MEASURES ASSOCIATED WITH

VOL. 18 NO. 4 (1995) 725-734

OUTER

MEASURES ASSOCIATED WITH

LA’I-i’ICE

MEASURES AND THEIR

APPLICATION

725

CHARLESTRAINA St.John’sUniversity

Department

ofMathematics

& Computer

Science Jamaica,

NY

11439

(Received March 18, 1994 and in revised form July 7, 1994)

ABSTRACT.

Consider aset

X

anda ^lattice/;of subsets of

X

such that

,X

^E/;. M(/;)denotes thoseboundedfinitelyadditivemeasures

on.(/;)

whichare studied, and

I(/;)

denotes thoseelements of

M(L)

whichare 0-1valued. Associatedwith aI.t _M(/;)or a l.tMo(/;)

(the

elements of

M(/;)

whichareo-smooth

on/;)

are outermeasures

t’

^and

it". ^In

^{termsof theseoutermeasuresvarious} regularityproperties of

Ix

^canbeintroduced,and theinterplay between regularity, smoothness,and measurabilityisinvestigated forboth the 0-1 valued case and the moregeneralcase. Certainresults forthespecialcasecarryoverreadilytothe moregeneralcaseorwithat mostaregularity assumption on

’

or it",whileothersdonot. Also,in thespecial caseof0-1 valued measuresmore refined notionsofregularitycanbe introduced which have no immediateanaloguesin the

general

^case.

KEY

WORDS

AND

PHRASES: Normal lattice, lattice regular measures, associated outer measures.

1991

AMS SUBJECT CLASSIFICATION CODES:

28C15,28A12.

1.

INTRODUCTION

Ourfirst aim in thispaperis

(see

Section

3)

toobtainfurther propertiesoftwo outermeasures

it’

^and

it" (see

^belowfor

definitions)

associated first with_itCI(/;)and_itE

Io()

andsubsequently with _itEM(/;) and _it

EMo(I;),

^andto apply these properties ^to characterize various classes of measures. Intheformercase wetherebyextend results of

[6,7,8].

Also in thecaseof CIo(.r_.)^we consider infurtherdetail thesubset

Is(/;) of/,,(/;)

ofslightlyregularmeasures

(see

below for

definition)

andhence extendthe work of

[6].

We

notethat in the caseof0-1 valuedmeasures such as_la

lo(Z;),

the associatedoutermeasure

it"

^isclearly

regular

and

S,,,

^the

"-measurable

^setscanbe explicitlycharacterized. This is nolonger the case if CMo(/;)andwe musthypothesizeregularity of

If’

^incertaincases in order^togeneralize thetwo-valuedcase. Also ingeneralthe characterization of

S,,

^{is notasexplicitasin}the two-valued case,and this furthercomplicatesthegeneralsituation. If

J(.r.,),

i.e.,isstronglyo-smooth

(see below)

and^if/;isa 5-1attice then

t’

"

and, hence,

S,, S,,,,

^and

S,,

^hasbeen characterizedexplicitly in

[5].

Werecall thisresultin Section 4 and buildonittoextend someofthe results in

[5],

seein particularTheorems 4.2, 4.3, 4.4.

We beginwith a reviewof thenotations

(section 2)

^{whichwillbe used}throughoutthepaper, aswellas areviewof the relevant definitionsneeded. Furtherrelatedmatterscan befoundin

[1,2,3].

2.

BACKGROUND AND NOTATION

We

introducethe necessarymeasure theoretic, and lattice definitions, and notethe known properties aboutlatticemeasures thatweshall need.

The definitions and notationsarestandard andareconsistentwiththosefound in, forexample,

[1,3,9].

^Wecollect theones weneed and some of theirproperties forthe reader’s convenience.

726

Let X

,,

be an abstractset,^.6a latticeof subsetsof

X,

which we will assumethroughout. We shall assume that

, X

E.6. ForE CA", E’denotesitscomplement. Wedenoteby:

(1) .,(.6),

^thealgebra generated by.6;

(2) 6(.6),

the latticeofallcountable intersectionsofsetsfrom.6;

(3)

.6’,thelatticeofcomplements^ofsetsfrom.6.

Weintroducethefollowingmeasure theoretic definitions.

Def": Thesetof allnonnegativefinitevalued, finitelyadditive

(f.a.),

bounded measures on

.(.6)

will bedenotedby

M(.6).

An

element E

M(.6)

^{is said to}be o-smoothon.6iff whenever

L,,

E.6, n 1,2,...,and

L,,

then

(L,,)

^0.

An

element

IX M(.6)

^{issaidtobeo-smoothon}

.(.6)

iffwhenever

A,, A(.6),

n 1,2 and

A,,

O,thenia(A,,) 0.

(Note

^thatthiscondition isequivalenttoI.t being

countably additive.) An

_{element Ix}

M(.6)

^{is said}tobe

strongly

o-smoothon.6iffwhenever

L, L,,

E.f.,, n 1,2 and

L,,

L, L N

L,,,

then

I.t(L inf{l.t(L,,)

n 1, 2

]..

An

element

IXEM(.6)

is said to be .6-regular iff for any

A.(.6),)- sup(l(L) IL ^CA,L

Thefollowingnotation isusedtodenotethe subsets of

M(.6)

determinedbythe aboveproperties:

M,,(.6)

^isthesetofmeasuresthat areo-smoothon

,;

M"(.6)

isthesetofmeasures that areo-smoothon

(.6);

J(.6)

isthesetof measures thatare

strongly

o-smoothon

,;

Ma(.6)

^isthesetof L-regular measures;

M.(.6)

isthe setof

L-regular

measures

of ^M(.6).

Wenotethat

J(.6) CMo(.6),

and

M(.6) NM,,(.6) CM(.6).

We

denote by

I(.6), l,,(z;), (z:), Is(L),

and

I,(L)

the subsets ofthe corresponding

M’s

thatconsist of the non-trivial0-1valuedmeasures.

We

shallwrite

IX v(z:)

^wheneverI.t,v aremeasures,orset functionssuchthat

L) v(L)

forall

L

E

Observe thefollowingenlargement,i.e., witheach

IX EI(z:)

^{there is avE}

la()

s.t.

IX v(z:);

and foreachIX

M(.6)

there isa v

M,(.6)

s.t.IX

v()

and

X) v(X).

For

theseresults and otherrelatedmatterssee

[5,7,8].

If IX

EM(,),

we defineasetfunction on

X

by:

For E CX, l’(E)-inf{u(L’)[E ^CL’,L ,.

The function

IX’

^{has the following}

properties:

(1) For

every

E CX,

0

’(E)

^<+o%

(2) .’() O, (3) IrE CF,

then

(4)

() -W ^o. iff.(),

(6)

If

IXM,,(.6),

we define a set function

IX"

^on

^X

^{by: For}

in

Y.i.l I.t(L’i)[E

^{C U}

^L’,L

^{.6, 1,2,...}

la"

^{isin fact}an outer measure. Wenotethat in the

i-1

caseof0-1measures, if

Io(.6),

^then

"

^0.

ASSOCIATED

Forthe’-regularmeasuresthe following holds:

(1) IX-MR(’)iffixM(’)andix(L’)=sup{ix(K)lK

^CL’,K

_}

forevery L

.

(2)

_{IX MR(f_,)}iffIX M(f_.,)^and

IX(A)= inf{ix(L’)]A

^CL’,L

E’}

^forevery A Thefunction

IX’

^{givesrisetoanothersetof}measures inM(’).

Deft’: AnelementIX M(’)^{is saidtobe}weakly regulariff foreveryL

.,,

IX(L’)

sup{ix’(K) KCZ’,K

" ^}.

Thesetofweakly regularmeasureswill be denotedbyM,(’),^andthecorrespondingsubsetof0-1 measuresbyI,(’).

Wenowrecallsomelatticedefinitions:

Def":

(a) A

^lattice

"

^{is said to benormal ifffor anyA,B}

,

s.t.A CIB

-

there exists C,D

_

^{c. s.t.ACC’,BC}

^D’

^{andC’fqD’}

_.

(2) A

lattice

"

^{issaid tobe}a delta-lattice

(b-lattice)

^iff

" ^6(’).

(3)

^{A lattice}

"

^{is said to be} complement generated (c.g.) iff for

L .:_,,

there is a sequence

L,

’,n 1,2 s.t.L fi

L’,,.

(4) "

^{is said tobe}countablyparacompact(c.p.)iff forany sequence

{A,,}

from

" ^s.t.A,, ,

^there

exists asequence

{B,, }

from

"

^s.t.

^A,,

^C

^B’,,,

^{n 1, 2 and}

^B’,, .

(5) If.:.,

^and’:, are two latticesofsubsetsofX,then

L

semi-separates

’2 B ’2

^andAfiB

=

^{impliesthereexistsC}

^’,B

^C

^C,

^andACIC

^=.

(6)

^If

’

^and

^’:

are two latticesof subsets of

X,

^then

’x

^{separates,2iff}

^{A, B}

impliesthereexistsC,D

’1

^{s.t.ACC, B CDandCClD}

=9.

(7)

^{z; is}complementediff

L

E" implies

L’

^:_.

(i.e., "

^{isanalgebra).}

(8) ,

iscountablycompact

(c.c.)

^{iff for}everysequence

{L,, }

^from

"

^s.t.

^{f L,,} ,

then there exists

L,,,L,2,...,L,k s.t.i.YllL,,, .

Wenotethatnormalityofalatticehasthefollowing equivalentformulations:

(a) L

isnormaliffixl(’),ixsv,ixsv2,vl,

vls()

^then

v-v

^2.

(b) L

isnormaliff

IX l(’),v Is(:_,)

^andIX-:

v(’),

^then_{Ixsv =v’}

IX’

^onL.

(c)

^{:. is}normal iffwheneverL

CL’ UL’2 whereL,L,L2,

^thenL-A

LIB

whereA,B

.

^and

AC

L’a,B

^C

L’2.

Wealsolistsomefurtherconsequences^ofnormality aswellas somerelationsinvolving the alreadynotedsetsofmeasures. Furtherdetails canbefoundin

[6,7,8]

^{aswellasbelow.}

(2.1)

^{If isa 6-lattice,}

IX@J(’)

then g

-

L(’,

^i-1,2

(2.2)

^If Ix

Mo(),

^then

IX(X) IX"(X)

^and Ix

IX"(,).

(2.3)

^If

,

isc.g,and

IxMo(’),

^then Ix

(2.4)

^If

"

^{isc.g.andnormal,and tx J(’), then Ix}

(2.5)

^If

"

^isnormal then

(2.6)

If

IXI(’)

then

S,,-{ECX[EDL,Ix(L)-I,LCL;

^or

(the

^setof

t’-measurable

^{subsets of}

X).

Consequently,L

CS,,

^iff I,().

(2.7) If

Io(L),

^then

S,,- ^E CX[E

^D

L,,L,

L,

la(L,)

1,n 1,2,.. or

E’

D

("1L,,L,

CL,p.(L,) 1,n 1,2

-I

(the

setof

"

measurable subsets of

X).

Wenotethat

S,,

^C

Sa,,

^if

^Io(L).

Note:

(1)

^Theconverse conditionof

(2.5)

isfalseinthefollowingsense: IN(L)

IR(L)

^doesnotimply^L isnormal.

Counterexample:

Let X OandletA,B CXs.t.ACIB O, A

LIB ,

_{X. LetL O,X,A ,B,At3B}

_}.

_{Then,L is}

a lattice thatis notnormal,butI,(,)- IR(f_,).

(2) We

notetheinequality,

t9() Io(,).

Counterexample:

Let

X

Obea set, L a latticeof subsets ofX. IfL isc.g.and normal, thentg(L)

I,(Z:),

and if L is c.c.thenI(L) Io(L).

Therefore,if

Io(L)-

(L),thenwehave:

I(L) I,,(L) (I)(L)

I.(,)

I,(L).

=}Liscomplemented.

Now,

take(X,

G)

^tobea

T

²topological space. LetL Z, thezero sets, i.e., foreach continuousreal-valued function,

f,

on

X,Z(f) {x ^X ]f(x) 0}.

WechooseZso that it isnotanalgebra. Now

Z

isc.g.and normal. Ifwelet Xbepseudocompact sothat

Io(Z)

I(Z),see

[4],

^{then there}is ap.

Io(Z)

^s.t.p.

(Z).

3, SOME FURTHER RESULTS ON0-1

VALUED

LATTICE MEASURES

There are severalrelationsthatexistbetween the 0-1 lattice measures thatcanhold whencertain conditions areimposedontheunderlyinglatticeof subsets. Inthis sectionweshallconsidersuch relations.

THEOREM 3.1. LetX

,

_{0 beaset,}

.

^alattice

^of

^subsets.

(a) IlL

^is

normal, t@Io(.),

^v

I(.),

andt

<v(L),

thenv

Io(L’).

(b) IlL

^is6-normal,tIo(L), ^vI(L), ^andkt^<v(L), ^thenv

(L’).

(c) IlL

^{isnormal,Ls.s.6(L),kt}

^C(L),

^v

^Is(L),

t v(L),^thenv

P(L’).

PROOF.

Wereferto

[7].

Weconsidernext/(L),introduced in

[6].

Werecall that_la/,(L)iff la l,,(r.)and whenever

L

Ls.t.

la(L’)

1,thenthereexists

L,

L s.t.

L’

D

t L,

and

la(L,,)

^{forn 1,2 Weobtain} somefurthercharacterizationsof

I(),

^somenewandsomeknown,but in alternateways.

Wefirst notethe following:

PROPOSITION3.1.

If

tI(L),thentI(L)and.

CS,,.

PROOF. We always have t^<

"

^{onf_.,,since la}

^{Io(L). (Recall:}

^IfIx

^Io(L),

^then

^la"

^0.)

Suppose that l.t(L)=0 ^for L ^E.g.,. Then lt(L’)= 1, and since _l.tI(L),L’D

L,,

where

L,

L,

D(L,,)=

^{forn 1,2 Therefore,L C}

L’,,,L,, , (L’

)=⁰forn 1,2 ^There-

LATTICE MEASURES

fore,

IX"(L)

⁰andsoIX

IX"

^{on ‘.}

NowletL

‘

^s.t.

^IX(L’)

^0.Then

IX"(/.,’)

0

(since IX" ^Ixon‘’),

^andsoL’

S,,,,hence ^L S,,.

If

L’)

1, thensinceIX

I,(‘),L’

D t3

L,,L,

fi‘,

IX(L,)

forn 1,2 Thereforeby2.7 ofSection2,

L’ S,..,

^hence

^L S,...

^Therefore,

‘ ^CS,,.,

^{whence.(‘)C}

^S,,,,

^andso

^IX-IX"

ThereforeIX

I(‘)

^since

IX"

is an outermeasure.

Theproofis nowcomplete.

PROPOSITION

3.2.

lf

ix

ls(‘)

and

if ‘

^{s.s.6(‘), thenIx}

PROOF.

By

Proposition 3.1,oneneed

only

show thatIX

I,(‘).

Let

L (i)

If

IX(L’)

0, the monotonicity ofIX^showsthatsup{ix(K)

K

C

L’,K

‘ ^}

^O.

(ii)

^If

IX(L’)-1,

then since IX

I,(‘),

there exists a sequence

{L,}

from

‘

^s.t.

^{L’D L,}

and

Ix(L,)

1,n 1,2 Let

B

t

L, b(‘).

^Then

L

t3B

O,

and since

‘

^s.s.

^b(z;)

^there

exists

Ko ‘

^s.t.

^{B CKo}

^and

^{L NKo-O,}

^so

^Ko

^{CL’. Since IX is finitely} additive, and

X -L’UK’o, IX(K’o)

0

= ^It(K-0)

^{1. Therefore}

sup{ix(K) K

CL’,K

,}

^1.

Hence,

for any

L U,,IX(L’)-sup{IX(K)IKCL’,K,},

^and so

IXI,(‘).

Therefore,

Theproofisnowcomplete.

PROPOSITION 3.3.

/f

IX

(‘),

^then IX^{canbeextended}uniquely^toa v

(6(‘)). (The proof

isomitted.)

We

nowgivesomealternate characterizationsof the measuresin/,(‘).

THEOREM

3.2.

Ix ls(L) iff

ix

.

^where

. ^{I(6(‘)) (where} _"

^isthe restriction

of .

^to

aCz)).

PROOF. AssumethatIx

I,(‘).

^Then_IX6

(,),

andsoby Proposition3.3, we can extend ix uniquely ^to a Z.

(6(,)),

defined by: For A ^tq

L. ^6(‘),

^where

L. , A,k(A) inf{ix(L.)

n 1, 2

}.

Let

D

f"l

L,. 5(,)

andsupposethat

(D’)

^{1. Then}

Therefore,

(2)

^k

,-61L’" -:,,.IX(L’,,)

^{by 2.1.}

It

followsfrom

(1)

^and

(2)

^that

X(L’,,) I

for somen6N.SinceIX

. ^{I, IX(L ’,)}

^{1.But Ix}

^6f/(‘)

and so

L’, D,,,.N K,,K,‘,IX(K,)-Ifor

m-1,2

Therefore,

_Z.

I.(6(‘)).

The converseisclear. Theproofisnowcomplete.

THEOREM

3.3. Let

Ix Io(‘).

Then

(1) _Ix ix"

^on

, _iff

_ix

_I.(.).

(2) ff

Ix

IX"

^{on ‘,then}

^L

^C

S,,,

^andIX^U

I().

PROOF.

(1)

^Suppose

IX-IX"

^on,.

LetL

.

^s.t.

^la(L’)-

^{1. Then}

^IX(L)-0-

IX"(L).

Hence,

there exists

g,, ,,n

1,2 s.t.

L

C 1.3

K’,,

and

IX(K’,,)-

0 for n 1,2 Therefore,

L’D K,,

where

K,, .,Ix(K,)-

for

-I -I

n 1,2,.... Therefore, Ix

/,(.).

Fortheconverse, suppose Ix

/,(z;).

SinceIx

IX"

^on

,,

^weneedonlyconsiderthecasewhen

L . ^s.t.I.t(L)

^{0. Then,}

^I.t(L’)

and sinceIX /,(.), there exists

K,, .,n

1,2 s.t.L’2) f3

K,,

and IX(K,,)- ^{forn 1,2} Therefore,

L

C

f K’,,

with I.t(K’,,)-0 forn 1,2 Therefore,

.Ix(K’,,)-

0

=

IX"(L)-0.

Hence,

_Ix

Ix"

^on

..

(2)

is immediate since

Ix-Ix"

^onz; implies IX

/,(.)

by part

(1)

^{and theresult now follows by} Proposition3.1.

Theproofisnowcomplete.

Note:

IXE(L), ^iff

IX’-IX-Ix"

^{on L’.}

THEOREM3.4.

LetIx(L),v Io(L),IX

v(L),

andS,,fqL-Sv,,NL.

^Thenv

(L).

PROOF. Assumethatv

(L).

^{Then there exists}

L0

^{EL and asequence}

{L,}

^{fromL s.t.}

L, Lo,Lo- L,,

but

v(Lo) inf{v(L,)ln

^1,2

}.

Since v is a0-1 measure, andmonotonic,

n-1

(1) v(Lo)-

^0andv(L,)- ^forn 1,2

(2) Now,

_IX v(L),^so

In(L0)

0, hence

IX(L’0)

^1.

Since

L0

^71L,,,with

L,,

L,v(L,,)- ^forn 1,2,...,itfollows from 2.7that

Lo @Sv,,-S,,,.

n-1

But, Ix ^v v"

Ix"

^{on Landv" v v’<}

Ix’ IX"

^on

^L’,

^with

Ix’ Ix"

^onL’sinceIx (L).

Now

v"(L,)-

^{for allnsince}

v(L,)-

^{for all}n,and

v"(Lo)-

1,since

v"(L’)-

O. Therefore,

Ix"(L0)-

I.

Now

(Lo)-

0, sothere existsN N s.t.

la(L,,)-

^{0 for all n}

N,

sinceIx ,(,). We ^have

L ’,,

’

^L

^’0,

^so

^I.t(L’,,)-

^{forn N. Therefore}

Ix"(L ’,,)-

^forn

N,

and so

IX"(L ’0)-

since

Ix"

^{is a}

regularouter measure. But

IX"(L0)

1, therefore

L0 S,,,,

^acontradiction. Thereforeit mustbe that v ,(:).

Theproofis nowcomplete.

THEOREM3.5.

Suppose

^that /(z;),v

(.,),IX

^<v(L). Then

PROOF. Since t /(,), then by Theorem 3.2, _la

L]

^{where k}

I.(6(L)),

and

(\ ^L,,/- )/]L,, ^{inf{t(L,, ,,n}

^1,2

^{}. By}

Proposition 3.3,vcanbe extendedtoaO (6(,)) where

(,lL,)-inf{v(L,) ^{]L, .,,n}

^1,2

^}.

SinceIX v(.,), L 9on6(L). Therefore

.

^{=.IX v on.,q(L).}

Theproofis nowcomplete.

4.

THE

GENERAL CASE M(L)

Inthis section we consider the non 0-1 measures on.,(.f_,).

In

particular,we obtain results pertaining^toregularoutermeasures, andresults which insurethatcertain elements of

M(.)

are regular.

Def": LetX

,,,

_{be a set,}

,

alatticeof subsets ofX. Let_ItM(,). For E CXwe define:

It,(E)-sup{it(L)lL

CE,L

L}.

Wenote thatthesetfunctions

ia’

^andIt,have thefollowingrelations:

(1) It(X)-

It,(E’(E)+

It’(E)

for anyE CX.

(2)

IfECX,thenE

S,,

^iffit’(E)-

^It,(E).

Weaddtheproofof

(2)

^forcompleteness.

PROOF OF (2): Seealso

[5].

Let

E S,,.

^Then

(1) It’(X) It(X) It’(E)

^/

It’(E’).

^Itfollows from

(1)

and Remark

(1)

preceding,that

(2) It’(E’) It,(E’).

Now, usingstandard argumentsinvolving supremum and infimum, itfollows from

(2)

^that

It,(E It’(E ).

Conversely,toshowthat

E S,,,

it will sufficetoshow:

It’(A’)>It’(A’CIE)+It’(A’CIE’)

where A.:.;.

Lete>0begivenandarbitrary. Thereexists

L

L s.t.

E

^C_

L’

and

(1) It(L’)

<

It’(E)

+

Similarly,^thereexists

K :_,

s.t.

K

C

E

and

(2)

It,

CE)-.

^<

ItCK)

SinceKC

E

C

L’

andIt^issubtractive,

(3) It(L’ K) It(L’) It(K)

Itfollows from

(1), (2), (3)

^{and thehypothesison}

^E

^that

(4) ItCL’-K)

^<e

LetA’

6EL’,andwritethedisjointunion

A’ nL’-[A’

n(L’-

K)] U(A’ nK).

SinceIt^isadditive,

(5) It(A’ nL’) It([A’ n(L’-K)])

+

It(A’ nK) By

monotonicity ofIx,and

(5)

and

(4)

^weobtain:

(*) ^{It(A’ nL’)}

^<

^{It(a’ nK)}

^{+ e}

Wehave

A’

NE CA’

NL’,A’ NE’

CA’

NK’,

andagain bymonotonicity ofIx,

(5) It’(A’ NE)

⁺

It’(A’ NE’)

^-:

It’(A’ NL’)

+

It’(A’ NK’).

^SinceIt

It’

^on

^L’,

^wehave

It’(A’

NL’)

It(A’

NL’),

It’(A’

NK’)

It(A’

NK’),^andsofrom

(6)

^weobtain

(7) It’(A’ CIE)

+

It’(A’

^CIE’)-:

It(A’

^CIL’)+

It(A’

^CIK’).

By (*)

^weobtain from

(7):

(8) It’(A’ E)

⁺

It’(A’

CIE’)^<

It(A’ CIK)

⁺

It(A’ CIK’)

^{+ e}

But, It^isadditive, andA’-A’f3(K t3K’),^{so we}obtain from

(8),

() ’(A’ n)

+

’(A’

n’)<

(A’)

+

.

Since e>wasarbitrary,_{and It}

It’

^on

^L’,

^weconclude from

(9)

that

It’(A nE

+

It’(A

hE’)<

It’(A

’)forany

A’

’.

Hence, E S,,,,

^{and theproofisnowzomplete.}

Def": LetX

,,

_{0 beaset, and let vbeafinite,}finitely subadditiveoutermeasure

defined for

^{allA CX. Let}

^{S,, {E}

^{CX v(A v(A hE)+}

^v(A

^hE’),

.for

^{all A}

^CX}

^{be the}

set

of

^all v-measurable subsets

of

^{X. We}

define

^{a set}

function

^{v by: For}

E CX,

v(E)-inf{v(M)lE

^CM,M

S}.

Itfollows thatv is itself afinite, finitely subadditive

(f.s.a)

^outermeasure s.t.

v(X)

v(X)^and v v for all

E

CX.

Deft’: LetX

,

_Obea set, vafinite, f.s.aoutermeasure definedforallsubsets ofX.

LetS,,

^be thesetof v-measurable subsetsofX.

(1)

We saythat vis coverregulariff forACXthereexists

M S,,

s.t.A CMand

v(M)- v(A).

(2)

^{We saythatvis a}

regular

outer measureiffv v.

Wenotethat

It’

^isregularifIt

I(L).

Also,ifIt

I,,(L)

^then

It"

^isregular. Wehave:

PROPOSITION 4.1. Let

X

0 beaset, v afinite,

f.s.a,

^outermeasure.

(a) If

^{vis coverregular,thenvisregular.}

(b) If

^{visregular, then}

E S,, iffv(X)- v(E)+ v(E’).

PROOF.

(a)

^Thisfollows fromastandard greatest lower bound argument, and themonotonicityofv.

(b)

Theproofissimilartothat instandardmeasuretheorywithamilde argumentatthe end.

Thiscompletestheproof.

Wenowapply Proposition4.1toobtain:

THEOREM4.1.

IfS(L’)

separatesL, andIt

M,,,(L)

QMo(C.’),thenIx

M(L).

PROOF.

Toshow thatIt M,(L)itsufficestoshow thatL

CS,,.

Let

L

L, and lete>0begivenandarbitrary. SinceIt M,,,(,),there exists

Ko

^z;s.t.

Ko

^C

^L’

and

(1) It(L’)-’

^<t’Cgo)

tCL ’).

Since

Ko

^{QL 0, and}

^5(L’)

^separates

,,

there exists

U,

V

5(L’)

s.t.

U- ^Q_-1

U’,,

^V- _-1

V’,, KoCU ^LCV,

^{and UtqV-}

-1

andwemayassume that

U’.

^q

V’. ,

^O.

Since It Mo(Lg,the choice of thesequence

{U’.

f"l

V’. }

from

L’

requiresthat

U’.

^f"l

V’.)

^0.

Therefore,

thereexistsN Ns.t.

It(U’,

^tq

V’,)

< forna:

N.

Now, It(U’,

t"l

V’,) I.t(U’,)

+

It(V’,)-it(U’, V’,)

forn 1,2 Thus, ifn

aN, (v’. u _{v’.) (u’.)}

+

(v’.)

(u’. u _{v’.) cCKo)}

+

’CL)-

E

=.it(U’. UV’.)zit(L’)+it’(L)-e

by(I).

MEASURES

Therefore,

t(U’.

^U

V’,,)

^a

la’(L’)

+

}t’(L)

^esincep. p/on^,’. Therefore,

(X)--t’(X)zt(U’. kV’.)z’(L’)+’(L). Henee,

^L

S,,

^=*Z;

CS,,.

^Therefore,

Theproofisnowcomplete.

THEOREM4.2.

Suppose Mo(.r).

(a) If

"

^on

^,

^and

"

^{isregular,then}

^,

^C

^S,,.

(b) If

t

"

^on

^,,

^and

"

^{isregular, then t}

^M(L).

(c) If

t

"

^{on f,and}

"

^is

^regular,

^andfs.s.

^6(),

^then

^M().

PROOF.

(a)

Let

L

.6bearbitrary.

(1) tCX) CL

⁺

tCL

^’).

Since

I"

^on

^{z;, la(X)} Ix"(X),

^p,(L) p,"(L),andbydefinitionof

",

^i.e.,

"

^<

^’,

(2) "(L’) ’(L’) t(L’),

since

’

^onz;’.

Itfollows from

(1)

^and

(2)

^that

(3) V’CX)

>

"(L)

+

"(’).

Hence

clearly

L S,,,

^since

"

^isregular. Therefore,

;

C

S,, ()

C

S,,.

(b)

Since

-" ^one,

^and

"

^{iseountablyadditive}

^on(L),

^{itfollowsthat}

^MO(L).

(e)

t

L

and let e>0begivenandarbitrary.

By

definitionof

",

there exists a

sequence {L. }

from s.t.

L

C

L’.

^and

(1) .x(L’.)<"(L)+esinee-" on.

Since

"

^<

’

^and

’

^on

’, "(L’.) (L’.)

forn 1,2,

(2)

erefore,

"(L’.)

p(L’.).

The countablesubadditivityof

"

^gives:

(3) " ^{’. E "(’.).}

Combining

(1), (2), (3)

^weobtain

Since C

S.

^by

^(a),

^{itfollowsfrom}

⁽⁴⁾

^that

(5) "

Since

^L.

0

^(L

wasarbitrary,

^{’ -ee"(L}

we

^’-

conclude from

^,

^whereL’D

(5)

^that

^{L,L, ,n}

^1,2,....

, _IL’D ^{L,,L, ,n}

^{1,2,... for}

^any

(.1 "’)-

up

" .,

By(b),

M(L).

Therefore, sinceLSS,S(L)itfollowsby(*)thatm

" hg)where" M(L).

Theproofis nowcomplete.

(c)

^istrueinparticularif

"

^{onL and}

"

^is

^regular

^{and L isa6-Lattice.}

THEOM

4.

If ^J()

^and

CS,,

^then

^M(L).

PROOF.

Since

Y () Mo(L)

CSw

"

^isregular. It followsfrom Theorem

a.2),

thatit suffices to showthat

"

^{on L.}

734

(1)

^NowsinceI.tJ(,),IX-

la’- IX"

^{on,. We}always haveIX-:

I.t" ^on..

If there exists anL

_.

s.t.

(L)< IX"(L),

^{then since}

, CSwlx(L’)> Ix"(L’)-

g(L’) by

(1),

a contradiction.

Therefore,it mustbethecase thatIx

IX"

^{on z;. We}conclude thatIx M(,).

The

proof

^{isnowcomplete.}

Note:

(1)

^If

’- ^{IX" on,’,}

^and

^if"

^{isregularthenI.t}

^{J(/;). (This}

result can befoundin

[5].)

(2)

If IX J(,)^then

IX’ IX"

^{on ,’.}

We conclude byextendingaresultin

[5];

namely,

THEOREM

4.4.

(a) If ^{L’ L’}

^where

^L,

^L

^for

^{alln and}

^if

^ixJ(L),then

l.t( ^{t(L ’, ).}

( L’,), t(L’ )whenever A ^{L’ ,foralln,} andifix"

^isregular,

thenixJ(.).

(b) /fix,,

,,- ,,-

PROOF.

(a)

^See

[5].

(b)

^{Weknow that}

IX"

^-:I.t^{on ,’.} Supposethere exists

L’

^z;’s.t.

"(L’)

^<

lx(L’).

^{Then thereexists}

L.

^s.t.L’C

6 _L’.

^and

Ix(L’.)

^<

Ix(L’).

-1 -1

But

L’- (L’ elL’).

^{Therefore, la(L’)-g}

elL’) Y .(L’ elL’)

by hypothesis.

Therefore,g(L’)^<

Y

^g(L

%)

<g(L’)which is acontradiction. Therefore,

g"

’

^on

^’,

^{and since}

^IX"

isregularit followseasilythatIX

J(.t;).

The

proof

^{isnowcomplete.}

ACKNOWLEDGMENT.

^{The authorwishes to}thank the referee forsuggestions^which improvedthe abstract, and thepresentationofseveralstatements.

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[2]

[3]

[4]

BACHMAN,

G.and

STRATIGOS, P.,

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CAMACHO,

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Extensionoflatticeregularmeasureswith applications, Jour.Indian Math.Soc. 54

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GILLMAN,

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M.,

Rings

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