f) depends only

on f and ia independent of the choice of J tn\,

fhe class o f functions whoso negatives a r e i n "U w i l l

b e d o n a t e d b y * i . e . - \J * {fl-f £ U } .

3.2.7 Theoron. A function f is in • U if and only if

thore exists a noninereasing sequenoe of f u n c t i o n s {f,^ of L

suoh that lis fn • f •

Proof. Let Jfn"J "bo a noninereasing sequence of functions

of L suoh that lim ftt « f. Then {•%] is also in L» is non~

decreasing^ and linC*^) » »f« Thus, f t • "U •

If f < - tJ « thon -f e 1/ . Urns, thoro exists a non*

d e c r e a s i n g s e q u e n c e { f ^ to L s u c h m a t l i s f n « -ft a n d f o r

e a c h fa fcL, -fn e L. thus* f - f a \ is a noninereasing sequenoe

of funotions of L suoh that lim(»fn) • f*

From 3*2»? it is obvious that LC - TJ • Also, f < - IT if

and only if *f t U .

3*2.8 t h e o r e m . Hhe c l a s s - { J is a v e o t o r l a t t i c e o f

funotions under m u l t i p l i c a t i o n b j r nonnegative real numbers.

P r o o f # tot f a m i g b e i n • X J , a n d l e t a a n d b b e a o n ^

negative m&l n u m b e r s . Obviously, af+bg t- XT . Mow , -f a n d

- g axe in U • Benoev -f A -g t U and ~(-f A -g) m f Vg i s i n

- U . Similarly, f Ag t ~ XT , and the proof is complete.

3-2.9 Definition, If f £ . U • thoa 1(f) « •«•*).

If f it in both U a»A - U t tho Aoflaltloa i« #o*iaiatoi*t

with 3*2.6, for f e - U lapllo* that -f^U 9 and f+(-f) * 0.

Henoe9 X(f+(-f)) « I(f)+X(-f) • 0. ®n*s» 1(f) • -X(-f).

3.3* Definition. Let f bet function on X* fhoi* the

upper integral, Iff)* la diftaid by 1(f) » sib X(g) where

g e U and g> f. If no rash g «xiita, thoa 1(f) «°° • 1!ho

lower integral, l(f)t is defined by J|(f) • -l(-f).

Since «l(»f) « ~glb 1(g) where g > »f Hat

geXJ * -(-lab 1(g))

where -g fr f sad -ge - U • aa equivalent Aoflaltioa of £(f)

oould l»# j ff) « lab 1(h) m t m h £ f and he - U •

3.3,1 Jheorea. the upper Integral has the following

properties*

i) l(f*g)*l(f)*l(g)i

U ) If o > 0, thoa !(©f) • ®l(f)| sad

III) If f f g, then I(f)£ 1(g).

Proof. X*t 1(f) » gib I(hf) where hfe V and hf f, and

lot 1(g) • gib l(hg) whore hg € U wad hg > g. I # t

• - N * . i h f + « £ u « * > w '+«} •

fhoa,

gib X(hf)+glb X(itg) « gib m t4h s) > gib I{hf*g)#

for if otherwise, thoro would oxlst hf and hg saoh that

X(hf+hg) < l(hf<fg) for ovoxy h^g e H. Bat thla Is laposslblo

•lno. hf+he£H. M o m , I(fKT(,)i T(f+g). If 9>o, thn

I(of) • «lto X(«h_)

• gib •X(h-)

• o l ( f ) .

vr-ii than 1(c) « «1» X(k,)2 dk Ill i - I(f>.

A l l l l l t t r pS9#f fltUNfS ttlftt 2 ( f )4 | ( | ) f

)*3«2 SflSSE* f «* X#

9mmt» tow 0 • 1(0) » I ( M ) * I(f)+I(«f)t fims*

-X(-f) f Iff I «ad iCf) £ 1(f).

3»3*9 JktSUOtt* If I b n 1(f) * J(t) » 1(f)* Pnmt* 1 ( f ) m gib ifg | «h«n « > f and g € U * Iff I

alium f e U • H»r t £ U » Share exists « SMaiftmftftlag

M I M A M [fa] c L otioii that l i s f B • f» Mm *»tme L m that

I ( f a ) • 11%) *» -X(-fj|) « 2(%H wiiy n,

K m K f f t ) • « % K

fima* X(f) ^ 11* I(f t t) • 1(f)* line#* I f f ) » 1(f) • I f f ) .

3*3** gtfVnlUffl f U i aam«flatt«a function m x

oueh Hist i{f) « j;(f) maA this mmmm mlue 1« flats©, th«n

f in IAU t» te intagHMU and 1(f) » I f f ) • |{f)«

3«3«5 Baflalttem* 2f f U « fuactios on X such that

X(r*) • !(#•> and 1(f) • (f) uH at 1*M* <n» of th«ae

values t i finite* th«m 1(f) m l ( f * ) - l ( r ) .

3«3*6 Bafinitiea* It t %m m fiction m X meh that bath

t+ tmA f* mm lak«gnbl«f Hiaa f le nil to fee intags*bl*«

®ta following thaavaa ati#w@ that Definition 3#3*# Mr

consistent Witt py®Tieti® atatanasitii mmmmAm X*

*5

3*3»7 &£££&• ** * fuaotion f on X le iatografclo,

thoa 1(f) • g(t).

Proof* If f i« liitegyabl®, then

- «f) - Kf+j-Kr-),

teat

X(f*)~I(f-)> X(f+-f) » I(f)> 1(f)

» j(f+-f-)> i(r*)-Kf)

so that 1(f) • £(f).

3*3*® SStSUEKE* *»r * fwiotioii f #» X# 1(f) » J(f) and

till® mint io finito If and only If for each £ > 0» thoro

«lit ge- U and h € U ouoh that g* f* h, 1(g) ami 1(h) am

flaitti and X(h)~X(g) < € •

Proof. for ®aeh e > 0t ouppooo that sueh g &M h oxiot*

«W X(gN 1(f) «inoo i(f) « Xttb X(k) nhoro k£-u and k±t.

Likewise, 1(f) f X(h). Bo&oo, 1(g) * 1(f) * 1(f) i 1(h), hut

X(h)-X(g) <€ for •••XT 6 > o* ®1UB# J(f) • 1(f).

Lit 1(f) « J(f), Sion gXb 1(h) « lub 1(h) Hhoro g£ • U »

li e U g f f - h. 0*9*, far etiflr € >0§ thoro oxiot « e ~ U »

he U •»«)! that I{h)-I(g)< € •

3*3«9 **aaa» If g aM h ar® funotions on 1 suoh that

lih, 1(f) and X(h) aro finito, and X(h)~X(g) for a

partioulor £ > o, thoa g+f h* and X(h*)-X(g*) < € .

Proof. ®*o proof consists of throe o&soe. If h £ 0 and

g> 0, thoa tho proof is trivial siaoo h * h+ and g » g*. if

h < 0 and g < 0* thoa tho proof U again trivial oiaoo h+ » 0

4*6

0# XI1 hi 0 MIA u Of %h®tt h • sud * 0 so 1sftu&

t < # f h • *• and X{h+M(g*) < I(h)-I(g)< £ *

3*3*10 Theorem. If I<f) « J{f) and this mine is finite,

then f Is integrable.

Proof. By 3*3*S» for aver? e > 0f there exist h e U and

ge - U stioh that « i t f It and X(h)-X(g) < 6 . Dftvg^ t+± h+

w that X(g+)f l(f+) £ I(h*)# and X(h+)-X<g*) by

3*3*9* i»8Wf J(f+) • l(f+). By a sinilar proof,

1(f) * 1(f).

Both common values mist be finite* and f is tntegrable by

Definition 3«3«6.

fhe HieoreiBS 3*3*7 sad 3*3*10 give the following nsoessary

and sufficient oondltlons that f be Integrable.

3»3*li fhooren* A funotlon f on X Is integxmble If and <OH*

only if 1(f) and £(f) exist* are finite, and are equal. Shis

somen -valus Is X(f)*

»T Theorem 3*3*11, It is ebrious that 3*3*8 also proridss

neoessaiy and snffleient eondltlon® for the Integrabllity of f.

She olass of functions on X whioh are integrable will be

denoted by 3.

3*** fhooren* The olass S Is a rector lattlee of

functions containing L, and I is a nonnegative linear fonotional

on 3*

frgoof* Let fe 3 and o be a real ztunber. If o> 0, H H UMMt

X<of) « eX(f) « 0.1(f) * K©f). Xf o < 0, then using 3*2*9*

J(of) - -I(-of) m o(«4)I(f) • ©Iff)

• • • ! ( • ? )

• I(.a)(-f) « l(ef)«

fh«s» «f e §• Lit f«g c. 3* then T(f+g)£ I(f)+X(g) and

-I<f+g> * *(-lWf#g))) « X(-f-g)i -X(f)*X(g).

asms,. Jff*g> > I(f)+I(g) a»t f(f+g) ± I(fHl(«) i i(f+g)« EtBM, I(f*«) * J(f4«> « X(f)*Z(g) - Z(f)+Z(g). and f4g*8. «M*Bf 8 Is a llaaar apaee and I li t Uniir fHaatlaaaX an 9* Qta**aa 3*3*1 «h«w* that 1 la naisnagatlTa an 8. Bar 3*3«3» & c. U«

f € L lspllaa f £ U • Haae#t 1(f) • j;(f) and this ralue Is flslti tm f £ L. Thua, f £®» and I»C8*

proof that 8 la a -raotor lattloa la ooaplata If for f € 8, than f*£ S* &at f e 8* ffc* araay € > 0» thai* exist a • gc - U aad an he U auoh that g f h and X(h)~X(g) < * • Xtow

X(g*) i I(f*> * T(f+) * Z<h+), and • ®ww* X(f+) m £it+$ mime £ ma ajrhltmsy ant. f+6 8*

3..1 Learn, 4 &«n»gatlve function f ob X is la U if and oalj If thara azlata a nosmagattra aawanaa [ fa] € I* auoh

oo that f « f« Zf auoh a aequanoe axlata, than X<f) « Xl(f )«

Pwof« 31m If part ©f th© thaoxwa la Iwwsdlat# sImm !» cU .

m

Let f H® ?mmm%!*• 4» U • %bmr® axiata a auh

&#©a?tasl»g aaqua&aa [%] t L amah that Ua » f« Oenaldar

g+ for mch a. mm obtaining • ssouaagafclv® Meq.mn&& mam n

Halt la f« lat ' l " «!• f 2 * <2*81i ^ ** «asai*lt

CO r u • % - ^ . f Hw» f • l i s 8b » Z ' i •"4 • i n B* r n s *"

1(f) - l t« I<%)

- Um I< t f . ) '»* 1 m

• ii« i : i t ' i i OO

- X K*i>« |:i

3*^«2 tMwm* Xf ff.J is * aaqittaaaa of aoimagafeiv© c «J Op OO

ftraotlens #f U , thaa f * ^ f n la la VttM 1(f) » ^ I ( f R ) .

Wrnmtm F&r aaah a thai?® #xiit« fgu J e L «ueh that

% • 21%ga W Bhw, m = l oo oo

' - 2 ; n-X m~ 1

1* fcte« sum ©f a @f ®®a»a®atiira funofclons of |» tad

wmt in> i»U» iwr aaat* a* Kf . ) • zL^ifhum) ** 00 ** m?l §

i ( f ) - Z « V -n~i •*

3#4«3 ffcaagaa. If J f a | la a aan&aamaalag sequanee In a

flMh that l i s tn m f, thaa t £ 3 and 1(f) * 11a X(fn) If aal

•aljr If 11a X(fn)< °o .

fwaf t ftr araxr a# f«- f« If 11a X(fn) • 00 # thmn

1(f) •<» . Bat this lapllaa that t 1® mt la 8« whioh la a

eoatyadletloa*

%9

x*t t i n X(fB) <00. stnoa {f^ea# t h i n i x i i t i m

hgC V mufo that f^f h% and X(hi)~X(fiX «|»« Batatollsh a

aatuanaa {%}€ U «t»«h that for a> 1, h^ • {*»-*a»l) and

h Stan T" FA and

^ X(h|)»I{f f t) 4 £g+» * * *"gii ^ ^ •

°0 oc,

I*t h * 2 1 h | . than he U by 1*%«3 aa& X(h) « l{h±)*

Alto, h-> f and X(h)«lla X(fB) < € • mm t n e B lapllas that

for every a there exists a -TJ atteh that e n - f n and

<€ . Ihnr 8^4 f f h and Z(h)*X(«^}^ 2* •

Thus, FES HY I.J.? and 11* «FB) » X(F).

3»*«* B>aoroa» th# fimotioiaal X la aa Integral on S.

?ROOF» BY 3**» X IS a aaaaagatlta l inear fuaotloaal OA

S. Z*t [ f B j fea * mmimmmixm sequenoe In a sueh that

11* tn m 0. Oortalal? XI* X(FB) > - 00 alnea X(FB) > 0 FOR

omtf n. mm {-FB} %m nondesre&sing, LL*(-FN) #* - f» and

itM X<-FB) < 00 . BAR fhoora* 3**#3» -F E S aad X<*FB) • X(.F)

@0 that FC 8 and 11* X(FB) « X(F) » 0. 1ISM, X la aa

latagral on a by 3*2.

St la important to asoertala the latagrafelllty of the

l imit of a aaquaaoa of faaotioaa l a l «v©a though tha sequence

saf a©t too imTemtiitog* The following two thaorowi shm

oortala ooadltloaa untax* whioh tha l l a l t la latagrafela*

5©

3«fe.5 ®|2t$eB« ^ ffnl Is a s#ttt#noe of nrnin*gatlv© . . . • t i i M I M i iiinm I I I H o m i [ # * J

functions Hi 8, thoa fcho function iaf tm U la 8» end ££& fn

is In 3 if 11a X(ftt) < , ta whloh oase

x c u s fa** Ala Kf n )«

Pm&t* mt % • f2A ®*o» {%} is * aoa^

insreasins sotuoaoo of noim#gatiir© fuaotlons i& 8 smote that 11m « laf fn » g. Utoa {-%] i s nondcoreaslng in S and

» -*• ! ( •%) * 0 fo r <Mh fi s® that ® Is la I tgr

3 . 4 0 . Lot h k » inf f a « Visa f hjj} Is i a 8, Is uonaoisatlTo,

is aondoovoaslngp sal lira hn « iim f a . Slaoo hfe 6 f n for , k i af X(hfc) b 2A& X(fa) << 0 0 * Honao, l i a X(hk) <• 00 whioh

iaplios X(lla K^) i s In S by 3 ^ . 3 . Uwrn

z ( l i a hk) • Kiia f n ) • Xla iCM-JUf t «*a>*

3«*.6 thoorsa* Lot 1m a sovioaoo «f function* ia

a* tt thor® exists g la 8 suoh that in g for oaoti a» Ml

f - l l a f t t , thoa X(f) • l i a I ( f n ) .

Proof, fflio sequence of fuaotloas [%*«} Is aoaftogatlvo

aal la I# Mom tn+g £2g. Soaoo, X(fa+«)£ 21(g) and

X(i|ft(fa4*)) * 21(g) < «> •

By 3* *5# is ia S» • lt»Cfn4g) » f+g Is

-la I# And X(ila(fn»g)) £ llm X(fa+g)« Inas#

K f ^ ) -

i M s x(fa+g)

• jjbU<*»)+X<«))»

md 1% follows that 1(f) £ ]JA X(fn)«

Slio sequence of fuaotioas J g-f a ] i s nozmogativo a»d ia

8 so that X<MS<e-fa>> < H i « X(«)+2lB But

si

I<3JS(g-fa)> . I(ll«(B-fn))

- I(g-f)

- K s ) - l ( f ) .

Warn—, - U | ( - I ( f a ) ) - - I E I ( f n ) , and

1(f)? I E X(fB)> Bwa, H g I(fB) > X(f )> I B I ( f a ) . 9MM-

f o n t 11a X(fn) * 1(f) .

Oia remainder of this ahapter will to© ooneeraed with

datalapla® a mmmm spaoe fro® the Integral Z«

3.5* Definition* 4 nonnegatlw function f U said to

be xeaaarable vlth reapeet t© I if f A i e B tor m x y g c @«

3»5»* SSS$SEe ** * g are nosmegatlre aaaamnatile

f*aetl9na» then f A g» f V g, of for a > 0» f 4 , mid f* t n

MASSIftllc*

fm®t* tmt f and § fee nonnegatlfe aeasmrabie funetlona,

and l e t h in 8. then hA(fAg) » (hAf )A (hAg) vhlah la

Is 8. Henae, f Vg la aeaeaxable* The funetlon

hA(fv/g) « (hAf)V(hAg).

whloh la In St so that fVg la iae*aitx«ble«

Let ©> 0. f\*r eaeh h£ 8» hAf £ s and e ( ~ A f ) £ S. Bat h

e(~~«Af) • hAaf. therefore, af € 8«

Haw f* » f VO and f • -(fAO) • 0* Slnee f VO and 0

are aeaaurable, f* and f are aeaeurable.

3*5*2 Theorem. If J f a j la a sequence af nonneg&tlira

aeaavtrable fitnatlena anah that 11a f a » f , than f la saaaurabXe*

5a

Proof* fpr oaoh a» (tnA*) € s for ge 8, ®tmst {fftA g}

is « soquoaao la 8 aad ll*tfaAg) « f Ag« Bat |fnA g\ |g|

so that fAg In S by 3«*«5»

3.50 Definition* 4 set 8 iaX t« said to bo aoasarablo

with rospoot to I if its oharaotorlstio funotion fg li

measurable. A mt S in X la said to bo latograblo with

rospoot to I if its oharaotorlstio fwotloa is latograblo*

3-5»^ A set 3 In X is said to bo a sot of

measure mm If it is latograblo an! X(Kg) «» 0*

3*5*5 $!StS£&* If 4 ami B an measurable sots* thou

&UB# Af)B aafl M are measurable.

Proof* l<et A and B bo lsoasnrablo sots* Hiwa

*UUB) " s a V %

aaA * % A % # both of vhioh aro measurable by 3*5»1*

*or g la 8* • <gA0MgAK AMgA(K A A B))*

Sa«sfc of tho terms Is la 8 so that th® mm mmt bo in 8* Soaoo*

i*B is measurable. Xa establishing too abore equality, thoro

are three oasos to ©©asider. If g £ 0» th® equality is sat-

isfied siaoo g » £+«*«« If %» thoa g > aaA g>

fho ©quality is again satisfied siaoo • % - % n »• ®*#

othor possible range of falnos for g is 04g<%* 2a this

oasot must bo 1 so that x£ 4« Then g A % « t

A A B lc. If xel Tliua,

(«aka).(«a*a0b) - [«| « » < ; .

51

Also,

® A % * S * ff* It, » I Of if *£ 1

aad tfao ofmaitr i> ootobliohod.

3*5.6 333&8SS3L* ** f*»] *® * a#fw»®* of •ooeuroblo 8it*»

then «•* W O atiMWWftWl®.

f m f * l « U * UA|| •»& lot AgU Oioa

% is ao*«tu»blo bgr %$•$• tho 0041101100 [%] is a soqttonoo

of aouramblo #ots oaA jl "j is o softioiioo of noxmocfttiYo

&«a@m»bl© tmmtlwm such that

• V V V ...VK V

Then

11» - 1U(K MV — v k ^ ) - iA,

n l la aMmxabl* tf 3. j.2.

u t A * nj^ «H' % « • n V «-» { % } *• «

imgilifi a«fttezu»@ of aeoattrablo fnaottowi sueh that

li* - IlmCE^A % A ,## A ^ ) - % t

and Ij, is MMunuMtblo by 3«j«2*

Zki oxdor for tho olooa CI of «o**ax*b2.o oofs to fox* m

<r-algebra, it is necessary to hwre X C d . A sufficient

oondLition for tho maaourabllltjr of X i* glvm in tho following

108880*

3.6. Lemma, Xf tho fnaotion i its aoaoux*blo( thon X

%« measurable.

Proof. lot i bo * aoftSMxmbio ftmotion. thon X is

me&eurabl© sine® « i#

3.6.1 'Ehooroau If the funotion 1 Is measurable, then

tho oXass ft of M M U i b l i sots is a <r •algebra.

Proof. frost 3«5«5» AUB and A Oil aro aoasuxmbXo if A

n i B u i aoaswmble# If [a^] 1s a sotaoztoo of aoaswablo

sots, thou UAjj la aoasuxablo bgr 3.5.6. If i Is x»oa*u»blo,

thoa X is asoasambXo, aaA the o©mpXe®ent of any Boasmsablo

sot ie then moasaxmbXo by 3*5*5*

n is m m possible to us# the iatogsal 1 to dofia® m

3,6*2 Definition. Let the fnnotiom 1 bo aoasnmble and

define the set fuaotion u on CI br

X(Xg)» If 8 is iatogxablo*

co s if E is In (X , bat B is not integrable.

3.6.3 Theorem, ®io sot funotion u as defined in 3*4.2

is a measure.

Proof, Oiie class fl of measurable sots is a tr-algebra

by i.*.U ®**»t #€ m W • ic%i « t w » 0* Also,

all values of 11 oust bo nonnegative sinee 0. t&t

bo a sofwoiioe of disjoint, measurable sots in d * She proof

is oosploto if 00 °°

*< U B*) * 51 t\zi ** fiz t SiO

lm% % « % u % u *•*1)% S»4 # « P i V 919x1

*02 *

si»oo ®$/l% * If and are in S, then their sua is

ia B and *(%} • «{%)+»*(%)• Kg,^ ant K ^ . If

»(8)

55

K o g I s i n S , t h o a a n d a 7 i I n 8 s i n s # % ^ A % 2 * * * *

% g 2 A m m I n 8 * 9 r a a » f o r l a S «

a ( % ) * n C % ) 4 n C l a ) *

H o a o o * t » ( C f c j ) » « > I f a a & o n l y i f a t l « u t © s o o f u ( S | ) »

1 1 ( 8 2 ) 1 1 ® n t I f t h i s l o t f c o o a o o #

t t ( ® 3 ) « « < % ) * » { % ) •

0 0

S t t o r o f o r ® * 1 1 I ® f i n i t e l y t d A l t l f t a a d ^ T ] m ( M & ) • l i a u f a ^ ) .

f h e s e q u e n c e i s a o M o o r o a s i n g * I t m * » K Q 9 m &

m ( % ) * ( % » ! ) • H o w u ( 0 ) I s o i t l i o r f i a i t o o r l a f i a i t o . I f

I t l a f i a l t o » K ( | a m s t b o I n S , t a d i s l a 0 f o r o a o h a *

H o a o o t b y 3 . * . 3 » l i s K K ^ ) » 1 ( 1 1 * X ^ ) - I C % ) « © t h a t

1 1 K u ( Q ^ ) * a ( 0 ) « X f « ( < * ) l a l a f i a i t o 9 t h e p r o o f I s a g a i n

d o p o a & o a t u p o n 3 « * K 3 * K l t h o r t h o r n o x l o t o * p o i l t l t t i a t o g o r

a a u o h t h a t u ( G ^ ) * ° o f o r K q ^ U l a i f o r o a o h a w i t h

l i a 1 ( 1 ^ ) - 0 0 # l a p « ( ( ^ ) « C O t t h o a m ( % ) • 0 0 f 0 r a l l

a > a a a d l i a a ( f l ^ ) » < » * I f i o l a 8 f o r o a o h a a a d

l i a I ( I ^ ) • 0 0 t t h o a l l m u ( C ^ ) - < * > a l a o o a ( C ^ ) « 1 ( 1 ^ )

f o r o a o h a * t h o r o f o r o , w h o t h o r a ( G ) l o f l a l t o o r l a f i a i t o *

0 0 0 0

E » ( * „ > . « » » ( V - » « > - » < u v

Cl&PfBB I?

o o m a x m OF ws mmmm

A theory of integration was developed in Ohaptar XII

without first defining a aaaawa. A aaasmr® ip*m (X,<3,u)

was than developed tmm the integral I# IT is of interest

to investigate tha rasults of using tha methods of

Chapter II on tha IMI«» spaaa (Xt<3 »m). In partioular,

will the resulting alassas of measurable and integr&ble

funotlons be the aaae as thosa arrivad at in Chapter III,

and will tha integral with raspaot to t*» J^(f), b# tha M M

as 1(f)? It will 1m shown that under certain conditions the

answer to tha above qaeetlen will be in tha affirmative.

theorem. Let (X, ® ,a) bs a measure space, let

be tha integral with respeot to « aa dafinad in 2.1.*, /. 1 and lat X M *• tha aat of funatiana (on X) whioh are

intestable on B. Then j' is an I-Integral.

Proof, /(u) is a linaar veotor spaaa bar 2.5.3 and

2.2.2. Definition 2.1.* shows that f e X.M implies

f+ € /<U).

thus* X. (*) 1* * 'vaotor lattioa of funations bjr 3.1.i. tha

integral is a linaar functional on £(u) by 2.*.3 and it is

nonnegatire by 2*2.3. fhe Monotone Convergence Theorem,

theorem 2.6, impliee that if fa| is a noninoreeaing sequence

5?

of fmaotioas in X (u) tuoh that 11m fR * 0, thon

11m (f» - fitm f. • 0* '1

Definition 3.2 is satlsflod, and tho proof is oomploto*

4*2« SSSSZH* fnnotlon 1 bo moasarablo* If f

lo & nonnogatlvo, Uttogsmblo function on X, thon for oaoh

roal axatoor a, tho sot S » >a} la moasuxablo*

Proof* Xf « it n«g&fciT#e then E * X which is aoasuvablo*

Xf * is positivet let

f f g « — _ A 1)

• &

whloh li intestable since

f f — - and —»/\i

a ii

• n integrable* How g >0 If zeM and g • 0 if x^i. x*t

hjj • Iabs whieh li also intestable. Zf z eE» 11a ® lt

and If z 19 then )% • © for each n. Thereforet 11m • Kg

whloh is measurable bjr 3«5*2« Ifcus# 11 is measurable* Xf

# * @ i lot h^ » lAnf whloh is lntegs«blo. Xf x j£g, thon

f(x) » 0 and hjj « 0 for each n. If x^l» 11m 1% « i, ttius,

li® hB ** %g and Is again moasurablo by 3*5*2.

Qm following theorem shows tho equlvalenoe of tho

Intogml X on S to tho integral with respect to u.

*.3« SftogSE* & fc* * veotor lattice of functions

on Xi lot X bo an Integral on &• and lot tho function 1 bo

measurable with respect to X. Sion thoro exists a (r -algebra

58

Q. of subsets of X and a ma&sure « on <2. suoh that eaeh

funetlon f on x 1« x-iategmble if and only if it is late*

gvable with resyeet to u . If f Is lntegzable, then

« f ) - / f

«•»*• / f U th« in tceml with n q a i t to a tmr X.

ffgoof* the alass d of sets measurable with respeot

to X i s a <r«*algebm If 3.6,1. I*t m be defined ©a <2 as

in 3-6.2« Bien u i s a m m « » on CLby 3*6*3*

Let f be a noxiBsgatlTet X-lategmble function on X*

then f i s aeasuvable with respeot to Q by 4*2* Let

1 *k,a " [*lf(*) I

where k and n a*e positive integers. Br theorem fe.2» KJt>n

is Measurable* If

then

It f 7-—#

n

n —f >i k

so that

V » " S , n A ( T f >

whloh is in S* Thus, * ( 3 ^ ) < «*> , Let

i n*

a |r, K»®

Obrlousljr, % £S*

5f

It is neoassary to ahow that llm * f • Suppose that

£(%q) » A for some X, Since f 1® noxmegative, A > 0. tf

A » 0, then the proof Is trivial. tterefore,*. It is assumed

that A >©• Bow A Is either an integer* or it is not*

Consider the ease where A is a positive Integer* and.

lot n be an integer such that n > A. Obviously, nA

« i for eaeh K from 1 * 1 through k « aA~lt and K- » 0

for k 2mA* Hius, for n >A,

r,* 1 %Cx@) * Z %L

i «a-I i "••"Ml > JLm \ Kn> a tc»» a

i )*f0

n

i fjfft Jfy»#i mmmm

Benoe, i

li* * lia(A» < — ) « A*

n

If A is not a positire lattgir, let p « fA]• Choose

€ > ©• Consider the positive real number o • A-p. The

inequality 0 < o <i is satisfied. Shore exists a positive

integer N suoh that 1 IT

— <»<-—• H H

ant there must exist a positive integer 1^, where 1 < kn <=n-lf

60

m&U that

JUS < q <

M 1

UkiwlM, for the Integer )H-1V there «mst exist a positive

integer %4,|» whers i ^ %+i f(N+l)-l, au©h that **•1 ^

i O < mmmmmrnm*

1+1 M*i

Hi general, for JHn9 there exist* a positive integer >mn»

11 jj+n s ltan»l, smell that kMiM

Irfm < © < "" »

H4» Mi The seqmenoe

y%#®7 I M4a J

is thus established, Jfow

kjiiM im J 3 « cB

m.A..rn*

MTS*

«M there exists a positive integer X* smelt that

ltw»n JKrrak

u . a s / ' l+a I <c fc

M

for all n > W< • Gheose n ?nax(p+l9 )• Then there

exists a positive integer l t 1 ? H€ » smeh that K+l • n.

Tlius, ^Iftl ^

mt n '

61

mm

hi 4 ** p*fo ~>p"f

a

m® that

up a p

a

"P+^n ^ . » p4»—' < JU

B &

X n

«a<x0> * ~ X **k a a A:l • i n ? 1 xplfCn % i\*

" T L ^ , n * r L > . » * T £ , V

I t *_ » — - C i l J - H ) « p*—2.

a a a

Bro», /%-A| < £ for n >aax(p+i, Sfe+H) and li» % « f.

War «aoh a, %£*"* th« ••queno* /%J is »©**«

4#oreading. m#a 11a li^) ± 1(f) <• 00 m that 1(f) • 11m !(%)

by 3.^3* 8**

' " • ' • T I / v t %«**.»>-Bat % 1« a alapla fuaotloa wltt* -i- b#la« th« vala# obtained

®a «a®h B» k • 1 to k • a2. Thu«,

nv 1 Kv 1 ( T. — u C Ek,n) - z — u ( B**a^ x ) - M *

$2

asd 1|%I - /%• % i» & mmrngj&tlw®* mmmmble fmmttm with to (X% mi. lis % « f. Bmm» br ttiooro* 2.6, 11* /gjj » /f.

1(f) * 11* lt%) * 11* /% • /f#

«M f in iatogxftblo with iroepoot to u.

l*t f bo any X»tntogzablo fuaotioa on I* Bion f* and

f* ®r® mmrn Bttlret Xw&ntogznblo function® whioh am

aoaummblo with r®sp®et to A fey &#2« Eoaoo® f • f*-f~ la

*oaatt*ablo with roapoot to CL "bp l*4»fc «ai She

fmottona f* a»& f~ must bo iatograblo with roapoot t© u»

lea©#,

1(f) » i{f+M(f) - /f4*- /f « ft*

Lot f bo a aonaogatiiro function on 1 whioh is into-

gvablo with roapoot to n. Aa in tho flrot part of tho

proofs lot

**.»" t* /'<*>> — j m

and lot

% x n «n--r X%»,„•

* *.«•! *

low Jf c «*» ao that *{Ek>n) oo fo* oaoh (k»n)« Soaoo»

Kg istaat be te a aM oottooquotttly la in i* few

lis llg ) « 11* /% • /f * 00

by Shoovos 2*5. 2hozoforo» ft! and li* IC%) « Kli* % ) • 1(f)

$3

fey 3.fc.3« Honoo, Jt • 11a !(%> • 1(f)* If f 1® tatj

funotlon on 1 whloh U lutogmhlo with *o«poot to u» thoa f+

and f mm X latogrrtblo smd

Jf • /f+- |f « Kf+MCf) • 1(f),

la oxdor to show Hie oqnlraloaoe of the I-iatogt*! to

tho $M$©gml with mmpmt to m» it m« neoeseary to hato X

noaouxahlo* Tho condition that the function I ho rao&eurable

woo ohown to ho oufflolont to guavontoo tho aoaauvahllltjr of

X. Although thlo thoolo Aooo aot do oo» It would ho lnto*»

estlng to Investigate the pooolhlllty of other sufficient

condition® mad. In particular, to pwrldo necesa&ry and

sufficient conditions under twitch X is measurable.

BIBLIOGBAPHY

Books

Monro®» M. B.# latrodnotloii to Measure ag| IftteMration, Beading, Mdison-wealey, 1953.

Hoyden, H» L., Real Analysis. Iteomlllaa, Hew Tork, 1963.

Budln, Walter. Principles of j fheamt.lftal Analysis, MOOimW-r

ffipgg t»nalatedj>y 1,^0. rung. G. I. steeherts and C< XoungrSTf.^telierfc and doapatiy, Hew fork, 193?.

r, Angus 1*, iitf ® i tt i

nylor. JagM *., O.n.r.1 ttwnr of B m e U w a and i&i Blaiadell, Mew York, *9o5*

Artiolee

Stone, K. H», "lotea on Integration,w Proceedings gf the National Aoademy gf aelenees. f@l„ ^ ( P ) ,

6^