Internat. J. Math. & Math. Sci
VOL. 15 NO. 4 (1992) 70|-718
701
MEASURES ON COALLOCATION AND NORMAL LATTICES
JACK-KANG CHAN
Norden Systems
75 Maxess Road
Melville, New York 11747, U.S.A.
and in revised form April 15, 1991
Received January 29, 1991
ABSTRACT
Let
1
and
2
be lattices of subsets of a
nonempty
X.
set
is a subset of 2" We show that
suppose 2 coallocates I and
any ,-regular finitely additive measure on the algebra generated
be uniquely extended to an
can
by
algebra
by
generated
in
contained
,
2"
-regular
The case when
,
measure
is
not
as well as the measure enlargement
on
the
necessary
problem
are
Furthermore, some discussions on normal lattices
considered.
and
separation of lattices are also given.
KEY
lattices, normal lattices, coallocation lattices,
WORDS
semi-separated lattices, regular finitely additive
measures, u-smooth measures, measure extension,
measure enlargement.
1980 MATHEMATICS SUBJECT CLASSIFICATION
INTRODUCTION
1.
Let
of
28C15, 28A12.
X.
X
be an arbitrary set and
If
,c ,
additive
finitely
and if
land 2are lattices
then any
coallocates
,,
measure on the algebra generated by
of subsets
,-regular
,
can
be
uniquely extended to an -regular measure on the algebra generated
by
This situation has been investigated by J. Camacho in
2.
We extend his results in several directions in this paper.
consider
the
case where
lis
below
for
We will
not necessary contained in
Theorem 3.1) and show that under suitable conditions any
definitions) gives rise to a
v6
MR().
(see
peMR()
We
[2].
will
(see
also
702
J.K. CHAN
measure
enlargement
consider
besides measure extension problems,
problems
(see e.g. Theorem 3.3) and will finally apply these
to the case of a single lattice
[8]
Szeto
,
results
M.
thereby extending results of
for measures on normal lattices.
We begin by giving some standard lattice and measure theoretic
Section
background
in
consistent
with
[1,4,6,7,9].
is
terminology
and
notation
Our
2.
In Section 3, we consider
the
coallocation theorem and a variety of consequences of it.
general
Section
is devoted to a more detailed discussion of normal lattices
4
to
that
[3].
of G. Eid
2.
This work extends to some extent
separation of lattices.
and
BACKGROUND AND TERMINOLOGY
section,
this
In
theoretic
we
some
summarize
notions and notations.
lattice
and
measure
This is all fairly standard
and
as previously mentioned is consistent with standard references.
2.1
Definition
X
Let
be a nonempty set and
e(X)
is a collection of subsets of
lattice
is the power set of
X,
, .
which is
X
finite unions and finite intersections, and
-=
’
L’
e
closed
’
Let
is a lattice if
DeHnHon 2.2
,I
and
2be
(i)
is
(2)
is a complement generated (e.g.) lattice
if it is closed under countable intersections.
e
YLe., HL1,L
(3)
such that
is a normal lattice
VL
(4)
L.6ff,
LIL=
L
if
Iln=xLn’.
if
=
is a countaly paraoompact (c.p.) lattice
V
L,L.
e, LIDL2D...
I’ I’’’"
3
(5) S
X.
any lattices of subsets of
is
6
s.t.
limn_L.= (Lni{)
Yn, L
nC.’
and
.
if
=
.’I
Sl-countably-paracompact (SI-c.p.)
if
A
under
Le
where L’ denotes the complement of L.
Let
X.
is.
MEASURES ON COALLOCATION AND NORMAL LATTICES
B 2,
eZ2
9AI,A2,...
s.t.
semi-separates
if
AB=
B
A6, Be2,
separates
s.t.
Vn,
,6,
BnCA
s,
if
L, cLuL,
where
function defined on the algebra
v AeA(Z), #(A) >
A
BcIand
s.t.
,eS
A finitely additive (f.a.) measure
ACB=O =
A’
2.3
Definition
(I)
and
if
oalloat.
L,eS,
Bn
BIDB2D...
6
703
(2)
0,
#()
is a finite nonnegative
generated by
,
such that
0, and (3)[finite additivity]
/.(A) +
p(AUa)
measure
0-I
A()
p
is a two-valued
taking value either 0 or i.
Usually,
simply refer # to as a measure on a lattice
we
that # is a finitely additive measure defined on the
A f.a. measure # defined on the algebra
(i)
-regular iff
o-smooth on
v A,A
(3)
e
A(),
o-smooth on
V
L:,L
The
A()
M()
MR(
M ()
Mu(
6
,
inf
,(A)
Or, equivalently,
(2)
v A e A(),
A(.),
,(A)
(’)
-
m
,6M(w)
algebra
mean
A(.).
is
,(L)
LcA, Le ).
’DA, {.6
).
.(A)O
as n--=
(Ln)0
as noo
notations for the collections of
will be used throughout
, ,
to
iff
L:DL.D..., (Ln)
following
.
iff
ADA,D..., (A)
,
A()
sup
measure
additive
finitely
.
f.a. measure on
A()
-regular
(pEM(S)
p
o-smooth on
(,eM()
,
u-smooth on
A()
measures
on
J.K. CHAN
704
we
Similarly,
IRo()
,
(.EM(.)
M (s)
A (s) and S-regular
I(s), I.(s), I(), Io(s),
o-smooth on
define
also
and
for non-trivial 0-i measures.
implies o-smoothness
If p is S-regular, then o-smoothness on
A(.). Thus, M(.) M.()Mo(.’) M(.’)M(s).
A(), we have M(s’) M(s) and I(S’) I(S).
Since A(’)
is o-smootho. A(e) (,eM()) iff, is cou.tablyadditive.
Furthermore,
on
,
.
.
i,2 e
Let
M(s).
(1)
i<
(2)
I< 2
and
,,.EM(s,).
of
"2t A
I:
Denote
I
A(I)
from
if
i
measure
from
#21
,(L’)
-<
_< .(L’)
21
on
i
and
Definition
X
such that
to mean the restriction
A(sl), then .is called
A(2) (or, less precisely,
on
to
2);
,21)
(or simply
a measure extension of
from
it
.);
and
a
2 e M R(s2)
I(X)
2(X),
A(I)
enlargement of i from
I to
YLe,
lattices of subsets of
regular measure extension, if
If
I(L) < .(L)
YLE,
if
on if,
Ic 2are
Suppose
"I
I(A) < .(A)
2.4
Definition
If
AEA(),
if
on ’,
p1<_
(3)
Define
to
2is
then
A()
(or, less
and a regular measure enlargement, if
a
called
precisely,
2E
M.().
2.5
e(X)[0,),
A real-valued function
is called a
finitely
supera4itive inner measure, if
, (()
()
o
(2) [no.decreasing]
V ACB C
X
A.B c X,
(3) [/inite superadditivity]
/ (A) _< /z (B)
AFB=
that is,
/ (AuB) >_ /z (A) + / (B)
e(X)[0,m),
A real-valued function
/T
is called a finitely
subaItive outer measure, if it satisfies (1), (2)and
(Y) [iqnite subadditivity]
Let
set E c
X
A.B c X,
AFB=
/ (AUB)
be a finitely subadditive outer
-measurable,
p=(TE) + (TnE’),
is said to be
#=(T)
/ (A) + / (B)
measure on (X,). A
if
YTc
X
_<
MEASURES ON COALLOCATION AND NORMAL LATTICES
705
We have the following theorem characterizing a normal
as a special case of the
lattice
coallocatio, property
THEOREM Z. 1
YL6 Sot. LC
is normal
L;uL’,
where
L1,L 2
=
Proof:
II
II
Suppose
LI,L2eff
LInL2=..’.X=LuL.
and
LIL,:, LIcI, L,c,
Thus, when
Let Le and
LcLuL
where
ave
we
L,L.
(LL)(LL)
L(L;UL
are disjoint.
By noality,
(L-L)C
:=,
Bl,ze
(L-L:)(L-L{
and
,u,=
Then
(L,)o(L,)
== LC
and
or
:=.
Consider
#.
such that
,u=X.
Define
L(ilu,)
LX
LO(L’UL?)
(LL’)U(LL)
LCL(L’uL)
(LL’)u(LL)
assumption,
Then by
L.
-
is normal.
L-L and L-L,
L-L: aria L-L
(L-L?)c,
Now,
(LnL?)
(LoL)
L?
cL
C
The following results are obvious
(1)
ff is normal
(2)
1
,
le coallocates itself
=
separates
Furthermore,
,
Ie semi-separates
ff coseparates
itself.
2"
we have the following measure theoretic
terization of a normal lattice
THEOREM 2.2
ff
iff
is normal
THEOREM Z. 3
Suppose
c
LP.
Then
z coseparates
ff =
ff2
coallocates
Proo/:
Suppose
L,C IuI,
where
LeZ,,
,eZ.
Then,
charac-
J.K. CHAN
706
;-L,n{., and ;-L,n,,
Define
= LI
. .
;,1"e,.
And
L{L)
LI{LI)’
C
C
Thus,
hence
;c
Similarly,
Hence
THEORES 2.4
H 0(’)
countablyparacompact
c
o()
Proof:
Suppose
E.sw,
c.p.
L.
NOW
,
t
L
s.t.
.(L.)
L
L.c,
, SO().
0
M(’)
Hence,
M O().
c
MEASURES ON COALLOCATION LATTICES
3.
In
this section we extend some of the work of
unique
the
e
n
MR(2)
that
it
belongs
extendability
,
where
and
the
are lattices of subsets of
=oallocatlon
lattices in order for the
and
to
a
MR(,)
of a measure # e
is not always necessary to assume that
to
[$]
c
main
theorem to hold (see Theorem 3.1).
X.
[2]
measure
we
nor
on
note
that
results
X
of
the
We first define
two
functions which form an inner-outer measure pair.
.
Definition 3.1
,
Suppose
For all
EcX,
and
are lattices of subsets of
#.
(E)
A(E)
We have the following
Let
e
M(I).
,
and
# e
define
and
THEOREM 3.1
X
m
EDL1, L,e,
SUp
/(LI)
inf
(.(L)
[Coallocatio. t%eorem]
and
We have
be lattices of subsets of
X
EcL
.
L,eff,
suppose
M().
(1)
(2)
(3)
MEASURES ON COALLOCATION AND NORMAL LATTICES
..
707
is a finltely superaddltlve inner measure
=
2coallocates "I
ff2coallocates ffl
.
(4)
I. is finitely additive on
."
o,,
X
In particular, if
e,
a.d
then
(X)
.(X)
(X)
o,,
s
(5)[a]
’
is a fintely saddtive outer measure
(6)
suppose
coailoces
(7)
suppose
coailocates
If either [a]
z
is -geasuEle
c
or [b]
z
semi-separates
then
eve
[1"]
X
Ec
I
[2"
is
element of
-measurable
is a fl.itely additive measure on
()
" efl().
"
is
3"
[4"]
Proof:
-regular on
() The proof is standard and is therefore omitted.
Let
(2)
A,B2 e
M2 coallocates
B
A, ,B e
(L,)
(AlUB 1)
(A 1) + (S 1)
S
sup{
M(AI)
AD A,
+ SUp{ M(B1)
BDB,
Taking sup on the left hand side,
sup(
.
.
is finitely subadditive on
additve on
S.
Together with (I), #. is finitely
(3)
The proof is also standard and is omitted.
(4)
Let
Le
,’(L’)
_<
,. (LI)
[i]
708
J.K. CHAN
Ae
Now if
.
_<
(L’)
Lc A,
and
.
inf
(A’)
then by monotonicity of
L’c A’,
A).e
.
sup{
-"
z
[i] and [ii]
If
X esl,
e, X=’
If
L,I
Let
..(A)
"
Suppose
.’. if
iI
ic 2,
Ac A,
L, c
.
eMR(Z,)
.’.
Now,
V
inf
(L,)
_<
inf
inf
<
.’.
(6)
""
on
then
,."
’(T)
TC
I,
X,
where
L,c
A’
A, A 6
(LI)
..
Ae Z,
(A)
(A,’)
Ae,
inf
Tc
.(A)
hence
L,cA, A..
LIcA’, AI
ACAI)
(*.*
L,cA:,
and (5)[a]=
s.t.
()
_< / on
,, "
Ec
we have
on
,.
X,
A, Ae
TcA, A,eS,
(4)
..(At)
Taking inf,
.’(At)
Z ’(AE) + ’(AE’)
z .’(TE) + .’(TE’)
"
(T)
_>
"
(TE) +
is finitely subadditive
/’(T)
(X)
A, e S,
sup
(,(A,’)
Suppose
Suppose
Now
L c
(A)
(A)
(A)
"
.
’(X)
(X).
,(X).
(A)
,()
then
Suppose
#
[ii]
XDLIe-I)
(L,)
on
s.t.
A,
(L’)
(L,)
(5)[b]
But
(X)
.
inf
Taking inf,
i.e.
and
and
sup
.(X)
,. "
e
A(X)
consequently,
(5)Is]
LI= X,
take
.
’(Tr(EuE’))
<
"
(by assumption)
(’" TeAl,
(TE’)
A(TE) + ’(TrnE’)
."
T)
[iiil
[iv]
MEASURES ON COALLOCATION AND NORMAL LATTICES
#" (THE) + #A (THE’)
#A (T)
[iii] and [iv]
which is the definition of
(6)
""
But
/(TnE)
’(TnE’)
/
X,
A-measurable.
E to be A-measurable, we have
By the definition of
’(T)
YTc
X,
is finitely subadditive, the above is equivalent to
>
(T)
(TnE)
In particular, take
(7)
E to be
Tc
709
t
Le
T
s coallocates
suppose
Le S.
YTcX,
(TnE’)
/
,
we have
S,.
To prove that
L is
’-measurable, we have to show,
by (6), that
VA62,
let
P,Q6
AL
P c
A
(.’. Q6) s.t.
Thus,
Pc
Now,
PuQ
and
PnQ c Pn (AoP.)
a
Qc
and
(AOL)
C
U
A P’
Q c
and
A
(AnP’)
sup ((Q)
A
.(P) + ,(Q)
D
C
A
PuQ e S
PnQ=#
,* (A)
.(P) +
.(A)
(7)[a]
Z
..(AInP’)
.(P) + ,.(AnP’)
Suppose
[v]
c
If
AP’
(AzUP)
(P) +
(A)
Z
(AnLz)
sup ((P)
B’(AL) +
We conclude, from (6), that
A()
A()
c
AL
(PcL)
DPe) + (AL)
"(A L)
eve
VAeM1
element of
,-murable se ).
"
.
on
)
is -measurable.
By a standard
Carathodory
710
J.K. CHAN
argument,
/’I/,
..
Suppose
L2 e
.(L)
is a finitely additive measure on
2
(L)
by (4))
sup ((L,)
S
But
L
L,c
sup
Hence,
L,c L, L, eS
(L I)
L,c L, L, eS,
’(L,)S ’(L), taking sup
(’(L) L. cL, LeS)
sup (’(L,)
’(L)
L, cL,
sup
"
which means that
S-regular
on
S.
is
Suppose
o.
Now any element of
Ui, (Ai n Bi’
Conseently,
S,-regular
"
e
A
S e
.
A(S)
A(,
.’[,
argument,
"2 semi-separates if,.
c
/’-me.asurable sets ).
-< ."
L]6]}
_< sup{
]
is #’-measurable.
By a standard
is a finitely additive measure on
.’(,)
/(L,):LICL2’
is also
MR(S,).
Taking sup,
sup(
"
S,
and
_<
SIc
is, of the fo
Caratheodory
A
L E "1
.(L.)
’(L)
Since
We conclude, from (6), that every element of
A(,)
by (5)[a])
"
semi-separates
# on
a’,
.
MEASURES ON COALLOCATION AND NORMAL LATTICES
B" (L z’)
But
is
Note
If
(LI). LICL
sup{
-regular
eace.,
-
.
then
then
,,
,
and
coa//ocates itse//,
is finitelyaddit, ve
and
^(x)
X
2=
(1)
A
(2)
," (L) +
Proof:
and conseently,
trivially semi-separates
Corollar.y 3.1
Suppose
Hence,
L,eS
-reguIar on S,
and
o,
’
e
711
.
(7)[a]
is norrnal).
(.
(X)
(7)[b].
(X)
(I) Direct consequences of Theorem 3.1.
(2) From Theorem 3.1(4),
Now
But
^’(X)
is finitely additive,
’(LuL’)
,. (X)
(X)
.
A(L) + ’(L’)
(X),
.’.
,’(L) +
^(L) +
(X).
(L’)
The coallocation theorem leads to the following direct
conse-
quences whose proofs are omitted.
THEOREM 3.
Suppose
e c e.
there exists a unique
.
Furtheore,
TBOR[M
u
MR(S,),
s.t.
Suppose
MR(e),
e MR
MR(,)
s.t.
is S-regular on
S, c S
THEOREM 3.4
u6
6
(el).
and
If
on
S.
I,
coallocates
,
then
ul16 MR(I).
IS
Note that
Regular measure enlargement on coallocation lattices
Suppose
ue
."
Regular measure extension on coallocation lattices
and
6
S u on
S(s,).
S
and
If
(X)
s coallocates
s
then
P(X).
[Regular measure enlargement on a normal lattice]
is normal and
eM().
< u on le and
s.t.
Furthermore,
Then there exists a unique
(X)
u(X).
if we impose a o-smoothness condition on
obtain the following
THEOREM 3.5
Suppose
e c
e
and
!e.
coallocates
e,
and
,
M (e,),
,
we
712
J.K. CHAN
MR(2)
v
.
v is the regular measure extension of
where
Then
Proo/:
,
"
.
"
, ,
on
A.c B
,
Since
A. lO.
Tus,
(A.)
Hence,
"
conseently,
fl is
.
regular measure extension of
A.c,
A.,
-regular
[Theorem 3.1(5)[b]]
suoe
[
--0
a
V
"
(B)
S,= S=
regular measure enlargement of
B
V
0
.., na
ts
is o-smooth on
may
assume
S,
<
(B)
3.5
.
V> O,
without the loss of generality, we
In particular, if
Corollary
unite
a(A.) + /2
<
[i]
hence by Theorem 3.2,
-regular on
1, since
a’(s:)
MR()
6
is the
is
Theorem 3.(7)
v
,
M(,)
S, 6 MR(S,)
is
e
S
noal, we
ave
,(). .(),
,
S v on
,
fl (X)
ts ehe
u(X)
Then,
(’).
Suppose
,
is
ve
MR()
$1 C $
and
coallocales
o.., o=, ..d .o.
where
v is the
unite
1
and
Suppose
e
.(,),
and
regular measure extension of ft.
(e.).
hen,
Prool:
Thom3.
.’. v
eo(), (fi:)
is o-smooth on
We
S,
--0
and since
(S.)
--0
is regular on
now give two applications of the results
lattices to topological spaces.
VS.e.
S, ve M(S).
on
coallocation
713
MEASURES ON COALLOCATION AND NORMAL LATTICES
1)
MEASURES ON A LOCALLY COMPACT HAUSDROFF SPACE
Let
X
all compact
of
tion
be a locally compact
X
K 0 or K, unless
cause
while
is compact.
K coallocates
that
6-sets,
K 0 is compact.
Thus,
collection
of all
does not belong to
either
Then K 0 c K
For any
K0
X
6
the collec-
K o is
is the
K
2
Note that in this case,
compact sets.
,=
Hausdroff space and
M R(K0)
, 6M(K0).
coallocation
the
which is
uniquely to a regular measure v
theorem, we can extend
be-
o-smooth,
is
By
be shown
can
and it
also o-smooth, because K is compact. Hence, v 6 O(K).
2) MEASURE ENLARGEMENT FROM ZERO SETS TO CLOSED SETS
MR
X
Suppose
Let
z
is a countably paracompact and normal topological
3 (zero sets) and
,
normal.
3 c
disjoint
closed
,
e2
because all zero sets are
sets
can be separated
3 is c.p. and normal.
Therefore,
, coallocates 3 [Theorem 2.3].
Let
MR(3
,
That is,
(closed sets)
closed
by
is
6-sets,
disjoint
Thus, 3 coseparates
zero
,.
extension
sets is
3-regular.
MR(,)
e
Theorem 3.1(7)[a] implies
c.p.
and
sets.
Hence
Then by Theorem 3.2, there exists a unique
measure
space.
on all
regular
open
MRs(3). By Theorem 3.5, the unique regular measure
extension is 6 MR(, nSa(,’). Now , is c.p., hence
[Theorem 2.4]. Then, v MR(,). This is the result of Marik [5].
Suppose
4.
6
NORHAL LATTICES
In
lattices
this section, we give further characterization
and
further consequences of a lattice
being
terms of associated measures on the generalized algebra.
Def n t on 4.1
Let
Ec
X,
be a lattice of subsets of
X,
and
p
M().
define
inf
,’(E)
inf
/"(E)
p.(E)
m
sup
y’(E)
-=
inf
It is clear that if
Ec’, 6
--z/(E.’) Ec U.__,ln, {....
,(’)
, MR(),
then
,
,’ on
A().
of
normal
normal
in
714
J.K. CHAN
THEOREM 4.1
,,u e
Let
.
M(s),
/z < u on
Proof:
Iz < u < u’<
< u on
It is obvious that
Ec’
u(’)
E,
.’.
I"
u
’on
on
<
on ’.
inf(,(’)
’e’
In particular, E e
u’S
S
THEOREM 4.2
I (),
e
Suppose
’(LI)
L 1,L e
Let E c
X
s.t.
),
’
on
.
Hence,
.,
I
’(L2)
and
i
Then
Taking inf
.
u’(E) S ’(E).
S u S u’S
,(’)
<
’e’
inf(u(’)
u(X)
,(X)
such that
’(LnL)
1
is .ormal.
Then,
Proof:
is not normal.
Suppose
I (s),
3,
.’. 3
Li,L2e
,
,(Lz)
NOW if
<
l,,e
LlcL
%(L,)
1
L,C ;’,
u, on
u
Similarly,
,
I.(),
<
u,()
,
,,
ul(L-)
and
then
on
u
({,’)
{,
S, then
’(L=)
gives a contradiction.
’ (L;nL=)
But
i.
u.
,
on
%(L)
but
u,
1
’ (LI)
, ’ (L;L=)
1
i.
I.
L;nL2
.’.
is normal.
Consequently,
THEOREM 4.3
uM(), ,oM(),
Let
’
on
< p 6
u_
(1)
p < u
(2)
is
u’
MR (’
_< p’
normal
p’ on
u= u’
Proof:
u_< p on
(I)
’
by Theorem 4.1,
(2)
Suppose
p < u on
p < u
,
u’< p’ on
.
is normal and
and
.
u
MR()
3 L s.t.
(
.’.
u({’)
By normality,
< u(L) + z
3
L.,L b,
and
Then
on
=
p(X)
u(X)
s.t.
and
s.t.
u=
u(L)
+
u’.
<
>
L
Lc
L;,
,.
Since
I.
we have
Then, by assumption,
<
0
on ’,
L.c
if
,<
s.t.
0
{, S,
Then
c L{,
L;CL{=
Hence
p’(L).
=0
__
MEASURES ON COALLOCATION AND NORMAL LATTICES
L
.’.
L’
C
LB
C
’
C
p(m.’) _< P(Lb)
p’ (L) _< p’ (n’)
(n) <
p’(L) < u(L)
(I’
(Lb)
gives a contradiction.
u’
/
and
e
u
715
< u(L) +
on S.
THEOREM 4.4
Let
<- IZ"
(I)
I
(2)
’=
X,
S be a lattice of subsets of
Mo(s
Then,
erywhere
on S’
",,(X)
<
(3)
<
(4)
(X)
(5)
.(L’) +
(6)
If
S is
(7)
If
S is 6-normal, then
/’ on
S
’(X)
(X),
’(L)
<
normal, then
e
The condition
NOT
YLeS
"
<
" ’
So(S
’=
onS.
is imposed, because when
is not o-smooth, then
measure and if
on S
Proo/:
"
is a
0-i
m 0.
(I) By definition of ", the inf encompasses more sets than that of
’
" ’
<
hence
’
(2) Take E
,
(X).
and
’(X)
Taking the
"(X)
#"(X) _< (X).
X
,(Li’) < ,(X), but
’
inf
and
,
U,,Li’,
,,(Li’
L’; X,
or
of the above, we have
" (X)
B LES,
In particular,
(X)
For suppose
consequently,
Now suppose
on S’
we have
U, L’ e
Since
’=
eS’
(3) From (I)and [i],
"(X)
everywhere.
pairwise disjoint Li, E,
lim,=,,(,)
LI
(X)
,
6
u()
[ii]
B(L) > #"(L),
p"(L) + 9(L’)
(X)
"
also
"(X) a (X).
(v "_< on S’)
< /(L) + p(L’)
<
,
"(LuL’) < "(L) + "(L’)
-<
.’.
We now show that
on S.
(4) [iland [ii]
by assumption)
contradicting [ii]
Together with (I), we have
/(X)
/z"(X)
/’(X).
<
" ’
<
on S.
716
.
Ye > 0,
(5)
J.K. CHAN
()
<
(L’)
L’
where
e-%,,
D
(X)
Taking inf, we have
/(X)
D.,
L’
If
>
.(L’)
(X)
/z.(L’) >
/()
>
/(L’)
since
< / on 2’’
/.
(’)
()
(X)
<_
.(L’)
[iii]
/F(L)
Taking inf, we have
/(X)
/.
[iii]and [iv]
=
<
(L’)
/(X)
inf
(#(’)
E -%’,
.
/’(L)
(6) L is normal, then Corollary 3.1(2) and (5) imply
,= A
Now
" ’
LcUn=IL.* L.2".
<
(7) From (I),
Then
Let
Di
_<
on 2", and from (3),
IL.,
.
LnD=
A
_< #.=
on 2".
"
2"is
-%’,
is normal
L2".
/F(L)
3 L e 2", s.t.
Suppose
everywhere.
[iv]
#’(L)
J
VLE2".
/F(L)
/z.(L’)
’
LC
(L) < #’(L)
3A,BE2", s.t.
LEA’, D cB’, A’nB’
/’ (L) <
/F(A’
<
/z(A’
<
/(S)
/" (B)
fmm
Lc
Taking inf, and since
/F(L) < inf{
__l/z(L)
gives a contradiction.
U._-I L.*,
LC__IL, L.2"
J
#’(L), or
"(L)
#"(L)
#’ on 2".
#"
I
THEOREM 4.5
Let
a lattice of subsets of
-%’, be
_<
s.t.
If
on
-%’,
is
Let
L
c.p.
-%’,
normal
or,
L.
-%’,,
LocA.’e.c.’e,
on -%,,’;
B
ne
Proof:
""
Suppose
-%’i c -%’,2,
-%’1 normal
(.)
_<
(A.’) _< (^.’)
eo(2"))
and
-%’i
and
M(2"),
M o (-%,,).
p
Vn
p(Ln)
THEOREM 4.6
Suppose
p
(X).
(one may assume, with the loss of generallty,
(’." p_<
Mo(2"),
and let
countably paracompact and normal, then
Proof:
-%’,’
;,(X)
2",
X,
-%’I
separates
-%’,2
normal.
is normal.
-%’,2
Then,
_<
Bnl
0,
.(e.)
0
).
or
peso(2").
I
MEASURES ON COALLOCATION AND NORMAL LATTICES
Let
u[
Extend
,
and
separates
.,
""
to
Since
.,
.’. 1
(L,)
For
%(L)
B
But
THEOREM 4.7
sppoe
p(X)
_
1
L,,
separates
IR()
but
1
s.t.
.(L)
Since
H.(..),
u is l-regular on
2
V L; e, u(L;)
sup(
is
u(LI)
Sl-regular on
sup(.(.)
S
L In
LI
O.
,LI, %(,)=
and
Taking sup,
A S fb on
,.(n,)
S:.
S, separates S
el, s.t.
LICL, ,C,,
onx.
LcLxcL
u.(L,)
rbon
Then
uEMR(),
= ,,
S
Similarly,
p
%, "b e
and
LxE,, s.t.
,
.
A S f. on
e,
’,., .d
.on.,
to
Ub
A(L)
A(L,) -,(L,)
(X),
Proof:
i.e.
and
B
unite.
.. .
Le s.t.
%eIs(,),
.
noal,
ub
the extension is
separates
.on S:.
.’.
u.=
to
We now show that
suppose
bn
[Theorem 2.2].
., I.(.),
.,
I(),
AE
.on :e,,
# _<
s.t.
Ub
is noal.
respectively.
B
,
I(.),
Extend
Ix(:e,),
v,I
Ub[
Suppose
t
L
%
x" 1 noal
on
"."
.,
I(e,.),
#
717
D.ES.
S:
S
is noal,
Then
718
J.K. CHAN
ACKNOWLEDGEHENT
I express my gratitude to my teacher and dissertation advisor,
Professor George Bachman of Polytechnic University
for
I
(Brooklyn, NY),
his guidance, encouragement, and proof reading of this
would
also like to thank the two Referees
for
their
paper.
valuable
comments.
REFERENCES
[1] A.D. Alexandroff, Additive set/unctions
i,
abstract spaces, Mat. Sb. (N.S.), 9($1), (1941),
563-628.
[2] J. Camacho, Jr., Extensions of lattice regular measures with applications, J. Indian Math. SO.,
54, (1989), 233-244.
[3] G. Eid, On normal lattices
a,d ;Vallman spaces,
Internat. J. Math. & Math. Sci., 13,
No.l, (1990), 31-38.
[4] P. Grassi, On subspaces of replete and measure replete spaces, Canad. Math. Bull., 27, (1984),
58-64.
[5] J. Marik, The Baire and Borel measures, Czech. J. Math., 7, (1957), 248-253.
[6] G.
Nbelin8, Grundlage, der A,alytische, Topologie, Springer-Verla$, Berlin,
1954.
[7] M. Szeto, Measure replete,ess and mapping preservations, J. Indian Math. Sou., 43, (1979),
35-52.
[8] M. Szeto,
0, maximal measures with respect to a lattice,
Measure Theory and Applications,
Proceedings of the 1980 Conference, Northern Illinois University, (1981), 277-282.
[9] H. Wallman, Lattices and topological spaces, Ann. of Math., 39, No.l, (1938), 112-126.