AMS WORDS

Internat. J. Math. & Math. Sci.

VOL. 21 NO. 4 (1998) 653-670

653

SOME REMARKS CONCERNING FINITELY

SUBADDmVE

OUTER

MEASURES

WITH APPUCATIONS

JOHNE. KNIGHT

Department

of Mathematics

Long

Island University

Brooklyn

Campus

University Plaza Brooklyn, NewYork 11201U.S.A.

(Received November 21, 1996 and in revised form October 29, 1997)

ABSTRACT. The presentpaperisintendedasafirst step toward the establishment ofageneraltheory of finitelysubadditive outermeasures. First, ageneral method for constructingafinitely subadditive outermeasureandan associatedfinitelyadditivemeasureonanyspaceispresented. This isfollowed by adiscussionofthetheoryofinnermeasures,theirconstruction, and the relationship oftheirpropertiesto thoseof an associated finitely subadditiveoutermeasure.

In

particular,the interconnections between the measurable setsdeterminedbyboth theoutermeasureanditsassociated innermeasure are examined.

Finally, several applications of thegeneraltheory are given,withspecialattentionbeingpaidtovarious latticerelatedsetfunctions.

KEY WORDS AND PHRASES: Outermeasure, inner measure, measurable set, finitely subadditive, superadditive, countably superadditive, submodular, supermodular, regular, approximately regular, condition

(M).

1980AMSSUBJECTCLASSIFICATION CODES: 28A12,28A10.

1. INTRODUCTION

Forsometimenowcountablysubadditive outermeasureshave been studied from the vantage point ofageneraltheory(e.g. see[4],

[6]),

butuntilnowthishasnotbeentruefor finitely subadditiveouter measures. Althoughsome efforthas been madetoexplorethe interconnections between certainspecific examples of finitely subadditive outer measures

(see [3], [7], [8], [10]),

there is still no unifying framework forthis subjectanalogoustothe oneforthecountablysubadditive case.

Thispaperis intendedtobe a first step toward the development of suchageneraltheoryfor finitely subadditiveoutermeasures. Ifsuccessful,such anabstract framework wouldunifythesubject,making further researchinto thisareamore efficient andtherebyenhancing furtherprogress. New examplesof finitely subadditive outermeasures could be establishedat will in any space andtheir characteristics readily inferred from aminimumof information.

Section 3 contains a discussion of the general concept ofa finitely subadditive outer measure.

Starting fromacoveting class and anappropriatesetfunction defined on that class,weshow howto construct a finitely subadditive outer measurein any space and howthis leads to a finitelyadditive measure inthatspace.

In

Section4weexplorethegeneralconceptofinnermeasure. Using a suitable finitely subadditive outermeasure,we show howto constructan innermeasure inany space. Wethen characterize the measurablesetsdeterminedbythis innermeasure and examine the interconnections between thosesets andthe ones determinedbytheoriginaloutermeasure. Thepropertyofsubmodularityplaysakeyrole in

654 J.E. KNIGHT

this investigation. The results obtained here make it clear that thestudyofinnermeasureisessentialtoa fullunderstandingoffinitelysubadditiveoutermeasures.

An

example ofthe value ofthegeneral principles developedin Sections 3 and 4 ispresented in Section 5. Wecan use agivenfinite, finitely subadditiveoutermeasurevand the collection

S

^ofits

measurable sets to define two new finite setfunctions v and

uo.

^The results of Sections 3 and4 immediately revealnotonlythat

’

^isa finitely subadditiveoutermeasureand that_{Vo is}an innermeasure, but alsotheir basicproperties andtheir interconnections with oneanother.

The importanceofageneral theoryof finitely subadditive outer measures beginstoemergeinthe discussionof theset functions

v

and vo, forhere we first see the marked contrastsbetweenthenew theory and that for thecountablysubadditive case

(see [6]).

Whenv and ,o are constructed from a given finiteoutermeasure,certain important related concepts which coincideinthecountablysubadditive case are actuallydistinctinthe finitelysubadditive case. Forinstance, ifvisacountablysubadditive outer measure and

q, ,.qo,

and

,So

denote the collections of measurable sets for v, v

,

and o, respectively, then

,S q, ,So. ^However,

^{if v is} finitely subadditive, we can only say that

S

^C

^,S, ,So.

Toobtaincomplete equalityinthelattercase,wemustimpose an additional condition on

,. At

the end of Section5,weseeotherstrongcontrastsbetween thenewtheory andtheold in the conceptof regularity.

InSection6 weapplyall thegeneralresults of the previous sectionstothe importantexamplesof measures definedin termsof a lattice, thus obtaining in a systematicmannerthefundamentalproperties of the associatedouterandinnermeasures.

(In

thisconnection, see also

[3], [7], [8], [10].

Foramore detaileddiscussionofthegeneraltheoryof finitelysubadditiveoutermeasures,see[5]).

In

^thenextsection weprovide forthereader’sconvenienceasurvey ofsomeofthemorespecialized notationand terminology that we shah usethroughoutthispaper.

In

the caseoflatticerelatedmeasures andoutermeasures,our notation andterminologyareconsistent with

[1], [2],

[9],

[11], [12].

2. BACKGROUND AND NOTATION

Throughoutthispaper,

X

willdenotean arbitrarysetand

7:’(X)

the collection of all subsetsof

X.

Weshallalwaysassume that

X .

^{Acollection C}

^7(X)

^{will becalleda latticeif}

^{A, B}

^impfies

that

A

tJ

B

and

A B .

Furthermore,weshall always assumethat

, ^X .

^{Thecomplement}

ofaset

A

C

X

willbedenoted by

A ,

^{andthecollection}

_’

^{willbe definedby}

_’ {L’

^C

Notethat is alattice.

4()

^willdenotethealgebra generated bya lattice and

6()

the lattice of countable intersections of setsfrom

.

^{Alattice willbe calleda6-lattice}

^if6() .

We shallalso need thefollowingdefinition and theorem in Section 6.

DEFINITION2.1. Alattice is saidtobenormal, if for all

A, B

such that

A

f3

B ,

^there

exist

C, D

such that

A

CC

, B c/’,

and

C’

f3

D’

THEOREM2.2.

A

lattice isnormal if and only ifforall

A,

L],

L2

^{such that}

^A

^C

L

there existA1,

A2

^{G such thatA]C}

L, A2

^C

L,

and

A

A]

A2.

If the propertyofcountablesubadditivityisreplaced byfinite subadditivityin the definitionofan outermeasurev,weshallsay thatvis af’mitely subadditive outer measure.

We now list some of the less common terminology that we shall userepeatedly throughoutthis work.

DEFINITION 2.3.

An

extended real valued set function

A

defined on a class of sets is superadditive on

,

^ifwhenever

^{E, F} , ^E

^t

^F ,

and

E

CI

F ),

then

A(E F) >_ )(E)+A(F).

If

{ Ei }i

^{C isanyfinitecollection}of pairwise disjointsetsforwhich

E

^and

i=1

i’-I i’-I

REMARKSCONCERNINGFINITELYSUBADDITIVE O MEASURES 655 we shall say that

A

is finitelysuperadditive on

.

^{Ifthis same statementholds for anycountable}

collection ofpairwise disjoint sets in

E

whoseunion alsobelongs to

’,

we saythat

A

is countably superadditiveon

’.

DEFINITION2.4. Let

A

beanextended real valued nonnegativesetfunction definedon aclass of sets

.

^{If for all}

^{E, F}

^6with

^E

^t.J

^F

^and

^{E F ,}

^{theinequality}

A(E

^t.J

F) + A(E F) _ ^{A(E) + A(F)}

holds,weshallsaythat

A

issubmodularon

.

^{If thegeverseinequalityholdsfogany}such pai ofsetsin

,

^{weshall saythat}

^A

is supermodular on

.

^{Ifstrictequality}holds under the same conditions, then

A

is modular on

.

^Obviously,ameasure defined on analgebra

.A

ismodularon

REMARK

2.. Sincesubmodularityon aclass implies finite subadditivityon

,

^{weshall refeto}

asubmodular, finitely subadditiveouter measuremogesimplyas asubmodularoutermeasure.

We closethis section withabriefsurvey of the measurenotationandterminologywe shallbeusing.

For a lattice

c (X),

^we denote by

M()

the collection ofall finite, nonnegative, finitely additive, nontrivial measures on

4().

^Thesubsetconsistingof all the0-1valued measures in

M()

^will

bedenotedby

I ().

DEFINITION 2.6. (a) A measure p

M()

is E-regular if ^fog all

A sup{(L) A

D

L }.

(b)

A

^measugep

M()

^is-smoothon

,

^iffogallsequences

{L,},__

then_#

(L)

^---,0.

() A

^measure#

M()

^isor-smooth on

.A(),

^iffo allsequences

{A,}__

C

t()

^fowhich

A. l ,

then

#(A,)

0. (Thisisequivalenttosaying

that/

^iscountably additive.)

(d) Ameasure/

M()

isstrongly-smooth on

,

iffoallsequences

{L},__

C fogwhich

( )

L,

and_n=l

L, e , then/

_n--1

L --"mf/(L).

Note:

An

alternativecharacterizationofthispropertyis: p isstronglyor-smoothon

,

^{iffor all}

L’ , _L’, sup/(L’,).

sequencest ,,,=, C forwhich

L, T

^and

^{[J L, e} ’.

^then

,,=1 -=1

Weshallusethe following notationstorefertothesemeasures:

Ma(f.).

^{the subset}ofall/:-regularmeasuresin

Mo().

thesubset of all measuresin

M()

^{whichare;-smoothon}

.

M (),

the subsetof all measuresin

M()

^{whichare,-smoothon}

Mo.().

^{thsubofall mauresin}

M()

^whicharestrongly^;-smoothon

.

M ^(),

^{thesubofall-regular}measures in

M ().

The correspondin ^sub,s of

I()

^{are d,oted} by

I(),Io(),I(),Io.(),

^and

I().

rspectively.

Note:

Ceary, M(Z:) C/o.()

C

Mo().

DEFINITION 2.7. For any lattice/: C

P(X)

and any measure/ 6

M(/:).

^{we definethe set}

function

p’

for all

E

C

X

by

.’(E) f{.(L’)IE c L’,L _{e }.}

DEFINITION2.8. If C

P(X)

^{is alatticeand}# 6

M().

wedefinethesetfunction

#"

^{for all}

EcXby

i--1 i--1

DIgFINITION2.. If12C

"P(X)

^{is a}lattice,

M(),

^and

’

^{is the}

^,

^functionolD,fruition2.7, thenweslllsaytlt islyrglriffor

every/./ ,

656 J.E. KNIGHT

Wedenote thesubsetof allweakly regularmeasures in

M()

^by

Mw().

Finally, the restrictionofasetfunction u toa certainclassofsets willbedenotedby 3. FINITELY SUBADDITIVE OUTER MEASURES

Inthissection weexaminethegeneralconcept ofafinitelysubadditiveoutermeasurealongwith its relatedpropertiesand its associatedfinitelyadditive measures. We beginwith

DEFI/qITION3.1.

An

extended real valued setfunctionudefined on

T’(X)

^{is calleda}finitely subadditiveoutermeasure on

7:’(X),

ifv satisfies thefollowingconditions:

(a) /(0)

0and

z/(E) _>

0for all

E

C

X.

(b)

^If

E, F

C

X

and

E

C

F,

then

v(E) < v(F).

)

(c) Forany finite collection

{E,}in__.l

^C

7:(X),

v

Ei <_ ,v(E).

i=1 i=1

If

v(X) <

^{oo,vis said tobefinite.}

DEFINITION 3.2. Let ^v be afinitely subaddifive outer measure on

T’(X). ^A

^set

^E

^C

^X

^is

t,-measurableiffor every

A

C

X, t,(A) v(A

^f’l

E) + v(A

^f’l

E’).

^Theclassofallu-measurablesets willbedenotedby ,5.

Althoughonecanshow

,S

isanalgebraandthat

v

^isa fmitely additive measure, these results do notdemand thatvbeafinitelysubadditiveoutermeasure.

THEOREM3.3. Let beanonnegafive real valuedsetfunctionon

7(X)

^{such that}

A())

^{0. If}

{E cXIVG cX,(G)=(GE)+(GE’)},

then

,5

is analgebraand

, _,[s

isafinitelyadditivemeasure.

Theorem 3.3 shows how toconstruct a finitely additive measure from a finitely subadditiveouter measure. The constructionofsuch anoutermeasure wouldnaturallyseemtobethenextproblem.

DEFINITION3.4. Let CC

T’(X)

be nonempty. We saythatCisacoveringclass if)_ECand for every

A

C

X,

thereis a finite collection

{ Ei }i"__1

^{CCsuch that}

A

C

i=1

As inthe standard theory ofoutermeasures, we cannow construct a finitely subadditive ^outer measure.

THEOREM 3.5. For any covering class

C

C

7(X)

and any finite, nonnegative set function definedonC suchthat_#

()

^{0, thesetfunction}

^A

^{definedfor each}

^A

^C

^X

^by

i=1 i---1

isafinite, finitelysubadditive outermeasure on

(X).

By

imposing^{certainconditionson}

C,

we canimprove uponTheorem3.5.

THEOREM 3.6. Let

C,/,

and

,

be defined as in Theorem 3.5, andsupposet,is asetfunction definedfor each

A c X

by

(a) If is closed under finite unions and / is finitely subadditive on

C,

then

A

v.

If/

is monotone, then

, _extends/

toa finite, finitely subadditiveoutermeasure on

7(X).

(b)

IfCis alattice

and/

issubmodularon

C,

then

A

visasubmodularoutermeasure on

7(X).

PROOF.

(a) C

^is closed under finite unions, so forevery

A

C

X,

thereis a

B C

such that

A

C

B.

Now

A

CBtgU...U$, sobythe definitionofA,

A(A) _</(B)-t-/(i)-t--..-I-/())---/(B).

Thus,

A(A) _< inf{/(B) A

C

B e t:} v(A). (3.1)

REMARKS CONCERNINOFINITELY SUBADDITIVE OLrIER MEASURES 657

Onthe otherhand,for eachfinite collection

{Ei }i"=l

^C(forwhich

A

C

t3

^Ei,

^{[3 E,}

^E

^C,

^sothe

definitionofvand the subadditivity ofpimply that ^{,-1 ,--1}

It followsfrom

(3.1)

^and

(3.2)

^that

A

von

’(X).

Let

A

EC. If_#ismonotone,then

#(A) < _{inf{(B) A}

_C

_B C} A(A),

so_#

<

AonC. Thereverseinequality isclear,so

A

_#onC.

Co) Suppose

A1,

A2

C

X

and let

>

0 be given. Cisclosed under finite unions, so thereexist B,ECsuch that

A,

CBi, 1,2, andwemaychoose the

B,

so thatp(Bi)

< v(Ai) +/2,

1,2.

Furthermore,

B1

^LJB2,

B1 n B2

^E

^C,

^sothe defirfitionofvandthesubmodularity

of/

^on

^C

^{imply that}

v(A1

^U

A2) + v(A1

^f3

A2) <_/(B1

^t3

B2) + ^p(B1

^f3

< (A) + (A) + .

Thus,v issubmodular. Since

pan (a)

implies

A

v,thedesired conclusionfollows.

REMARK. Let/I

and/2

be Inite, nonnegativesetfunctions, each definedona covering class CC

7(X),

vanishingat

,

and yielding, accordingtoTheorem3.5,finitely subadditiveoutermeasures

1

and A2, respectively. If

I 2

^{on{,then ),I}

A2

^on

7(X).

Sincethe concept ofregularity for finitely subadditive outer measureswill be important in later sectionsof thispaper,welist hereforconveniencethe following theorems and definitions.

DEFINITION3.7. Ifvis afinitely subadditiveoutermeasure,

A c X,

and

E

/q,weshall say that

E

is a measurable cover for

A

if

A

C

E

and

(E) v(A),

where

vlsv.

If there is a measurable cover for every

A

C

X,

weshall say thatvisregular.

THEOREM3.8. Ifvisafinite,regular, finitelysubadditiveoutermeasure,then

E ,S

if andonly if

v(X) v(E) + v(E’) (see [6]).

Withthe machinerywehave nowsetup,wecan easilyconstructfinitely subadditiveoutermeasures and their associated finitely additive measures in any space. We specify C and p and the finitely subadditive outermeasure andits associated finitelyadditive measure are automatically defined. To determine the properties of these set functions, we need only examine the properties of C and _#.

However,

we can learn substantially more about finitely subadditive

outer

measures in general by examining the related conceptofinnermeasure.

4. INNER MEASURES

Weturnnow to adiscussion of thegeneralnotionofaninnermeasure definedon

7(X)

^{and the}

relationship ofitspropertiestothoseof a finitely subadditiveoutermeasure.

DEFINITION4.1.

An

extended real valuedsetfunction pdefinedon

7(X)

is an innermeasureif it satisfiesthe following properties:

(a) p(O)

0.

(b)

Forall

E

C

X, p(E) >

O.

(c)

pis monotone.

(d)

^{p is}countably superadditiveon

’(X).

We maystatetheproblem ofinterestasfollows: if weuse a finite, finitely subadditiveoutermeasure von

7(X)

^todefinea newsetfunction p on

7;’(X)

byp(E)

v(X) v(E’)

^{for all}

E

C

X,

then when ispan innermeasure? Underthestated conditions, p^wigpossessthefirstthree properties ofaninner

658 L E. KNIGHT

measure, but to assure countable superadditivity, v must possess additional properties, as the next theorem shows.

THEOREM4.2. If

,

is a finite, finitelysubadditive outermeasure,and p is defined on

’(X)

by

p(E) u(X) u(E’)

^{for all}

E

C

X,

then:

(a) p()

^0.

(b) Forall

E

C

X,

0

_< p(E) <

^oo.

(c) p ismonotone.

(d)

^p

_

^uon

^7(X).

(e)

IfE6

S,

then

p(E)

(f)

If

,

_issubmodular,then p isasupermodularinnermeasure.

PROOF. Theproofsof parts

(a)

through

(e)

^arenotdifficult, soforthe sake ofbrevity,we omit them hereandproveonly part(f).

vissubmodular,so if

E, F

C

X,

then

u(E’ u

By

the definition of p and the finitenessofp andv,statement

(4.1)

implies

(X) p(E F) + (X) p(E

U

F) < v(X) p(E) + (X) p(F).

Hence, p(E

U

F) + ^p(E

^f3

^{F) >_ p(E)} + p(F),

so p issupermodular.

If

E

63

F

),thenp issuperadditiveondisjointsetsandbyinductionp isfinitely superadditive also.

Nowlet

{E,},=I

^C

7:’(X)

^beacountablecollectionofpairwise disjointsets. Sincepismonotone, then for all n,

i= i= i=1

Lettingn oo,wehave

p

Ei >_

p(Ei) p(E).

=1 =1

Therefore,p isaninnermeasure.

REMARK

4.3. When v is a finite, submodular outer measure, the setfunction p defined in Theorem4.2will becalled theinnermeasuredeterminedbyv.

Wenotealso thatbyan argument similartothe oneusedtoshowthat p issuperrnodularwhen^vis submodular in Theorem

4.2(0,

wecan also showthat^t,issubmodularwhenever pissupermodular.

DEFINITION 4.4. If

A

is an inner measure on

7:’(X),

we shall say that a set

E

C

X

is A-measurable iffor every

A

C

X,

wehave

(A) (A n E) + (A n E’).

Wedenote the classof allA-measurablesetsin

7(X)

^by

,.

Thenexttheorem follows immediately from Theorem 3.3.

THEOREM 4.5. If

A

is a finiteinnermeasureon

(X),

then,54 isanalgebraand

AffiAls

^isa

finitely additive measure on

q.

Wenowhavetheimportant

THEOREM

4.6. If

,

isa finite,submodularouter measureandpistheinnermeasure determined by v,then,S_a

{E

C

X Ip(E) v(E)}.

PROOF.

Let^,S

{E

C

XIp(E) u(E)}

andchoose

E e

8_a. Letting

A X

in Definition 4.4, wehave

p(X) p(E) + ^p(E’).

^Since

^X

⁶,$,, Theorem

4.2(e)

impliesthat

p(X) v(X),

^andbythe

definition of p,

p(E’) ,(X) ,(E).

^Sinceallquantitiesare finite,

REMARKS CONCERNINGFINITELYSUBADDITIVE OUTER MEASURES 659

p(E)

p(X)

p(E’) v(X) (v(X) v(E) v(E).

Thus,

E

E

,.q,

so

,S,

C,S.

To show the converse, let

E e

8. Clearly

p(E’)= v(X)- v(E)= p(X)- p(E),

so by the definitionofp,

p(X) p(E) + p(E’). (4.2)

Sincepissupermodular,then for any

A

C

X,

(A E) + (A E) > (A) + (E), /,(A E’) +

p(A

E’) >/,(A) +/,(E’).

Adding and applying

(4.2)

gives

p(A

13

E) +

p(ACt

E’) + p(A

t.J

E) +

^p(AU

E’) _>

2p(A)

+ p(X).

Again, because pissupermodular,weobtain

p(A

E) + ^p(A

^U

^{E’) <_ p([A}

^U

E]

^U

[A

^U

E’]) + ^p([A

^U

E]

^CI

[A

^U

E’])

p(X)

+/,(A).

Nowstatements

(4.4)

^and(4.5)imply

p(A^Ct

E) + ^p(A

¹³

^E’) + ^p(X) + ^{p(A) >_}

^2p(A)

+ ^p(X).

(4.3)

(4.4)

(4.5)

Hence, E e %,

^{and,therefore,}

,Sp

^C3. Combiningthisresultwith the reverse inclusionshown above, wehave

8.

COROLLARY4.7. Under thehypothesesofTheorem4.6,

,S c 3.

PROOF. ImmediatebyTheorems 4.2 and4.6.

DEFINITION ^4.$. Ifa finite, finitely subadditive outer measure v satisfies the condition that

E ,.q

ifandonlyif

v(X) v(E) + ^v(E’),

^{weshallsaythatvsatisfiescondition}

^(M).

Fromthis we obtaindirectly the important

THEOREM4.9. Ifv isa finite, submodularoutermeasure on

(X),

and pisthefinite inner measure determinedby v,thenvsatisfies condition

(M)

^ifandonly

ifS =S, {E

^C

XIp(E)=v(E)}.

PROOF. SincebyTheorem 4.6 andCorollary4.7wealwayshave

,S

C

,S,,

^{we needonlyshow}

the conclusionholds for

,Sp

^C

Assume that v satisfies condition (M) and let EE

8,.

^{By the definition of p, we have}

v(E) p(E) v(X) v(E’),

and sincev isfinite, we obtain

v(X) v(E) + v(E’).

The hypothesis now implies that

E

6

,Su,

sothat

,S,

^C

Onthe otherhand, suppose

,.q,

^C

,Su.

Clearly, if

E

6

,Su,

then

v(X) v(E) + v(E’),

soit will suffice to prove the converse implication. Assuming this latter equality holds for any

E

C

X,

the finitenessofvand the definitionof p imply thatp(E)

v(X) v(E’) v(E). Hence, E

E

,S,,

^soby

hypothesis

E

%,andconsequentlyvsatisfies condition

(M).

Itnow seems naturaltoinquireastowhenafinitelysubadditive outer measureissubmodular. One condition that ensures thisisgivenby

THEOREM4.10. If isa finite,regular,finitely subadditiveoutermeasure, thenvissubmodular.

PROOF. Clar.

Weconclude this sectionwith adirectresult ofthis theorem and Theorem4.2.

Thus, for all

A c X, p(A13E)+p(A13E’)>_ p(A).

Combining this with the fact that p is superadditive,wesee thatfor all

A

C

X,

,(A) (A E) + (A E’).

660

COROLLARY4.11. Let

,

_beafinite, finitely subadditiveoutermeasureand p thesetfunction definedfor all

E

C

X

by

p(E) (X) (E’).

^Ifuisregular,then p is aninnermeasure.

5. AN APPLICATION TO

THE

SETFUNCfIONS

Beginningwith afinitely subadditiveoutermeasure andthe collection

,S

ofu-measurablesets,we candefine two new setfunctions po and

o

^on

o(X).

Weshall then show howtoderive theirbasic properties using thegeneraltheory ofSections 3and4. Wethenfollow thiswithafull discussion of the interconnectionsbetween theproperties of

’

^{and ’o andalso oftheir}relationshipto theoriginalouter measure

,.

Throughoutthissection,

,

willalwaysdenote a finite, finitelysubadditive outermeasure on

(X)

and

,.q,

thesetof all,-measurablesets.

DEFINITION5.1. Forall

E

C

X,

wedefinethesetfunctions,o_{and’oasfollows:}

vo(E) sup{r,(M) E

TmOmM S..

(a)

For evy

(b) ^v isafinite, submodularouter measure on

)(X),

and if

E

68,then

v(E) r,(E).

()))o

is afinite,supermodularinnermeasure on

(X).

^Forall

^E

^C

^X, o() (x)- (’).

I)ROOF. (a) Clear.

(b)

8

is a latticeand

))]s

^{issubmodularon8,sobyTheorem} 3.6(b), is asubmodular outermeasure on

(X).

^SincebyTheorem

3.6(a), vJ8

^{r,, then}

^{v(E) v(E)}

^{for any}

^E

^(f

^8.

Clearly,v isfinite.

()

^Since

u[8,

^isafinitelyadditivemeasure on8,thenfor

E

C

X,

Definition5. implies

supI(x)

v(x)

v(x) v(’).

Part (b)above and Theorem

4.2(0

nowimply that),ois asupermodularinner measure. Clearly,Vo isalso finite.

REMARK

^5.;. Wedefineav=measurablesetrdin8^toDefinition 3.2 anddenotethe collection of all thesesetsby 8,0.

By

Theorem 3.3,8,0isanalgebraandiv’ v

]s,

i a finitely additive measure.

Similarly,we definea

vo=meamwable

^{setaccordingtoDefinition4.4}and denote the class of all suchsets by

8o.

^Thisleads directly^to

THEOREM5.4. (a)

8o

^{is analgebraand}

o VolS,,

^{is afinitely}additive measure on

8.

(b) E

6

8

^ifandonly

^ifvo(A) _< r,o(ACE)+vo(AnE’),forallA

C

X.

(d)

(e) I a 8 (or 8,),

^thn

o() v()

(f)

v satisfies condition

(M)

if andonlyifS,

8o.

I)ROOF.

(a)

Clear, byTheorem 4.5.

(b)

This followsby the superadditivityof

o.

() An

^immediateconsequence of Theorems^4.6and

5.2(b,c).

(d) Let E

6

8

^andchooseany

^A

^C

^X.

^Givena

>

0, there exists an

M

6

8

^suchthat

^A c M

and

v(M) < v(A) +

^{a. Since}

^A

implies

REILKS CONCERNINGFINITELYSUBADDITIVE OUTERMEASURES 661

m(A) + > re(M) m(M n E) + ,( n E’)

> _,o4 n E) + ,(A n E’),

andtherefore

(A) _> t(A n E) + t(A n _E’).

Hence

E

E80, and thus

8 ^c 80.

Corollary⁴7 gives,S,,oC

,S,,o.

(e)

^Aclear consequence of parts

(c)

^and

(d)

above and Theorem

5.2(a).

(f)

Follows from Theorems4.9and

5.2Co,c).

The following inequalitiesare notdifficulttoshow and arefrequentlyuseful.

LEMMA5.5. If

E

and

F

aredisjointsetsin

(X),

then

to(E) + to(F) < to(E

⁰

F) < to(E) + t(F) < t(E

^U

F) < t(E) + t(F).

With thislemmawe canshow

THEOREMS.6.

(a)

^If

E

E8,then for all

A

C

X, to(A n E) + (A’ n E) t(E).

Co)

^fig

c E

(

S,,,

then

to(G) t(E)- f(E- C).

The following important characterization theorem reveals therelationshipsbetween

,5,

8o, and whentsatisfies condition(M).

THEOREM5.7. Iftsatisfies condition(M),then:

(a) Forany

E

C

X,

if

to(E) t(E),

^then

E

(b)

E ,q

ifandonlyif

to(E) f(E).

(c)

PROOF.

(a) Suppose to(E) t(E)

for some

E

CX.

By

Theorem

5.2(a,c),

Buttis subadditive, so

t(X) t(E) + t(Et),

^{and sincet}satisfies condition

(M), E Co)

Clear.

(c) By

^Theorem5.4(c),^if

E

E

8o,

^then

to(E) ’(E). By

part(b) above,

E

,5so

80

^C

8.

Theorem5.4(d)now implies,5

,q,

REMARK

5.8.

By

Theorems

5.7(c)

and

5.4(f),

^t satisfies condition

(M)

whenever t does

However,

weshall soon obtainafar stronger result.

THEOREM5.9. tosatisfies condition

(M).

PROOF. ByDefinition 4.4, ff

E 0,

^then

to(X) to(E) + to(E’).

To show the converse, suppose

to(X) to(E) + to(E’)

for some

E

CX. Since t isfinite, Theorem

5.2(c)

implies

to(E’) + ’(E) t(X).

^Therefore,

,o(E’) + ,(E) ,(x) ,o(x) ,o(E) + ,o(E’).

Allterms arefinite,so

v(E) to(E). By

Theorem5.4(c),^itfollowsthat

E e ,o.

Wenowsee that

E ,5

ifandonlyif

to(X) to(E) + to(E’),

^sotosatisfies condition

(1.

We now wish to examine the concept ofregularityfor finitely subadditive outermeasures. To facilitatethisdiscussion,wedefine aclosely associated property.

DEFINITION 5.10. A finite, finitely subadditive outer measure t on

P(X)

will be called approximately regular ffThe concepts of regular^t and

’

^onapproximately regular

^P(X).

coincide in the standardtheory of countably subaddifive outer measures, but this neednot be the case in the theory offinitely subadditive outer measures. Nevertheless, a finite, finitely subadditive outer measure which is regular will also be approximately regular. This latter conceptprovidesuswith aconditionontweaker than full regularity butwhichstillguarantees that^tsatisfiescondition

THEOREM5.11. fftisapproximately regular,thentsatisfies condition(lVl).

PROOF. Since itis clear that

t(X) t(E) + ^t(E’)

^if

^E

^E

^,5,

^{weneedonlyshow theconverse.}

662 J.E. KNIGHT

Suppose t(X) t(E) + t(E’)

^{for some}

E

CX. If

A

68,thenusing

E

and

E’

astest sets in Definition 3.2, we have

t(E) t(E A)

^/

t(E A’), ,(E’) (E’ n A) + t(E’ n i’).

Adding theseequationsand using the subadditivityoft,we obtain

,(x) [(E n A) + (E’ n A)] + [,(E n A’) + ,(E’ n A’)]

> (A) + (A’) (X).

Hence,

,(E n A) + ,,(E’ n A) + ,(E n A’) + ^{,,(E’ n} A’) t(A)+t(A’).

Since allquantities are finite, we can subtract theinequality

t(A’) < t(E n A’) + t(E’

^Iq

A’)

^from

(5.1)

to obtain

t(A) >_ t(E

^f3

A) + t(E’

^f’l

A).

Sincetis subadditive andt v

,

^weactuallyhave

,,(A) ,(A n E) + ^{,,(an E’).}

Wehave shown that thisequalityholdsforall

A

8,but we can readily show thatitalsoholds for all

A

CX. Thus,

E

E

80

^{8,and theproofiscomplete.}

In

anefforttoshedmorelighton thesetfunctionv

,

we make thefollowingdefinition.

DEFINITION5.12. Iftisafinite,finitelysubadditiveoutermeasure on

P(X),

^{then we}candefine a new set function^z/’for all

E

C

X

by

t(E) inf{t(M) E c M

6

Recalling that whent isafmite, finitelysubadditive outermeasure, thenv is afinite, submodular outermeasure and

8f

^{isanalgebra,wecan}readilyapplythegeneral theoryof Sections3and4 tothis definition toestablish thenextlemma.

LEMMA

5.13. Thesetfunction isafmite, submodularoutermeasure on

7(X)

^andpossesses the following properties:

(a)

^IfE 2,,0,then

t’(E) (E).

(b)

(x) v(x) (x) <

(c)

For every

E

C

X, v(E) _< too(E).

(d) 8

^isanalgebraandYoo

vools

^{is afinitelyadditivemeasure on}

^8.

Thisnewsetfunctionrevealsimportantnewinformation about theoutermeasuret andenables us tostrengthensignificantly the conclusions of Theorem

5.4(d,f).

THEOREM5.14.

(a)

^v isapproximately regular.

(b) satisfiescondition(M).

()

S. 8o.

PROOF.

(a)

For any

A

C

X,

Theorem5.4(d,e)implies

(A) inf{(M)IA c ^M e _< f{(M) A

C

M e

=(A).

ItfollowsimmediatelyfromLemma

5.13()

^that

t(A) (A)

forall

A

C

X,

sov isapproximately regular.

(b)

Clear, byTheorem 5.11 and part

(a).

(c) A

^directconsequence of pa.et

(b)

andTheorem

5.4(0.

REMARKSCONCERNINGFINITELYSUBADDITIVEOUTER MEASURES 663 6. FURTHER EXAMPLES AND APPLICATIONS

Inthis section weshow the efficiencyofthegeneral theorydiscussedabovein Sections3, 4, and5 whenappliedtosome familiarlatticerelatedsetfunctions. Weshall soonseethat thenatureand special properties of thesesetfunctions arerevealed almost instantly. No longerwill weneedto examine each function individually to ascertain its nature or to labor tediously in an effort to uncover its special properties. Mostof these facts will be virtually self-evident once the function is defined.

Throughoutthis section,

X

willdenoteanarbitraryset and a latticeof subsets ofX. Weshall alwaysassumethat 0,

X

6

.

^{All othernotationand}terminologyinthissectionwillconformtothat introducedinSection 2. Beginningwiththesetfunction

#’

^{definedinthatsection wehave}

THEOREM 6.1. Theset

function/’

asgivenbyDefinition 2.7 isasubmodularouter measure possessing the following properties:

(a) ,’(x) <

(%) /’

=/on

’.

(c) < #’

^on

A().

PROOF.

’

^{is a lattice, so it is a covering class which is} closed under finite unions. Since /

e M(),

it is finiteand nonnegative on

’,

and

p(0)

0. Furthermore,/isnecessarily submodular (hence,finitelysubadditive)on

’,

andmonotone. Consequently,Theorems3.5and3.6imply that

#’

^{is a}

submodularoutermeasure on

7;’(X)

^such

that/’(X) <

^oo

and/’

#on

’. Property (c)

follows easily as aconsequenceof the definition

of/’

and the monotonicity of p.

REMARK

6.2. Obviously,

’ ^#’ls

^isafinitely additive measure on

,S,,,

^the

#’-measurable

^sets.

Inthespecialcaseforp

e I(), p’

isalsoregularandtherefore satisfiescondition

(M).

Weturnour attentionnow tothe familiarset

function/

toseehow thetheoryrevealsnotonlyits familiarcharacteristics, but alsoitsstrong connectionto

#’.

DEFINITION6.3. Let C

7(X)

^{be alatticeand}p6

M().

Forall

E

C

X,

wedefinetheset function_#on

7(X)

^by

(E) ^{sup{,(L) L}

⁽²

^{E, L} e }.

Fromthisdefinitionand Theorems 6.1and 4.2,wesee

that/i

^{is theinnermeasuredeterminedby/’.}

Furthermore, Theorem4.2 also revealsthat/i issupermodular, and if

E ,.q, then/i(E) I’(E).

CombiningthisresultwithTheorem4.6andCorollary4.7shows that

S,C %, {E ^c XII(E)=/’(E)}.

Thisresultopensthedoortoan even stronger and more interestingresult,namely THEOREM6.4.

,,, ^{{ E}

^C

^X [/(E) =/’ (E) }.

PROOF.

In

viewof thecommentsprecedingthistheorem,weneedonlyshow that

,S,,

^C

.,,.

Let

E

6

,S,,.

^Given

>

0, there exists^an

L

6 such that

L

C

E

and

.() < .(L) + /2.

(6.)

Also,there is an 6 such that

E

C and

/(’) ^{</’(E) +/2. (6.2)}

Combining

(6.1)

^and

(6.2)

^withthe fact that

E 6,9,,,

^{weseethatthereexist}

^{L e}

^and

’

⁶

^’

^such

that

L

C

E

C and

< ,’() + / ,() + / < ,() + .

^(.

Toshow that

E

is

#-measurable,

^let

W .

^Then

^A

^Cl

^E’

⁽²

^{W Cl/./,}

^sobythe monotoNcity of and Theorem 6.

l(b),

^wehave

66 J.E.

Ontheother hand,

A’

N

E

C

A’

f’l

,’,

so

Now/

^ismodular

on/:’,

so

(6.6) Combiningthisresultwith(6.3)and the monotonicity

of/z

yields

(A’ L) +, (6.7)

where the equality followsbymodularity. Itnowfollowsfrom (6.4),

(6.5),

(6.7),^Theorem

6.103),

and thefiniteadditivitiy of that

nm)+ i/(A’ nm’) _< (n’ nL’) ⁺

#’(A’ NL’)

_</(A’ n L) + (A’ n L’) +

(’ +

=/(A’) + .

Clearlythisholds for all

>

0, so

’(A’ E) + ,’(A’ E’) < ’(A’),

for any

A’

E

:’.

Itnowfollows easily that(6.8)holdsfor any

A

C

X,

andtherefore

E

E

8,.

This leadsimmediatelytotwoimportantresults concerning COROLLARY6.5.

If/z

6

M(.),

^then

Iz’

satisfies condition

(M).

PROOF. Thisfollows immediately byTheorems 6.4 and 4.9.

COMMENT. Wesee now

that/z’

^{is an}important exampleof a finitely subadditiveoutermeasure whichsatisfies condition

(M)

butis notnecessarilyregular.

THEOREM6.6. Ifp

M(),

^then

8, n {L e I/(L) (L)}.

PROOF. Clear from Theorem 6.4.

REMARK

6.7. Somesimple but importantconsequences ofthis theorem are:

(a)

p

MR()

^if

andonly

if/z =/z’

^on

,

^and

Co)

p

e M(/:)

ifandonly

if,4(/:)

C

8,.

Thislatter resultisthekeythatfinallyreveals conditions under

which/z’

isregular: namely,

/

^is regular irE’is6 andp

Weturnourattention now to thewell knowncountablysubadditive outer

measure/z"

defined earlier in Section 2. Some important facts to recall about

/’

^are

that/" <_

p on

’,

p

_</’

on

:,

and

#"(X) Iz(X) <

oo,

where/z Ma(:).

Also,

,5,

^isaa-algebra

and/"[s,

^{is acountablyadditive}

measure. Inthe special case forp

I,,(:),/’

^isalways regular.

While we require that /z

Mo()

in order to ensure that

/z"

^{is not trivial, we can improve} significantly uponthe above-mentioned results for

#"

^byimposing a stronger condition on p,namelythat

/

e Mo,(E).

THEOREM^6.$. If_#

e Mo, (),

then

#" #’

^=/zon

^’.

PROOF. Fromthe introductory remarks above

about/z"

^{andbyTheorem6.1,wealwayshave}

"<=/

^on

’. (6.9)

RMARKSCONCERNINGFINITELYSUBADDITIVE OUTER MEASURES 665

Toshow thereverseinequality,let

L’

6

’,

andchoose anye

>

^0.

By

the definition of

#",

^there existsasequence

{L, }

n--1

c

suchthat

L’

C

[3 L:

^and

E#(L:) ^{< #"(L’)} +

^e.

(6.10)

Itisclear that

L’ L’

N

L’ ^{[.J (L’}

^CI

^L,),

^{and ifN}is a positiveinteger,then

n=l n=l

U

N

^(L’

^f

^{L’) T} U ^{(L’ L) L’}

⁶

^{’. (6.1}

^l)

n=l n=l

The monotonicity and the finitesubadditivity of#imply that,forany

N,

.

_n=l

^{(L’ L’) <}

_n=l

^{.(L’ L’) _< .(L’)}

_n=l

<_ _.,(L) ^{< ,"(L’) +.}

n=l

Consequently,applying the hypothesisto

(6.11)

and

(6.12)

yields

< "(L’) + .

(6.12)

Since wasarbitrary,thisimpliesthat

#(L’) _< #"(L’),

^andconsequentlythat_#

<_ #"

^on

^’.

^Combining

thiswith

(6.9)

completes theproof.

Animmediateconsequenceof this resultis

COROLLARY6.9. If p6

M (),

then

#" p’

on

’.

PROOF. Clear,since

M()

C

Mo,(,).

If_#

Mo ()

^and

#"

^isregular,then the converse of Theorem 6.8holds.

THEOREM6.10. Let_{# 6}

Mo(). If#"

^isregularand

#" #’

#on

’,

then_{# 6}

Mo,().

L’ ’

PROOF. Let

{

,,,=1C beany sequencesuch that

L: T

^and

[3 L’ ^{e ’.}

^Let

^{L’ [J} L’.

n=l n=l

Then

L ^{T L’}

⁶

^{’. By}

hypothesis,

"

^isru|afand

"

^#on

^’,

^soitfollows that

(L’) "(L’)

" ^L’ n"(L’.) ,m ,(L’.).

Hence,

Thismeansthat#

e Mo. ().

Since

#"

^isalways regular for_{# 6}

Io.(),

the following corollaryisan immediateconsequenceof Theorems6.8and6.10.

COROLLARY6.11.

If#

⁶

Io(),

then_#

I,,. ()

^ifandonly

if#" #’

#

on’.

REMARK. Beforeweconclude our discussion of the outer measure

#",

wenotetwoadditional results ofinterestconcerning regularity for

#".

^{First,in a}spirit paralleling Theorem 6.10,

if# Mo()

and

#"

^isregularwith

#"

#on

,

^then#

M().

Onthe otherhand,if_#

M,(),

^then

#"

^is

regularand submodular.

(6.13)

666 J.E.

(E) i{p(L’) E

C

L’, L e

on

.

^{Ifwedee a}

^-mle

^inthe

^u

^way,

^en

^is

^gm

^d

^[&

^{is a}

ad&five me. Uoately, p is me, isnot

qte

outerme, lacg

oy

the

pro

ofte

badditi. s

defeis

rem,

how, ffis

no.

OM.15. If is a

no

la d p6

M(), :

(a) p is tely mbadditiveon

’.

)

^{is atelymbadditiveoutermre on}

P(X).

(c) c A() c &.

(d)

= ^{p’ on.}

(e)

ismbmodul.

PROOF.

(a) L

L1, de

>

⁰

v. ^By

the deflation of

, e sts

a

s A ch

^at

^A

^C

L

^U

L

^d

,(L Q) < ,(A)+ . ^(.14)

Sce

E

is

no, e st A ^E,

^{k 1,2,inch at}

A

^C

L

^d

^{A A1 UA2.}

^The

mb&fi of p

(A) S (A) + (A2). (.l)

Kg t

piis monotonedpi p on

,

^itfoHows

^{om (6.14)}

^d

^(6.15)

^t

,,(L L) < ,(A) +,(A) + S ,(L) +,(Q) + .

Ts

holds forl

>

0, we

,,(L L) S ,,(L) + ,,(L).

Fte mb&fiW

^{now foHowsbyduon.}

)

^S

E’

is a

covg

clclosedunderfite

om

dpiis m,

p (a)

ove

d

r 3.a)

ply

t

is a

e, e

mbad&fiveoutme on

P(X).

(c)

LdA,dcomidyE,FEmchtECA

L dFCA E .

From

s

it isclat

F E

d

E

U

F c A .

^{Now on}

^,

^soby

^{e ad&fi}

^{of on}

d

e

dfion

ofi

^we

(A’) (A’) (E F) (E) +,(f).

Thechoiceof

F

w i,mitfoHows

tt

(A’) (E) + p{(F) F

^C

A’ E’,

F6

} .(E) +.(A’ E’)

.(E) + (A’ ).

Clly

L

C

E’, A’ n L

C

A’ E’,

d

moaotoid of

^pfies

^at (A’) a .(E) + (A’ n).

iy,

sce E

w

i

^choir,

^(6.16)

(6.16)

REMARKS CONCERNINGFINITELY SUBADDITIVE OUTER MEASURES 667

(A’) > sup{p(E) E

C

A’

A

L’, E

6

} + (A’ n L) ,,(A’ L’) + (A’ L)

(A’ L’) + (A’ L).

This statement holds for all

A’

6

’,

but it is not hard to show that it also holds for all

A

CX.

Consequently,

L

6

,St,

^{whichmeansthat}

(d)

^Let

L .

^Thenbypart

^{(c), L ,}

^{and since}

^S

^{is analgebra,wealso have}

^{L’ ,.q.}

^Since

l&

^{is a finitely additive measure, it} follows that

(X)= (L)+ (L’),

^and from Theorem 4.2,

/(X) =/,(L’) +/’(L).

Consequently,

(z) + (z’) ,(z’) + ^,’().

Since _#ion

’

and all quantifies are finite,weconclude that

(L) =/’(L).

Thus,

=/’

on

.

(e) By

^part

(c),

wehave

’

^C

^,9,

^therefore

^[s

^satisfiesthe modularlaw on

’.

Fromthe remarks preceding Theorem 6.13, =/ion

’,

so_#iisalso modularand, hence,submodular on

’.

Theorem 3.6(b)nowimpliesthat issubmodularon

’(X).

Thesecondnew setfunctionwewish todefineandexamineinthe light of the general principlesset forthin Sections 3and4isthesetfunction

,

^whichhasastrong relationshipto

.

DEFINrHON 6.14.

If/ M() and/’

^{is the} outer measure ofDefinition 2.7, then for all

E

C

X,

we definetheset

function/

^on

’(X)

by

/(E) ^sup{/’(L) E

D

L

6

From this we can readily show that if is a lattice and _#

e M(),

then for all

E

C

X,

#j(E)+-(E’)=I(X).

Furthermore,/j^isfinite,nonnegative,

monotone,/(8)

=0,

and/j(X) =/(X).

Weshalldefinea/j-measurable^setinthestandard manner and denote the class of

all/-mle

^sets

by

,5,,.

^Asonemightexpect, Theorem3.3 assuresusthat

,5,,

^{isanalgebraand}

that/jl&,

^isafinitely

additivemeasure. Therelationship

of/

^{totheothersetfunctions}we have discussedissummarized in THEOREM6.15. If_#

e M(),

then:

(a) < <

’

^on

^’(X).

(b)

< < ,’

on

.

(c) , ^{g <} , ^{< ;,’}

^on

^’.

(a)

^,f

e &, men ,(E) ().

Theproofsof thesestatementsarestraightforwardandwillbe omittedhere.

Justas fails to be anoutermeasureunless isnormal,_so/

similarl);

failstobe an inner measure without thissame additional condition on

. ^However,

^{if is}normal, thennotonlydoes become countably superadditive and, hence,aninnermeasure,but otheruseful informationisrevealed also.

THEOREM6.16. If is anormal lattice

and/

⁶

M(),

then:

(a)

/jisasupermodularinnermeasure on

’(X).

0,) & ^{{ c x ,() g()}.}

PROOF.

(a) By

Theorem 6.13, is a submodular outer measure, so from the observations followingDefinition 6.14and the finiteness of

B,

itfollows

that/(E) =/(X)- B(E’) -(X)-’(E’),

for all

E

C

X.

Theorem4.2 nowimplies

that/j

^isaninnermeasure,and since it isnothardtoshow that isalsosupermodular,theproofiscomplete.

(b)

^ImmediatebyTheorem 4.6.

(c)

A consequenceof Theorem6.13andCorollary4.7.

In

additionto these results, we can also improve upon the conclusion of Theorem 6.15 if is normal.

668

.

^{E. KNIGHT}

THEOREM6.17. If/;isanormallattice

and/ M(),

then:

(a) < <_ _<

on

’(X).

(b)

p=/i

_</ p’

on

.

(c) p----pj----_</’----pon

AsinTheorem 6.15, the proofs of thesestatementsare straightforward and will b omitted.

Because weaklyregularmeasuresaredefined in termsof the values of theoutermeasure

p

onthe lattice

,

^wherep

M()

(seeSection2),it is nowpossibletosimplifTthestudy of these measures by treating them asanaturalapplicationofthesetfunction_pj.

For

example,wehave

THEOREM6.18. Ifis alattice

and/ M(),

^thenp

Mw()

^ifandonly

if/ ^on’.

PROOF. If# Mw(),

^{thenbythedefinition}

ofj,

^wehaveforall

L’

(L’) ,p{’(2,) L’ ^/, } ^(L’).

Hence,

j p on

Conversely,ifpffi/on

*,

then itfollowsthat for all

L ,

(L’) (L’) ,p(’(/,) L’ ).

Thisimplies thatp

Mw().

Usingthistheorem,itfollowseasily that

MR()

^C

Mw(),

^{butwe canstrengthenthisresult if is}

normal.

THEOREM6.19. If is anormal lattice, then

MR() Mw(f.).

PROOF. Itwill sufficetoshow that

Mw()

C

M(),

solet

Mw(f.)

and

L’ ’.

Clearly,

(L’) + (L) (X) (L’) + (L),

sobyTheorem6. ^$and the finitenessofall terms,wededuce that

(L) p(L). Hence,

p on

.

Since isnormal,Theorem

6.17(b)

impliesthat

#’

^on

,

sop

p’

on

.

^{IecallingRemark}

^6.7(a),

wesee thatp

M().

Asafinal illustrationof how ourgeneral principles maybe appliedtospecificsetfunctions inorder to simplify their study, we applythe results ofSection 5to the well knownset functions

discussed earlier. We shall see that thisapplicationalsorevealsnewandunexpectedinformation about thesetfunctions and

P,ecall from ourearlierresults that

p

is afinite, submodular^outermeasurewhichsatisfiescondition

(M), ,5,,

^isan algebra,and

,.q,, ,S (E

C

XII,(E) l’(E)}. Letg

^u

l’

in Definition5.1, wehave for all

E

C

X,

uo(E) sup{’(M) E

D

M ,S,,}.

The following resultsare immediateconsequences ofthegeneral principlesderived in Section5.

PROPOSITION

6.20.

(a)

^u isafinite, submodularoutermeamre.

(b)

Uo is afinite,supermodularinner meamresuchthat

uo(E) l’(X) u(E’),

^{for all}

^E c X.

(c) or

dl

c x, Uo(E) <_ ,(E) <_ ,’(E) <_ :().

PROOF.

(a)

and (b) are obvious consequences of Theorem

5.2(b,c).

To prove part (c), we note that by Theorems

5.2(a)

and

4.2(d),

^if

E

C

X,

then

uo(E) < p’(E) <_ u(E)

and

p(E) < p’(E) _< u(E). It

remainsonlyto show that

uo(E) <_ #(E).

Consider any

M ,.q,,

^{such that}

^M

^C

^{E. By}

^{Theorem 6.4 and the} monotonicity of

.’(M) ,(M) <_ ,(). Hnc,

o() ,up{.’(M)IE

V

e S } _< .().

Because satisfies condition

(M),

wealso have

REMARKS CONCERNINGFINITELYSUBADDITIV OUTER MEASURES 669

PROPOSITION6.21.

(a)

IfE6

S,,

^then

to(E) p,(E) tz’(E) u(E).

(b) If

ECX

and either

/zi(E)=/z’(E)

^or

Vo(E)=v(E),

^then

ES

u, and

to(E) =/i(E) =/’(E) v(E).

PROOF. Because /’

^{satisfies condition}

(M),

^Theorems

5.7(c)

and 6.4 imply that

,Svo

^,Sm

,S, .

^Inparticular, this means that

{E ^{c XIp,,(E)} ---/’(E)} {E

^C

Xlto(E t(E)},

so

that/i(E) =/’(E)

^ifandonlyif

to(E) v(E).

Clearly then, if

E d,

both equalities hold and by Proposition 6.20,

to(E)=/i(E) /’(E)=v(E).

Conversely, if

/,(E)=/’(E),

^then

E ,.q,,

^so by the above argument,

to(E)

p,i(E)

t’(E) v(E).

^Similarly,if

to(E) v(E),

^thenbyTheorem

5.7(b), E q,,

and then the sameequalitieshold.

NOTE. Similar conclusions holdfor any

E

in

,.q, ,.qo,

or

,.q,,

sincethese collections coincide.

Also,we notethat in additiontothepropertiesdiscussedabove, itfollowsfrom Theorems 5.9 and5.14 thatv isapproximatelyregularand that both to andv satisfycondition

(M).

Because

’-qd ,S,

^wecan give alternate definitions for^v and_towhent

=/’:

t(E) inf{/zi(M) E

C

M to(E) sup{/i(M) E

These alternative definitions lead to some unexpected results about and /j: namely, if

/

MR(l:),

then

=/’

^fand/j=/i to. Consequently,when p

Ms(l:),

then is afinitely subadditiveoutermeasureand_pjisan innermeasure,even if is notnormal.

ACKNOWLEDGEMENT. The authorwishes tothankthe Research ReleasedTime Committeeand theTrusteesof

Long

Island University fortheirgenerouspartial support of this research workthrougha grant of released time from teaching duties.

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