Some topologies on the set of lattice regular measures

VOL. 15 NO. 4 (1992) 681-696

SOME

TOPOLOGIES ON THE SET OF LATTICE REGULAR MEASURES

PANAGIOTISD. STRATIGOS

Long

IslandUniversity

Brooklyn,

New

York 11201

(Received December 20, 1989 and in revised form November 27, 1990)

Abstract.

We

considerthegeneral setting of

A.D. Alexandroff, namely,

an

arbitrary

setXandanarbitrary latticeofsubsetsof

X,

,..(L))

denotes the

algebra

of subsets of

X

generated

by

,

and

MR (L)

the setof all latticeregular, (finitely

additive)

measures on

.(L).

First, weinvestigatevarioustopologies

onMR (.)

and on variousimportantsubsetsofMR

(L), compare

thosetopologies,and considerquestionsof measure

repleteness

wheneverit isappropriate.

Then,weconsidertheweak

topology

on

MR(L),

mainly whenL is6andnormal,which is theusual Alexandroffframework.This moregeneral settingenables us toextendvariousresultsrelated tothe special case of

Tychonoff spaces,

latticesofzero sets, and Bairemeasures,and to

develop

asystematicprocedure forobtainingvarioustopologicalmeasure

theory

results onspecificsubsets

ofMR(L)

intheweak

topology

with L aparticular topologicallattice.

Key

Words andPhrases: Lattice;6 andnormal lattice;

separating

and

disjunctive

lattice;lattice semi-separation.

Measure;

regularity, o-smoothness,x-smoothness, andtightness ofameasure; tight setof measures. Lira sup-topology; weaktopology; Wallmantopology; relativecompactness. Repleteness, measure

repleteness,

and

strongly

measure

repleteness.

1980MathematicsSubject Classification:

28A60, 28A33,

28C15.

1.

INTRODUCTION. We

considerthegeneral setting of

A.D.

Aiexandroff

[2],

namely,

anarbitrary

set

X

and anarbitrarylattice ofsubsets of

X,L..(L)

denotes the

algebra

of subsetsof

X

generated byL and

MR (L)

thesetof allbounded,latticeregular, finitelyadditivemeasureson

JI(L).

First,weinvestigatevarioustopologies

onMR

(f_,)

andonvariousimportantsubsetsofMR

(L),

compare

thosetopologies,and considerquestionsof measurerepletenesswhenever it isappropriate.

It

should be noted that the firsttopologyweinvestigatewas first considered

by

Blau

[8]

and

by

Kallianpur

[14]

on specificsubsetsofMR

(L)

andbythelatter in atopologicalframework.

We thereby

generalize thoseresults and indeed we obtainBlau’smainresults asspecialcases.

Next,

we considertheweak

topology

on

MR(L),

mainly

whenf_,is6 and

normal,

whichisthe usual Alexandroff framework.Thismore

general

settingenables ustoextendvariousresults of

Varadarajan

[20]

and togive applicationstootherspecific topological

spaces,

ratherthantojust

Tyehonoff spaces,

lattices ofzerosets,and Bairemeasures,asisdone

by Varadarajan. Our

emphasishere is tojust givean

example

In

summary,our aim isnotjustforgeneralization, buttogiveasystematic

procedure,

in a

general

setting, namelythatof

MR

(),

forhandling

any

of those

special

settings.

We

adhere to standard

terminology

whichcan befound

e.g.,

in

[2,4,5,11,20],

and we review some of the more

important

terminology andnotation used

throughout

the

paper.

Section

1. Terminologyand notation.

a)Consider any

set

X

and

any

latticeof subsetsof

X,.

We

shall

always

assume,withoutlossof generalityfor ourpurposes,that

,X

.

Thedefinitions ofthe followingterms are foundin

[4]"

is 6,

separating,

disjunctive,regular, normal, Lindel6f, compact,

countably

compact, countably

paracom-pact.

A

subset

otX,

S,

is said tobe

-compact

ifand

only

if the lattice

S

Cis

compact.

Thecollection of

-compact

sets isdenoted

by

K.

For any

topological

space

X,

the collection of closed sets is denoted

by

9",thecollectionof

clopen

setsby and the collectionofBorelsets

by

.

b)For

an arbitrary

ftmctionf,

the domain

off

isdenoted

by

Df.

For

an

arbitrary

subset of

X,

S,

the characteristic function of

S

isdenoted

by

sos.

A

function

f

from

X

to

R

U{+/-}

is said tobe -contin-uous if andonlyif for

every

closed subsetofR

U{+/-oo},C,f-l(C).

The set whosegeneralelement is afunctionfrom

X

to

R

U{+/-oo}

which is -continuousand boundedisdenoted

by

Cb().

Theset whose goneralelement is a zero setof isdenoted

by

2;().

Thesetwhosegeneral elementis theintersectionofan arbitrary subset ofisdenoted by t. The

algebra

ofsubsetsof

X

generated by

isdenoted

by

.().

c)Consider any algebra

ofsubsets

ofX,.

A

measure

on.

is definedtobea function

p,

from to

R,

such that

I

isfinitelyadditiveandbounded.

(See

[2],

p.

567.)

Thesetwhose

general

elementisa measure

on.l()

isdenotedby

M().

For

an arbitraryelement

ofM(),

Igthe

support

of is defined tobe

f{L

f

Ix ()

-I

Ix

(X)}

and isdenotedby

S().

An

elementof

M(),lg

issaid tobe

-regular

if andonlyiffor

every

element

of,q(),E,

for

every

positive number, e,there exists anelement of

,L,

such that

L

C

E

and

E)

p(L)l

<

.

Thesetwhose

general

element is anelementof

M()which

is-regularisdenoted

by

MR(). An

element of is said tobe

-(o-smooth)

if and

only

ifffor

every

sequence in(),

(A,),

if

(A,)

isdecreasingandlimA, then lira

p4,)

0. Thesetwhose

general

elementisanelementof

M()

which is

-(o-smooth)

isdenoted

by

M(o,). An

element

ofM(),

Igis said tobe

-(z-smooth)

if and

only

iffor

every

netin

,(L,,,

if is decreasingand

limL-

@,

then lira

(L)-

0. Thesetwhose

general

elementisanelement of

M()

which is

-(z-smooth)

isdenotedby

M(’c,). An

elementof

M(),

is said tobe-tightifandonlyif

t

/M(o,)

andfor

every

positive number, e,there exists an

-compact

set,

K,

suchthat

Ix .(K’)

<e. The setwhose

general

elementisanelement of

M()which

is

f,-tight

isdenoted

by

M(t,). A

subsetof

M(),

A,

issaid tobe-tight if and

only

ifA

CM(o,)

andAisnormbounded and for

every

positive number,

e,

there exists an

-compact

set,

K,

such that whenever

tt CA,

then

I1

.(K’)

<e.

Thesetwhose

general

element isanelement

ofM(),

tt,suchthat

.(r$).

{0,1 }

isdenoted

by

I().

For

an

arbitrary

element

of.(),A,

{Ix

:_IR()/tt(A)

1}

isdenoted

by

W()

and

{Ix

_IR(o,L)/A)

by

Wo().

d)

issaid to be repleteifandonly ifwhenever

I

CIR(o,),

then

$(t),,

.

is said tobe measurerepleteif and

only

if

MR(o,)-

MR(,).

issaid tobe

strongly

measure repleteif and

only

Consider

any

set

X

and

any

latticeofsubsets of

X,f‘.

Reviewthedefinitionoftheweak

topology

on

MR(L).

Consider

any

elementof

MR(f‘),0o. Now,

consider

any

elementof

Cb(,),f,

and

any

elementof

R+,e;

then, consider

{t_MR(f_,)dffd-ffd[

<e

and denote it

by

N(lao,

f,e).

Further,

consider

{N(lao,f,e)/fCb(f‘),e

R+}

and denote it

by

S.

The following fact iswell-known: There existsa

topology

on

MR (f‘),’,

suchthat

S,

isasubbaseforthex-neighborhood system of and

:

isunique.

Next,

note thefollowing:

a)

For every

netin

MR(f‘),

(o),

for

every

element of

MR(f‘),v,

liraI.q-v in

:

if and

only

if for

every

element of

Cb(f-,),f,

lim

f

fdlg

f

fdv.

b)

For every

element of

Cb(f‘),f,

the function

?

determined

by

](t)-ffdlggtMR(f‘)

is z-continuous.

Moreover,

:

istheweakest

topology

havingthisproperty.

For

thisreason,

:

iscalled the weaktopologyon

MR(f.).

Assume

f‘is6 and normal. Thenthenormedvector

space

MR (f‘)

is

isomorphic

withthe conjugate

space

of

C,(f‘).

(See

[2],

p. 577,

Theorem

1.)

Consequently

the

topology

r coincideswith

o(MR (f_.),

Cs(f.,)).

Denote

the

conjugate space

of

C,(f‘)

by

C,(f‘)’.

Then,

since

o(Cs(,)-,Cs(f‘))

isthe

weak"

topology

on

C,(f‘),:

coincides with the

weak*

topologyon

MR(f‘).

Now,

recallthat the relativizationofthe

weak"

topology

on

IR(f‘)

istheWallman

topology.

(See

[21].)

In

the

sequel,

denotethe relativization of the

weak*

topology

to

M/R(_,)

by

w*

and the Wallman topology by

W. Let

usdiscard thecondition"f‘is 6 andnormal".

We

will consider othertopologieson M/R

(),

therelativizationofeach of whichto

IR(f‘)

is

W. Also,

we will discover various

properties

of thosetopologies,

compare

them,andinvestigate

questions

of measure

repletenes

wheneveranopportunity arises.

Section3.

First,consider thefollowingstatement: If r.is6 and normal, then for

every

netinM/R

(f‘),

(Ix,,),

for

every

elementof

M+R (f‘),

v,

lira v in

w"

if andonlyiffor

every

element of

a;,L,

li’-

I.h(L

v(L)

and lim

I.q(X)-

v(X).

(See

[20],

p.

182,Theorem

2.)

Before stating Theorem

2,

Varadarajannotes that its

proof

iswell-known andgivesthefollowing reference:

([3],

p. 180,

Theorem

2).

However,

it is tobe noted thatAlexandroff’s theorem pertainsto

sequences,

andits

proof

israther

long

and not

adaptable

to nets. Elsewhere,the theorem mentioned in

Varadarajan’s paper

isstatedwithout

proof

or is

proved

ina

topological

setting.

A

proof

ofthe statementmentioned aboveisgivenin

tliis

section.

Now,

topologize

M/R(f.,)

asfollows:

a)

Consider

any

net

inM+R(f‘),(g.,,,),

and

any

element

ofM/R(f‘),v.

.t,,)

is said to

converge

to vif andonly ff

(i)

For every

element of

f‘,L,

lim

ILt,,(L)

v(L)

and

(ii)

lira

[a(X) v(X).

Thestatement

"(,,)

converges

to

v"

isalso

expressed

as lim

g,

v.

[5)

Definean

operator

on

M+R(f‘))

asfollows: Consider

any

element of

M/R(f‘)),A. Now,

considertheelement of

’(M/R

(f‘)),.

-,

determined

by

{v

M/R(f‘)/there

existsanetinA,

(,,),

such that lim

g,,

-v}.

Show the operator

"-"

isaclosure operator.

To

do this,

show

theKuratowskiclosure conditions are satisfied.

(See [15].) (Proof

omitted.) Hence

the operator

"-"

is aclosure

operator.

Observation. Consideranyelementof

M/R(),po.

Now,

consider

any

element of

,L,

and any element of

R+,e;

then,consider

{Ix

EM+R()/Ix(L ’)

>

po(L’)-

e and

IX(X)-

IXo(X)I

<

}

anddenote itby

B(po,L’,e).

Further,consider

{B(p,L’,e)/L

Ef,e _R

/}

and denote it

by

S,,

_{o. The following}statement is true:

$o

is a subbaseforthe’F-neighborhood system of 0.

(Proof

omitted.)

(For

theo-smooth case,see

[1].)

Remark.

J.

H.

Blau

[8]

works with the relativizationof’fro

M+R(o,)

and calls it the

A-topology.

Special cases. 1. Consider thepair

IR(), W())

andtopologize

M/R(W())

inaccordancewiththe method described above and denote theresulting

topology by

.

2. Consider thepair

(IR(),tW())

and topologize

M/R(tW())

in accordancewith themethod described aboveanddenote theresultingtopology by

.

3. Considerthepair

(IR(o,), Wo())

andtopologize

M/R(Wo())

in accordance with the method described above and denote theresulting topology by

.

Proposition

3.1.

Therelativizationofto

IR ()

is

W.

Proof. Consider

any

netin

IR (),

(1,

and

any

element of

IR(),

v.

a)

Assume

lim v in and show lim

IX

vin

W.

Considerany

member

of the base for the

W-nighborhood

system of v,

W(L)’,

such thatv

W(L)’.

Then

v(L’)

1. Since lirn IX,,- v in

gv(L’)

lirn

(L’).

Consequently

lim

(L’)

1.

Hence

(p

is

eventually

t’’(L)

in

’. Hence

lira

IX-

v in

W.

b)

Assume

li

p v in

W

and show limI-q vin’T.

(i)

Consider any element off_.,L,

and

show

v(L’)

limp(L’).

Consider the case:

v(L’)-1.

Then v

E

W(L)’. Hence,

sincelimp vin

W,

(l

is

eventually

in

W(L)’. Hence

lira

p(L’)

1.

Con-sequently

v(L’)

lim

I(L’).

(ii)

Note liml.q(X)-

v(X).

Consequently

liml-q-v in"/’.

Consequently

e

relativizationof’Tto

IR (f.,)

is

W.

Proposition 3.2. ’Tis

T.

(Proof omitted.)

The followingthree

Lemmas

areneeded inshowing thatiff.,isnormal, then

’T

is

T2.

Lemma

3.3. Iff.,isnormal,thenfor

any

elementof

M/(f.,),

IX,for

any

twoelementsof

M/R(f.,),v,v,

if :Vl,V

on

and

IX(X)-v(X),v2(X),

then vl -v2.

Proof.

Assume

,

isnormal.Consider

any

element of

M/(,),

Ix, and

any

twoelementsofM/R

(L),

v

,

v2,

such that

IX

v:,v2on

,

and

X)

v(X), v2(X).

To

show vl v2,assumethe

contrary.

Thenthere exists

anelement of

,A,

suchthat

v(A)

,

v2(A).

Consider

any

such

A.

Assume vx(A)

<

v2(A). (Note

this

assumption

doesnotaffect

generality.) Then,

sinceVl,v2

E

M/()

and

vx(X)

v2(X),

thereexistsa

positive

number, ix, such that

Vl(A)

<

v2(X)

<

v2(A).

Consider

any

such Since

v(A)

<

ctv2(X)

and

v(X)

v2(X),v(A’)

>

(1

ct)v(X).

Hence

v(A’)-(1

ct)vx(X)

>0.

Hence,

since

vx

M/R(),

there exists an elementof

,B,

such that

B CA’

and

v(B)>v(A’)-(v(A’)-(1-ct)v(X))-(1-ct)vx(X).

Consider

any

such

B.

Since

B

CA’

and,isnormal,there exist elements

of,C,D,

such thatA C

C’

and

B

CD’

and

C’D’-.

Consider

any

such

C,

D.

Since

C’t3D’-,

CUD-X.

Consequently

la(C)

+

p(D)

tt(X). Hence It(C)

(1

ct)X)

or

D)

=-

txix(X).

Considerthecase:

C)

>

(1

-c0X).

Then,

since

(X) v2(X), v2(C)

a=

(1

-c0v2(X).

Hence,

since

v2(A)

>

tv2(X

andA C

C’,v2(A UC)

>

v2(X). Hence,

since this statement is

false,

p(D)

.

crUX).

Then, since

X)-vl(X),vi(D)>CtVl(X). Hence,

since

Vl(B)>(1

-tx)v(X)

and

B CD’,v(B UD)

>

vx(X).

Hence,

since thisstatement isfalse,theassumptionis

wrong.

Consequently

v

v2.

Lemma

3.,t. Consideranyrealvector

space E

andanysublinear functionalon

E,

p.

There exists a

linearfunctional on

E,

ap,

suchthat,:p.

Lemma

3.5.

Denote

thegeneralelementof

A(L)

by

A,

and the

general

elementof

M(z)

by

Ix.

Consider the set whose

general

element is a functionfromXto

R,

,

such

thatfisA(L)-simple

anddenote it

by,5(,q(,)).

Consider thenormedvectorspace

5(.fl(,))

and denote itby

E.

Denote

the

conjugate space

of

E

by

E’.

Thereexists a function from

M(Z;)

to

E-,9,

suchthat

M(,))-E-and

isanisomorphism.

Spe-cifically,

a)

for

every

ia,

q()

issuchthat for

every

1",

q()(.t]

13)

for

every

elementofE-,ap,

-l(ap)

issuchthat forevery

A,

9-1(p)(A)

Theorem

3.6.

If

,

isnormal, then’/’is

T2.

Proof.

Assume

,

isnormal.

To

show’Tis

T,

assume thecontrary. Thenthere exists a net in

M/R(,),(I.t,,),

such that there exist two elementsof

M/R(),vl,

v2, suchthat liml.q-vt,v2 and vl,v2.

Consideranysuch

(Ix,

v,

v2.

Next,

proceed

accordingto thefollowing

plan:

Frst,

obtain an elementof

M/(z;),

k,such thatk v1,v2 onLand

L(X)

v

I(X),

v2(X).

Then,

show v v2,thusreachingacontradiction.

For every

ct,consider

q(I-q)

(see

Lemma

3.5)

and denote it

by

,.

considerthe function

p

on

E,

such

thatp(f)

li-"

q,(f).

Notep

is asublinear functional.

Hence,

by Lemma

3.4,thereexistsa linear functional on

E,

ap,

sucl

that

xp

x

p.

Consider

any

such

Show

ap

isbounded.

Note

for

every

.f, ap(f)

p(f)=

li--’

9,0O

li--

oOl

i---

Iffdl.,

i--(lllllo}})-

l}l }11-

(x)

III.

Since

liplx-v,,lipla,(X)-v1(X).

Hence

lip

l.t,(X)

vt(X).

Consequently

for

every f, ap(f)

v1(X)

lljl.

Moreover,

for

every

f,

ap(-f)

p(-f)

i--

o(-/3,

i---

o(-Y)l

(X)ll/ll.

I-In

isbounded.

Now,

consider

-(p)

(see

Lemma

3.5)

anddenote it

by

k.

Show

.

6M/(L).

Note

for

everyA, A)

-(ap)

(A)

ap(t). Next,

show

ap

is ageneralizedBanach limit. Showforevery

f,

lim

(f)

ap(j

)

s

li-

,(]’).

Note

for

every.f, V(jr)

p(f),

usethe fact:

li___m

,(f)

lim

(-q,ff)).

Fix.f.

Note

p(-]’)

lim

,(-D

lira

(-(f)).

Consequently

ap(

D

-ap(-]’)

lim--’(-q,(f))

li__m(f). Consequently

for

every

f,

lirn

,(f) ap(f)

li"’

,(f). Hence

ap

is ageneralizedBanach limit.

Now,

notefor

every A,

Jim

,(g:t)": ap(:.). Consequently

for

every A,

k(A

lira

,(x:.)

limlx(A

0.

Hence

k

M/(L).

Next,

show

.

-:_v,v2onL.

Note

for

every

element of

L,L,k(L)

ap(ttt.)

P(L)"

lim

,(tCL)

lira

l.t,(L ).

Since lim l.t

v,v

2,foreveryelement ofL,

L,

lira

l.ta(L

vi(L),

v2(L ).

Consequently

.

v,v

on

L.

Finally,show

X(X)-

v(X),v(X).

Note

k(X)-

ap(x)

and

lirn,(Cx) ap(g:x)-:

lira

,0cx).

Since liral.t,

v,

v2,lira

lt(X)

v1(X),

vz(X).

Consequently

X)

v(X),

v(X).

Summarizing:

.

6

M’(.)

andvl,v26/M+R

(L)

and

.

v,

v

Then,

since Lisnormal,

by

Lemma

3.3,

v

v2. Thusacontradictionhas beenreached.

Consequently

7"is

T.

Corollary_

3.7. IlLisnormal,

thenIR(L)

isclosed.

Proof.

Assume

L isnormal. Recall that the relativization of’Tto

IR (L)

is

W

and

IR (L)

is

W-compact.

SinceLisnormal, by Theorem

3.6,

’Tis

T2.

Consequently

]R

(L)

isclosed.

IR(y)

isknown as theWallman compactificationof

X

and isdenoted

by taX.

(See [21].)

(2).

Consider

any

topological

space

X

such that

X

is/’

andlet.g 2;. Then

IR (2;)

isclosed.

IR (2;)

isknown as theStone-Cech compactificationof

X

and isdenoted

by

fiX.

(See [10].)

(3).

Consider

any

topological

space

X

suchthat

X

is

TI

and 0-dimensional and letZ,-C.

Then,

IR(C)

isclosed.

IR (c)

isknown as the Banaschewskicompactificationof

X

and isdenoted

by

[ioX.

(See [7].)

Corollary_ 3.8. If z;isnormal,then 9(

U{M+R (Z:)},

where 9( is the collection

of’T-compact

sets,is a latticeand is measure

replete.

Proof.

Assume

z:

isnormal. Then’/’is

T:.

Hence,

since

every

element

of

isclosed, is a latticeand iscompact.

Consequently

U{M+R()}

ismeasure

replete.

Proposition 3.9. Ifris6andnormal,

thenM/R()

is closed.

Proof.

Assume

.

is6and normal. Consider

any

netin M/R

(Z:),

(I-q),

and

any

elementof

MR (,),

such that

lim-v.

Showv

M/R(r.).

Since v

_MR(:_,),

is suffices toshow for

every

element of

.,L,v(L)

O.

Assume

there exists anelement

off.,,L,

such that

v(L)

<0. Consider

any

such

L. Then, by

A,

lexandroff’s Representation Theorem,

([2],

p. 577,

Theorem

1), v(L)

lim

ffdv.

Hence

for

every positive

number, e,there exists an elementof

Cs(,),fo,

such that

f0

a:

c.

andfor

every

element of

C,(z;),f,

iff

and

f

j,then

{v(L)-ffdv[

<

.

Let

e

-v(L)

andconsider

any

element of

C,(Z,),f

suchthat---. Then

Iv(L)

-ffodv

<

-1/2v(L).

Consequently

ffodv

<

v(/.)

<0. Since

lim

v,

limff0d

-ffodv.

Hence,

sinceforeveryct,

ffodl

O,ffodv

aO.Thus a contradiction has been reached.

Consequently

v

.M+R().

Hence

M/R(.)

isclosed.

Proposition

3.1.

Iff.. is6 andnormal, then

,

the collection of

w’-closed

subsets of

MR(Z),

is measure

replete.

Proof.

Assume

f_.is6and normal. Then,since

every

normbounded subsetof

MR

(f_,)

is

w’-compact,

(MR

(),

w’)

iso-compact.

Hence

(MR

(:_.),

w’)

isLindel6f;otherwisestated:

.7

is Lindel6f.

Consequently

9ismeasure

replete.

Corollary_

3.11. If

z:

iscountably compact,

6,

andnormal,then

(MR(o,.f.,),w’)

isLindelif.

(Proof

omitted.)

Examples.

(1).

Consider

any

topological

space

X

suchthat

X

is

countably

compact and normal,and letz; -.9". Then

(MR

(o, Sr),

w’)

is Lindel6f.

(2).

Consider

any

topological

space

X

suchthat

X

is

pseudocompact

and

Ta_,

andlet/; 2;. Then

(MR(o,2;),w’)

isLindel6f.

Proposition 3.12. If/; is

(separating)

anddisjunctive,then

M/R(r.)C

IX],

where

IX]

isthe vector subspace of

MR(L)

spanned by

X.

(An

explanation of

why

the word

separating

is enclosed within

parentheses

isfound in

([4],

p.

1502)).

Proof. Assume/;is

(separating)

and disjunctive. Consider

any

elementof

M/R(/;),v. Note

toshow v

E

IX],

bythe definition of

"-",

itsuffices toshow there exists anetin

IX],

(kto),

suchthat lira

I

v. Such anet is obtained asfollows:

Consider

any

finitepartition

ofX(relative to.(L)),P.

Set

P

{Ak;k

1 ,n

}.

For

eachk,consider

any

elementof

Ak,x.

Then, consider

v(A,)t.

Sincev

ff.M/R(.)

and/; is disjunctive,

v(A)l.t

of

Pl-

Now,

consider

(lxp;P

3).

(The

ideaofconstructingsuch anetisduetoVaradarajan

[20].)

Show

limlx

v v.

(Proof

omitted.) Hence

v

IX].

Hence

M/R(.)C

IX].

Thefollowingtheorem settles thequestionof coincidence of the

topology

r/-andthe

topology

w"

(when

,

is6 and

normal),

raised at thebeginning ofthissection,andalsogeneralizesanimportant result ofBlau

([8],

p.

27,obtained

by

combiningTheorems4,

5,

and

6).

Theorem 3.13.

a)

For every

net in

M+R (r.),

(Ixo),

forevery element ofM/R

(Z;),

v,

if lim Vt, v in

’T,

then limI-q vin x.

b)

The collection ofx-closed sets is contained in the collectionofr/’-closedsets.

c)

IfLis/5andnormal,thenfor everynet

inM/R(),

(lxo),

for

every

elementof

M’R

(.),

v,

if lim Ix,- v in

w’,

then lim

IM

v in’T.

d)

If

.

s

b andnormal,then’Tcoincides with

w’.

Outlineof aproof,

a)

Copsideranynet in

M/R(),(Ix,

and

any

element of

M/R(L),v,

such that limIG v in’/’.

Note

toshow lim Ix, v in

x, by

the definition of

x,

itsuffices toshowfor

every

element of

Cb(.),f,

lim

f

fdlg

f

fdv.

Assume

for

every

element of

C,(.),h,

ifh<1, then

limfhdlx-fhdv.

Then for

every

element of

Cb(,),f,

limffdlM-ffdv.

Therefore it sufficesto show forevery elementof

C,(,),h,

ifh <1, then

limfhdtM -fhdv.

Consider

any

element of

C7,(,),h,

such thath<1. Showlim

fhdlM fhdv

li_hdl.q.

(Proof

omitted.)

b)

(Use

a).)

c)

Assume

Z;is6andnormal. Consider

any

net in

M/R(L),

(IXa),

and

any

element of

M/R(/;),v,

such that lim

-

v in

w’.

(Recall

w*

coincides with

:.)

Showlim

I

-v in

(i)

Consider

any

elementofZ;,

L.

Show lim

(L)

v(L).

Since v

M/R(),

there exists an element ofL,

L,

such that

’2)L

and

v(/’)<

v(L)+’.

Consider

any

such

.

Since

L C/’

and is and normal,there exists anelementof

Cb(L),f,

such that

f(L)

{ 1].

and

.t’(/)

C

{0}

and 0

f

1. Consider anysuch

f.

Now,

noteforevery a,

I.q(L

fL

fdl

ffdl.

Hence

lim,

(L

lim,

-

fdF.

Since

lim,

I.q v in

w’,limffdl

-ffdv.

Consequently

li--

l(L)

,:ffdv

-f

fdv v(/’)

<

v(L)

+e.

Hence

1-

I,q(L)

v(L).

/,

(ii)

Showlim

p=(X)

v(X).

Since lirap= vin

’,

limfldt

f

ldv.

Hence

lira

I(X)

v(X).

Consequently

lira -v in’L

d)

(Use

a)

and

c).)

Corollary

3.14. Ifis

countably

compact, 6 and normal, then for

every nonncgativc

number,

k,

the subspacc

{

.M/R(o,)IX)=

k}

is

T2

and

compact.

(Proof

omitted.)

Thiscorollary generalizesanotherimportantresult of Blau

([8],

p.

31, Theorem

11).

Examples.

(1).

Consider

any topological space

Xsuch thatXis

countably compact

and

normal,

and letL -.9". Then for

every

nonncgativc number, k.,the subspacc

{F.M/R(o,.’T)IX)-k}

is

T2

and compact.

(2).

Consider

any topological space

X

suchthat

X

is

pscudocompact

and

T_,

andlet L Z,. Thenfor

every nonncgativc number, k,

the

subspacc

{

M’R(o,Z,)I

(X

k}

is

T2

and

compact.

The following theorem generalizesa resultof Kallianpur

([

14],

p. 948,

Theorem 2.

I).

Theorem

3.15.

Considerthecondition:

For every

clement of

9CK,

for

every

clement of

L,L,

if

K

f’)L

,

thenthereexistsanclementof

C#(),f,

suchthat

f(K)

{

1}

and

L)

C

{0}

and 0s

f

I.

(R) If satisfies condition (R), is separatingand disjunctive,

T2,

and

strongly

measure

replete,

then

u"(a,) ""t/u’(o,,’.)"

t) By

Theorem3.13,

b),

’/’isstrongerthatx.

I)

Show

x/u.mo.z

isstrongerthan

’T/M.Rto,,

.

Consideranyelementof

M/R(o,/;),,

and

any

memberof thesubbase forthe

q’/u.Rto.fneighborhood

systemof

,B(Ib,L’,e).

(See

the observationfollowingthe definition

of’T.)

Since/;is

strongly

measurereplete,

M/R(o,,) CM/R(t,/;). Consequently I M/R(t,/;)

Hence

there exists an elementof

K,

such that

(l).

(K’)

<_i. Consider

any

such

K.

Since/;_is

T2,K

Ct/;.

Since/;isstronglymeasure

replete,

it ismeasure

replete. Consequently

.M/R(x,,). Hence,

since /;isseparatinganddisjunctive,by

([4],

Theorem

2.5),

there exists an element of M/R

(x,

t/;

),

ixt,suchthat

Ixt/az)-

and

Ixt

is unique.

Now,

note

Ixt-

(!)"

ont/;.

Consequently IXt-

()"

on

K.

Hence,

since

(). (K’)

+

(ix0)"

(K)

(X), (). (K’)

IX(K’).

Consequently

IXl(K’)

<

.

Since

t

-M/R(,),

there exists an element

of/;,Lx,

such that

L

C

L’

and

IXo(L’-Lt)<

.

Consider

any

such

Lt.

Thenconsider

K

tqLt.

Note

K

tqLt

.

Set

K

fqL

K.

Note

IXt(L’ K)

IX(K’)

+

IXo(L

L)

<_-i+_i"

"

Since

K

K

f3L andLt

CL’,K

CL’.

Hence,

since/;satisfies condition(R), thereexists anelement of

C(/;),f,

such that

f(Kt)

{

1

}

and

f(L)

C

{0}

and 0

]"

1. Consider

any

such

Now,

consider the following member of the subbase for the

x/cRto,,-neighborhood

system of Ixo:

N(ixo;f,

1;{).

Show

N(Ixo;f,

1;{)

CB(,L’,e).

Consider

any

element of

N(;]’, 1;{),v.

Note

Further, notesince

vr(ixo;f,

1;),

o(X)l

<

.

Consequently

v

B(IXo,L’,e).

Thus

N(;f, 1;[)

C

B (I,L’, e).

Hence

:/u’<,,.z)

isstrongerthan

Consequently

q’/u’mo.-)"

Example. Considerany topological

space

X

such that

X

is

locally

compact, Lindel6f,and

T,

andlet

/;-. Then, by

([4],

p. 1516, Application

2),

7

is strongly measure

replete.

Consequently

M

/R

o,

7)

Section

4.

In

this section conditionsare obtained under which certain subsetsof

M/R(/;)

are

sequentially

closed in the

weak*

topology.

Theorl

4.1. If/; iscountably paracompact, andnormal,then

M/R(o,/;)

issequentiallyclosed

(in M"R (/;

)).

Proof.Assume/;iscountably paracompact, andnormal. Consider

any sequence

inM/R(o,,),

(Ix,),

and

any

elementof

M/R(/;),v,

such that limIX,-vin w

.

Showy

M/R(o,,).

a)

For every

n,consider

Ix,/tz

ad

denote it

by

.

Observation. Consider

any

element

ofM/R(o,/;),

ik.

Now,

consider

k/,ttz)

and denote it

by

..

Since

.

M/(o,/;),ik

M/(o,z(/;)). Hence,

since

Z(/;)

is

complement

generated,

Z M/R((/;)).

Conse-quently

Z.

.M/R(o,Z(/;)).

By

the aboveobservation,forevery n,

Ix,0

M/R(o,2;(/;)).

b)

Consider

v/

anddenote it

by

v

.

Observation. Consider

any

elementof

M/R(/;),k. Now,

consider

k/tz

and denoteit

by k

.

Since c.is andnormal,

Z(/;)

semiseparates/;.

Hence

,0

M/R(Z,(L)).

By

the aboveobservation,v

.M/R(Z,(/;)).

Observation. Consider

any

element

ofM/(z;),

..

Now,

consider

,tr.

anddenote it

by

Z0.

Further,

consideranyelement of

Cb(f),f.

Note

ffd. -ffdZ

.

By

the above observation, for

every

elementof

Cb(),f,

for

every

n,

ffd

-ffd

andffdv -ffdv

.

Consequently

limffdtt

.

-ffdv

.

Then,

by

([3],

p. 209, Theorem

3),

v

EM/R(o,2;(Z:)).

Sincez:is6andnormal,

2;(

separatesZ:.

Hence,

since

z:

is

countably

paracompact,

z:

is

2;(Z:)-countably

paracompact.

Consequently

v

.M/R(o,,).

Examples.

(1).

Consider

any topological space

X

such that

X

is

countably paraeompact

and normal, and let 9

r.

Then

M/R(o,2")

issequentially closed

(in M/R(.9")).

(2).

Consider

any

topological

space

X

such that

X

is

T

and

countably bounded,

andlet

-.

Then

by

([16],

p. 268,

Lemma

10),

2semiseparates2". Also,notethe condition

"X

is

countably

bounded"is equivalentto

"9

is2;

countably

bounded."

Consequently

M/R(o,.9")

is

sequentially

closed

(in M/R(.9")).

The following

Lemma

willbe needed onseveraloccasions.

Lemma

4.2. Ify_._is6and,semiseparates t,, then for

every

net

inM/R(o,),

(l,

for

every

element of

M/R(o,.),v,

if

limit-

v in’T,thenfor

every

element of

t,,F,

li’-

I(F)-:

v’(F),

(equivalently,

for

every

elementof

(t)’, U, lim(Ix).

(U)

=,.

v,(U)).

P..0..qt.

Assume

L is15and Lsemiseparatest;. Consider

any

netin

M/R(o,f),

(to),

and

any

element of

M/R(o,L),v,

suchthatliraI.q vin,2r. Consider

any

elementof

t,,F.

a)

Showv’(F)-inf(v(L)/L

.fandL

DF}.

Since/;

is6andv_M/R(,),v’(F)-inf{v(L’)/L

andL’

F}.

Consider

any

element

of,L,such thatL’

2)

F.

ThenF

L

.

Hence,

sincesemiseparates tL, thereexistsanelement

of/;,,

such

that

2)

F

and

tL

.

Consider

any

such/.

Then

F

C/

C

L’.

Hence

inf{vCE)//

EL

andE

F}

sv()

sv(L’). Hence

inf{v(/)/

andL’

D

F}.

Consequently

inf{v()/

z:

and

F}

s

v’(F).

Further,

note

v’(F)

.

inqvg)/

z:

and

F}.

Consequently

v’(F)-inqv()/

:

and/Y,

DF}.

b)

To

show

li- p(F)

v’(F),

assumethecontrary,

namely,

assume

1- I(F)

>

v’(F).

Then,

by

the result of part

a),

li-- p(F)

>

inf{v(L)/L

.

and

L

2)

F}.

Hence

thereexistsanelement of

,L,

suchthat

L

F

and

li-

(F)

>

v(L).

Consider

any

such

L.

Then,since lim I.h vin

’T,

li--

I.h(L)

v(/.,).

Con-sequently

li-’

I(F)

>

li--’

I.(L).

Further,

notefor

every a,

since

L

2

F,

p(L)

p(F).

Hence

li--p(L)

.

li--

p(F).

Thusa contradictionhas beenreached.

Consequently

1" I.(F)

-

v’(F).

Theorem4.3. Consider

{,;x X}

and denoteit

by

D(,).

If

(z

is6 and

z:

semiseparates

tz:,orfor

every

subset of

X,

S,

if

S

is

countable,

then

S

),

and

,

is

separating, normal,

and

disjunctive,

then

D(L)

issequentially closed in

M+R(o,z:).

Outlineofaproof.

Assume

(z:

is6andz:scmiseparates t, or for

every

subsetof

X,

S,

ifSis

countable,

then

S z),

andetc. Since

z:

isdisjunctive,

D(z:) CIR(o,).

Consider

any sequence

in

D(L),

(p),

and

any

element of

M/R(o,),v,

such that lim

,

vin

,r.

Since is

normal, by

Corollary

3.7,

IR(:)

isclosed.

Consequently

v

IR(o,.).

Consider

{x,x,z, ...}

and denote it

by

A.

Assume

for

any

twovalues ofn,i,j, if/j,then xi

x.

Case I.

There exists an element of

X,

y,

such that for

every

element of

,L,

if

y

L’,

then

A f(L’

{y

})

,,

.

Consider

any

such

y.

Then v

.

Consequently

v

IR(o,.).

A f(L’

{y

})

O.

Then

ct)

v*(A

0 and

[) A

t/;.

)

Since

A

t/;and lira

I

v in’/’and

(,

is6and semiscparates t/;, or for

every

subset of

X,

S,

if

S

is countable, then

S

/;),

by

Lemma

4.2,

li--l.(A)

ffi

v’Ot).

Note

li--,(A)

1.

Consequently

v’(A)

1. Thusacontradictionhasbeenreached.

Consequently Case

IIdoes not occur.

Examples. Consider

any

topological

space

X

such that

X

isnormal and

T1.

(1).

Let/; _9-.Since is 6and9- semiseparates

t9-(-

9-)),

and 9"is

separating, normal,

and

disjunctive,

D(r)

issequentiallyclosed in

M/(o,9-).

(2).

Let/; 2;.Since

(2;

is6and2;_{semiseparates}t2;

(- 9-)),

and2;isseparating, normal, and disjunctive,

D(2;)

is

sequentially

closed in

M/R(o,2;). ([20]).

The following three

Lemmas

will be needed in obtaining conditions under which

M/R(,/;)

is sequentially closedin

Lemma

4.4.

For every

element of

M/R(ff,),v,

if thereexists a

subset

of

X,

Xo,

such that

XoN/;

is Lindel6fand

v’(Xo)

v(X),

thenv

M/R(%/;).

Proof. Consider any element

ofM/R(,/;),

v.

Assume

thereexists asubset

ofX,

Xo,

such

thatXo

f"l is Lindelfand

v’(X0)-

v(X).

Consider

any

such

X0.

To

show v

_M/R(,,/;),

consider

any

netin

sch

that

(L,

isdecreasing and

limLa

O

and showlira

v(L)

0.

ct)

Since

v’(X0)

v(X),

X0

is v-thick.

(See

12],

pp. 74,

75.) RecallS(X0

CI/;)

Xo

CIA(/;)

and consider the function v which is such that

DVo-(XoCI/;)

and for

every

element of

Xo

CIA,

vo(Xo A) v(A).

Recall Voiscalled theprojection of v

toXo.

Note

for

every o.,

v(L)

vo(Xo

Since

OLo}

is decreasing,

(X L}

is decreasing and, since

limL,,-

,

lim(X

NL)-

f.

Further,

note lira

v(L)

lira

v0(X

o

CIL).

[B)

Note

v0

-M/R(o,XoCI/;).

Since

XoCI/;

isLindelOf,

M/R(o,

XoCI/;)CM/R(,XoCI).

Conse-quentlyv0

-M/R(*,XoCI/;).

?)

Consequently limv0(X0

CIL)-

0.

Consequently limv(L)-0,

andv

M/R(x,/;).

Lemma

4.5. If/;

’s

6,

thenfor

every

elementof

M’R

0:,

/;

),

p,

for

every

subsetof/;,

{L;A },

if

{Lct

EA

}

isa filterbase,then

p’(L,),

inf

p(L).

([18].) (For

thespecial caseof/;-2;ina

T3{_

space,

see

[20]$

Lemma

4.6. Ifatopological

space

satisfiesthe countablechainconditionand isparacompact, then it is LindclOf.

(Well-known.)

Theorem4.7. If/;is5and/;semiseparatest/;,t/;isparacompact,and/;isseparating and disjunctive, then

M

/R

(,,/;)

is

sequentially

closedin M/R

(o,/;).

(For

related theoremswith

,

2;,see

17].)

Proof.

Assume

Lis 5and/;

semiseparates

_{t/;,t/; is}

paracompact,

_{etc. Consider}

any

sequence

_in

M/R(,/;),(),

andanyclement of

M/R(o,/;),

v,

such that

limlA,

-vin’T. Show v 1. Show

IOS(l,)

is LindelOf.

ct)

Show

US(p,,)

satisfiesthe countablechain condition. Consider

any

subset of

(t,)’,

{0;x

CA

},

such that

{O,S(,)0;ct

EA

}

is

disjoint

and

A

and for

every

OS(p,,)f"10,,

,

O.

Show

{OS(p,,)f0;o,

EA}

iscountable.

For

every n,

consider

{c-A/S(l,)O }

and denote it

by

an nsuch that

S(la,’)N0a.

Consideranysuchn. Then

ctCAn.

Hence

ctC

UAn.

Consequently

A

U

A,,.

Note

foreveryn,

{S(la,’)

fq

0,;a

CA,"

}

_{is disjoint and foreveryct,if}ct

CAn,

then

S(Ix,’)

fq

0,

,

;

hence,

since

S(t,’)

satisfiesthe countable chain condition

(because

In,"

CM+R(v,)

and/;is

separating

and

dis-junctive),

{S(la,’)

f30,;a

CAn}

iscountable;hence

An

iscountable.

Consequently A

iscountable.

Hence

{US(t,’)

f3

0,;et CA

}

iscountable. Thus

US(t,,)

satisfiesthecountable chain condition.

[)

Since

US(t,’)

Ctz;andt/;isparacompact,

US(p,,)

isparacompact.

Hence,

since

US(p.n)

satisfies thecountable chaincondition,by

Lemma

4.6,

US(p,,)

is LindehSf.

2. Show

v’(\

S

(,’))

v(X).

SinceU

S

(,’)

Ctand lim

n

v in’Tand is5and semiseparates

/

tL, by

Lemma

4.2,

li-lx(?S(la,))v’(?S(lxk)).

Note

or

every n,

Ix,(S(I.q))’:l(

US(I.q));/_,

since

CM/R(r,)

and,is8, by

Lemma

4.5,

la:(S(lan))--tXn(X);

consequently

n(X)"=

Ix’,(?S(Ix,)).

Hence

li--,’(X)

,"

,/l(S()]"

Since

limnn

-v in

’T,

limn(X)-v(X).n

Consequently

v(X).

l,n

,(S(g)).

Hence

v’(,US(n))-v(X).

3.

Consequently

OS(n)ff-,

is LindelOf and

v’(US(n))-v(X).

Hence,

by Lemma

4.4,

\n /

vC

M+R(x,,).

Thus

M+R(x,.)

issequentiallyclosed in

M/R(o,,).

Example. Consider

any

topological space

X

such that

X

isparacompactand

Tx,

and letz;

y.

Since

.q-

is andsemiseparates

ty(=

5-),t5

is paracompact,and .q" is separatinganddisjunctive,

M*R(’,99

is sequentiallyclosed in

M+R(o,y). (This

result is alsotruein normalmetacompact

spaces;

see

[17].)

Section

f;.

In

this section conditionsare obtainedunderwhichtightness impliesrelativecompactnessin the

weak*

topologyand vice versa.

Theorem$..1. Ifis and semiseparates t,and isseparating, disjunctive,andnormal,thenfor everysubset

ofM/R(o,,),A,

ifAistight,thenA is

w*-relatively

compactin

M/R(o,L).

Proof.

Assume

is6and scmiseparates t,and isseparating,

disjunctive,

andnormal. Consider

any

subsetof

M/R(o,),A,

suchthat

A

istight. Then,

by

definition,

A

isnormbounded.

Hence

there exists apositive number, k,such that foreveryelement

ofgn(z), p,

if

p CA,

then

pll

k. Considerany such k.

Now,

consider

{p

C

MR

(,)1

Pll

<

k}.

Note

A

C

{p

e

MR

(,)1

oil

k

).

Hence

"

C

{p

MR

(z,)AI

oil

).

a’hefollowingfact is well-known:

{p

C

MR

(,)1

oil

iscompact.

Hence,

since the

topology

w"

is

T, {p

MR(:)/[

Pll

isclosed.

Consequently

"

C

{p

MR(:)/II

oil

-).

Con-sequently

X

iscompact. Thereforetoshow

A

is

w-relatively

compactin

M+R(o,,),

it suffices toshow

A CM+R(o,L).

Case A

.

ThenA

CM+R(o,L).

Case A

,

.

Since L is 6 and

normal, by

Proposition

3.9,

M+R(,)

is closed.

Hence,

since

A

CM+R(o,,),A CM+R(L).

Consider any element of

A,v.

Then v

CM+R().

Hence

to show v

CM+R(o,,),

it suffices to show for

any sequence

inZ;,

(Ln),

if

(/-,n)

isdecreasingand

limL,,

,

then lim

v(L,’)

0. Considerany

sequence

inL,

(Ln),

suchthat

(Ln)

isdecreasingand

limLn

.

Now,

consider any positive number,e. Since

A

istight, by definition,there existsanelementof

ft.,K,

such that for

every

lim

,,

v. Consider

any

such

<l.t.

Note

for

every

(). (K’)

<e. Since isseparating, disjunctive,and

nrmal,

K

C t.

Consequently

K

/tL.

Consequently

K

tL_{and lim th}

v,

and,C,is6andfsemiseparates tZ.

Hence,

by

Lemma

4.2,

v.(K’)

lim(g0.

(K’).

Consequently

v=.(K

’)

s

.

Since

,)

isdecreasingand

IimL,-O,

NL,-O.

HeRceKN/NL,

/

-0.

Hence,

since(L,>

is

decreasing

andK6

thereexistsa

\, /

value of

n,no,

such that forevery n,if n

no,

then

K

L,,

O;

equivalently

L,

C

K’.

Consider

any

such

no.

Thenfor

every

n,ifn:-

no,

then

v(L,)

:

v.(K’)

.

Hence limv(L,)

0.

Consequently

v

_M/R(o,.6).

HenceA

C

M/R(o,).

Consequently A

is

w’-relatively

compact

in

M/R(o,f).

Example. Consider

any

topological

space

X

such that

X

is

TI

and normal, andletZ 9". Since 9"is

and.7

semiseparates

tg-

9-),

and9-isseparating, disjunctive,andnormal, for

every

subset of

M/R(o,9-),

A,

ifA istight, thenAis

w’-relatively

compact

inM/R(o,9").

Thefollowingtheorem alsogivesconditionsunder whichtightness impliesrelativecompactness.

Theorem

5.2. Ifiscountably paracompact, separatinganddisjunctive, 6 and normal,thenfor

every

subset of M/R

(o,Z), A,

if

A

istight,then

A

is

w’-relatively

compact

inM/R

(o,).

(Proof

omitted.)

Examples.

(1).

Consider

any topological space

X

such that

X

is

countably paracompact,

T1,

and normal,and let Z-9". Then foreverysubset

ofM/R(o,.7"),A,

ifA istight, thenAis

w’-relatively

compact in

M/R(o,).

(2).

Consider

any topological space

X

suchthat

X

is

T,

and let -2;. Then for

every

subsetof

M/R(o,2;),A,

ira istight, thenAis

w’-relatively

compactin

M/R(o,2;).

([20],

p. 205, Corollary

HI.)

(3).

Considerany topological space

X

such that

X

is

Tt,

and let

-.

Thenfor

every

subsetof

M/(o,),A,

ifA istight,thenA is

",’-relatively

compactin

M/(o,).

The followingtheoremgivesconditions under which relativecompactness implies tightness. Theorem 5.3. Consider the condition:

For every

elementof

K,

there existan element

of,,L,

and an element of

6/,

such that

K

C

L’ C/.

(C)

If

,

is

separating

anddisjunctive,satisfies condition

(C),

is

strongly

measure

replete,

6 and normal, thenfor

every

subset

ofM/R(o,f.,),A,

ifA is

",’-compact,

thenA istight.

Proof.

Assume

,

isseparating and

disjunctive,

satisfies etc. Consider

any

subsetof

M/R(o,,), A,

such that

A

is

",’-compact.

Assume

A

,,

O.

c)

Show

A

isnorm bounded.

Note

for

every

elementof

A,

It,

t,II

-pO0-fld.

Considerthe element of

Cs(,),f,

which is such that

f-1.

Then,

by definition

(see

p.

’3),

for

every

element of

MR (f.,),

Ix,

f(tt) -ffdtt.

Since ris6andnormal,

f

is",’-continuous.

Hence,

since

A

is

,,’-compact, f(A

isbounded.

Consequently

A

isnormbounded.

I)

Showfor

every positive number, e,

thereexistsanelement of

96K,

such that

for every

element

of

A,

(X-

K)

<e. Consider

any positive

number e.

Now,

consider

any

element of

A,

Since isstronglymeasurereplete,

t

M

/R

(t,).

(See

Theorem

3.15.)

Hence

thereexistsanelement of

K,

suchthat

(X K

₀

<e. Consider

any

such

K.

Since satisfies condition(C),there exist anelement of

,L,

andanelement of

6,

such that

/

C

L,’

C/,.

Consider

any

such

L,/’,.

Since r isseparating, disjunctive, and

normal,

K

Ct.

Since is

strongly

measure

replete,

it ismeasure

replete.

Consequently I

-M/R(,,). Hence,

since is

separating

anddisjunctive,

by

([4],

Theorem

2.5),

thereexistsanelement of

M/R(,t),l,

such that

ix/0:)-

i and

I

isunique.

Moreover,

_I1

I"

on t:.

Consequently

I

TOPOLOGIES ON THE SET OF LATTICE REGULAR MEASURES 693

Observation. Consider

any

elementoff_,,L. Since is6andnormal, for

every

element of

Cb(,),f,]

is

w’-continuous.

Now,

considerthe function

V

determinedby

V

inf{f/f

_

Ca(f,)

and

f-.

rL.

By ([9],

p. 85, Corollary

10.4,

(b)),

V

is

w’-upper-semicontinuous.

Hence

{v

MR()/v)

<

e}

is

w’-open.

Note

for every element of

MR(,),v,V(v)-inf{f(v)/f_Cb()

and

f":L};

further,

note if

v0,

then, by Alexandroff’s Representation Theorem,

inf{f(v)/f_Cb(f)andf..}-v(L).

Consequently

{v

_M/R(.)/v(L)

<

}

is

w’-open.

By

this observation,

{v _M+R(o,)N(L,)

<

e}

is

w’-open.

Since

p(L,)

<,!

E

{v

M

R

(o,)/v(L,)

<

}.

Denote

this set

by

W,.

Note

{W:lx

EA

}

is an

open

cover of

A.

Hence,

sinceAiscompact, thereexists asubcover of

A,5,

such thatS is finite, and nonempty

(since A

is

nonempty).

Consider

any

such,$ and denote it

by

{W;n

1

/}.

Also,consider

LJ{/,;n

1,

...,/}.

Note

U{/,;n

1

/}

.

Denote

thisset

by/.

Now,

consider

any

elementofA,

.

SinceAC

U{W,;n

/},

there exists a valueof n,

no,

such

L

that

E

W,

o.

Consider

any

such_n0.

Then/

E

and

(X-/)- h(X-/’)

Ix(X-/,o)

’

h(

X-

,o

/)

Consequently

A

istight.

Remark. This theorem is a mildgeneralization ofaresult of Varadarajan

([20],

p. 205,

Theorem

29)

andimplies readilytheProhorovtheorem.

Examples.

(1).

Considerany topological

space

X

suchthat

X

is

T1,

locally compact,

normal, and

.9"

is

strongly

measure

replete,

and letf-.9". Thenfor

every

subset of

M/R(o,.7"),A,

ifA is

w’compact,

then

A

istight.

Remarks.

a)

Allconditionsaresatisfied,ifXis

locally

compact,

T2,

and LindelOf.

(See

[4],

p. 1516,

Application

2.)

b)

Allconditions aresatisfied,ifX islocally compact,

T2,

and paracompactand

separable.

(See

[4],

p.

1516, Application

2’.)

(2).

Considerany topological

space

X

such that

X

is

T2,

locally compact,

and2;is

strongly

measure replete,andlet Z:-2;. Then for

every

subset

ofM+R(o,Z,),A,

ffAis

w’-compact,

thenAis

tight.

Remark. Conditionsfor2;tobe

strongly

measure

replete

arefoundin

[6], e.g.,X /o(W(2;)).

Thefollowingtheorem is ageneralizationofTheorem5.3.

Theorem 5.4. If isseparatinganddisjunctive,measure

replete

and

Cech-complete

(implying

in particularthatf_._is

strongly

measure

replete)

(see

[4],

p.

1516),

thenfor

every

subset

ofM/R(o,f.,),A,

ffA is

w’-compact,

then

A

istight.

(Proof omitted.)

Remark. In theconcretesituation of

Tychonoff spaces,

withf-br,these resultscan be found in

[13]

and

[19].

Section6.

The followingtheorems describe arelationshipbetween

convergence

in

’T

and

convergence

in

’,

convergence

in

,

or

convergence

in

T’.

(For

the definitions of

,

and

’T’,

see

p.

4.)

[Denote

the

general

element

ofM/R(L)by It.

The followingthreefunctions,associated with

p,

occur in the theoremsjust mentioned,

namely,

and

it’.

For

their definitionssee

([4],

p.

1501 and

p.

1508).]

Theorem6.1.

For

every

net in

M/R(),(tto),

for

every

elementof

M/R(),v,

lim

p-v

in ’Tiff lira

-

in

.

(Proof omitted.)

Corollary_ 6.3. If is

countably

compact,thenfor

every

netin

M/R(,),

OX),

for

every

element of

M/R(,),v,

lim

-

vin ’Tiff

limit-

in

.

(Proof omitted.)

Theorem6.4. If is

(separating),

disjunctive,

6,

andnormal,thenfor

every

netinM/R

(),

0x),

for everyelementof

M

/R

(), v,

lim v in ’Tiff lim in

’.

Proof.

Assume

is

(separating),

disjunctive, 6,andnormal. Consider

any

netin

M/R(,),

Oto),

and

any

element of

M

/R

(),

v.

a)

Assume

lim

v in’T. Since

,

is

normal,

tW()

isnormal.

Consequently

tW()

is6 and normal.

Hence ’J’is

the

weak"

topology.

[See

(Theorem

3.13,

d)).]

Thereforetoshow

limit-

in

,

it suffices to showfor

every

elementof

C(tW()),g,

limfgd(.t

-fgd:v.

Since is

(separating),

disjunctive, 6, and normal,the function which

maps

the

general

element of

Cb(,),f,

onto the element of

C(tW(.)),f,

which is such that

DI-IR()

and for

every

element of

IR (),

k,]()O ffd.

is onto.

(^

isalso avector-space

isomorphism

andnorm

preserving.)

Consequently

to show

lim(t-

in

’,

it suffices to show for every element of

Cb(L),f,

lim

f

fd

f

fdft.

For

this

purpose,

considerthefollowing

Lemma.

If is

(separating),

disjunctive, 6,andnormal, then for

every

element of

M/R(),

for

every

elementof

Cb(),f,fd.-ffd

(Proof

omitted.)

Consider

any

element of

Cb(),f.

Note

for

every

bythe

Lemma,

j’Jdt

-ffd.

Since is and normal, ’/" is the

weak"

topology.

[See (Theorem

3.13,

d)).]

Therefore, since limlg,-v in

limff

d

-ff

de.

Consequently

limff

d

-ff

de.Again

by

the

Lemma,

ff

dv

-f

d.

Consequently

limf]d-ffdYv.

Consequently

limt-

in

.

b) Assume

lim in

.

_{Then lim th v in}

.

(Proof omitted.)

Theorem

6.

For every

net in

2t/R(),(l,

for

every

element of

M/R(),v,

limp-v inq’iff lim

I’

v’

in,ie.

(Proof

omitted.)

ACKNOWLEDGMENT

Theauthorwishes to

express

hisappreciationto

Long

Island

University

for

partial

support

of the presentwork

through

agrantof released timefrom teachingduties.

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