Outer measures associated with lattice measures and their application

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 a_{I.t _M(/;)or a l.t}Mo(/;)

(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 withitCI(/;)anditE

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 ordertogeneralize thetwo-valuedcase. Also ingeneralthe characterization of

S,,

is notasexplicitasinthe two-valued case,and this furthercomplicatesthegeneralsituation. If

J(.r.,),

i.e.,isstronglyo-smooth

(see

below)

andif/;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 usedthroughoutthepaper,

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.

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’,thelatticeofcomplementsofsetsfrom.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 tobe 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 saidtobe

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 factan outer measure. Wenotethat in the

i-1

caseof0-1measures, if

Io(.6),

then

ASSOCIATED Forthe’-regularmeasuresthe following holds:

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

CL’,K

_}

forevery L

.

(2)

IX MR(f_,)iff_{IX M(f_.,)}and

IX(A)=

inf{ix(L’)]A

CL’,L

E’}

forevery A Thefunction

IX’

givesrisetoanothersetofmeasures inM(’).

Deft’: Anelement_{IX M(’)}is saidtobeweakly 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 tobea 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 tobecountablyparacompact(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; iscomplementediff

L

E" implies

L’

:_.

(i.e.,

"

isanalgebra).

(8)

,

iscountablycompact

(c.c.)

iff foreverysequence

{L,, }

from

"

s.t.

f

L,,

,

then there exists

L,,,L,2,...,L,k

s.t.i.YllL,,,

.

Wenotethatnormalityofalatticehasthefollowing equivalentformulations:

(a)

L

isnormal_{iffixl(’),ixsv,ixsv2,vl,}

vls()

then

v-v

_2.

(b)

L

isnormaliff

IX l(’),v

Is(:_,)

and_IX-:

v(’),

then_{Ixsv =v’}

IX’

onL.

(c)

:. isnormal iffwheneverL

CL’

UL’2

whereL,L,L2,

thenL-A

LIB

whereA,B

.

and AC

L’a,B

C

L’2.

Wealsolistsomefurtherconsequencesofnormality 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

(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)

doesnotimplyL 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 thereis 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), and_kt<_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),then_tI(L)and.

CS,,.

PROOF. We always have t<

"

onf_.,,since la

Io(L).

(Recall:

If_Ix

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

There-LATTICE MEASURES fore,

IX"(L)

0andso_IX

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(‘), then_Ix

PROOF.

By

Proposition 3.1,oneneed

only

show that_IX

I,(‘).

Let

L

(i)

If

IX(L’)

0, the monotonicity ofIXshowsthatsup{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 _IXcanbeextendeduniquelytoa 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. Assumethat_Ix

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.Since_IX

.

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.(.).

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}

/,(.).

Forthe_{converse, suppose Ix}

/,(z;).

Since_Ix

IX"

on

,,

weneedonlyconsiderthecasewhen

L

.

s.t.I.t(L)

0. Then,

I.t(L’)

and since_{IX /,(.),} 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’since_Ix (L). Now

v"(L,)-

for allnsince

v(L,)-

for alln,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

}.

4.

THE

GENERAL CASE M(L)

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

In

particular,we obtain results pertainingtoregularoutermeasures, 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’

and_Itissubtractive,

(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).

Since_Itisadditive,

(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

()

’(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,if_It

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. Since_It 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

U’,,

V-

V’,,

KoCU

LCV,

_and

UtqV--1 -1 _-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’.)

E

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)

"

L.

(L

’

-ee"(L

’-

,

whereL’D

L,L,

,n

1,2,....

Since 0wasarbitrary,weconclude from

(5)

that

,

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)

C_Sw

"

isregular. It followsfrom Theorem

a.2),

thatit suffices to showthat

734

(1)

Nowsince_I.tJ(,),IX-

la’-

IX"

on,. Wealways 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;. Weconclude thatIx M(,). The

proof

isnowcomplete.

Note:

(1)

If

’-

IX"

on,’,

and

if"

isregularthen_I.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.ton ,’. 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 followseasilythat_IX

J(.t;).

The

proof

isnowcomplete.

ACKNOWLEDGMENT.

The authorwishes tothank the referee forsuggestionswhich

improvedthe abstract, and thepresentationofseveralstatements.

REFERENCES

[1]

[2]

[3]

[4]

BACHMAN,

G.and

STRATIGOS, P.,

On general latticerepleteness and completeness, Illinois

Jour.

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Math.27,no.4

(1993),

535-561.

CAMACHO,

J.

Jr.,

Extensionoflatticeregularmeasureswith applications, Jour.Indian

Math.Soc. 54

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233-244.

FROLIK, Z.,Prime filterswiththec.i.p., Comm.Math. Univ. Carolina 13

(1972),

553-575.

GILLMAN,

L.andJERISON,

M.,

Rings

of

Continuous Functions,Van Nostrand,Princeton,

NJ,

1960.

[5]

GRASSI, P.,

Outermeasures and associated latticeproperties,lnternat.

J.

Math. andMath. Sci. 16, no. 4

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SIEGEL, D.,

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[8]

SZETO,

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