707
Internat. J. Math. & Math. Sci.
Vol. i0 No. 4 (1987) 707-724
ON MEASURE REPLETENESS AND SUPPORT FOR
LATTICE REGULAR MEASURES
GEORGE BACHMAN
Department of Mathematics
Polytechnic Institute of New York
Brooklyn, NY 11201
U.S.A.
P.D. STRATIGOS
Department of Mathematics
Long Island University
Brooklyn, NY 11201
(Received
ABSTRACT.
October 31, 1984)
The present paper is mainly concerned with establishing conditions which
.assure that all lattice regular measures have additional smoothness properties or
that simply all two-valued such measures have such properties and are therefore Dirac
measures.
These conditions are expressed in terms of the general Wallman space.
The
general results are then applied to specific topological lattices, yielding new conditions for measure compactness,
Borel measure compactness, clopen measure replete-
ness, strong measure compactness, etc.
In addition, smoothness properties in the
general setting for lattice regular measures are related to the notion of support,
and numerous applications are given.
-
KEY WORDS AND PHRASES. Support of a measure, repleteness, realcompaetness, and
completeness, mease completeness, measure compactness, and Borel measure compactness,
clopen measure repleteness, strong measure repleteness, strong measure compactness, etc.
1980 AMS SUBJECT
i.
CLASSIFICATION CODES.
28A60, 28A32.
INTRODUCTION.
In an earlier paper [5], we obtained conditions for o-smoothness, T-smoothness,
and tightness of lattice regular measures. This was done in a general framework for
a set X and a lattice of subsets of X, i, which was just disjunctive and at times
separating.
which are not
The general approach was adopted so as to fit many topological lattices
6
or not normal.
This approach was made possible by utilizing gen-
eral lattice regular measure extension theorems (see
[4]).
Thus it was possible to
bypass the general Alexandroff Representation Theorem [2] in which a delta normal
lattice is needed. The results were then expressed in terms of IR(i)-X, where
IR(i)
is the general Wallman space associated with the set
X
and the lattice
These results generalized known results pertaining to Baire measures and
where 8X is the Stone-ech compactification of the Tychonoff space X.
[.
8X-X,
In particu-
lar, our general approach lead to new results pertaining to smoothness and tightness
G. BACHMAN and P.D. STRATIGOS
708
T
of closed regular Borel measures in just
mX-X, where
of
mX
X
is the Wallman compactification of
and also to clopen regular
BoX-X,
spaces expressible in terms of
o-dimensional T
Borel measures in
topological spaces expressible in terms
BoX
where
X.
is the Banaschewski compactification of
In the first part of this paper we utilize the framework of the previously mentioned paper and obtain new results for lattice repleteness, measure repleteness and
strongly measure repleteness.
We then apply these results to specific topological
lattices and obtain new conditions for measure compactness, Borel measure compactness,
and clopen measure repleteness and similar facts for strongly measure compactness,
strongly Borel measure compactness, and strongly clopen measure repleteness. (See, in
particular, Theorems 2.4, 2.6 and their consequences and associated examples.)
It is advantageous to be able to characterize various repleteness properties in
We pursue this in general in the second part of
terms of support of certain measures.
the paper.
L
We cite here just one of the more important results (see Theorem 3.3):
L
is separating and disjunctive, then
o-smooth, L-regular measure (which is not the zero measure) is nonempty.
has many applications.
If
is measure replete iff the support of every
This result
Thus, in this part of the paper, we concentrate on various as-
pects of support of a measure.
2.
TERMINOLOGY AND NOTATION.
Most of the terminology used in the present paper goes back to Wallman [10]
I.
Some of the more recent terminology appears inNoebeling [7]
and Alexandroff [l], [2].
For the reader’s convenience, in this part we
and Frolik [6], as well as in [5], [8].
will collect some of the special terminology which is used throughout the paper.
X
Consider any set
of
X
L
generated by
L
ated by
measure on
is dnoted by
A
o{LI.
M(L).
Igl(X)}
<An>,
if
L
<An >
M(L),
for
,
M(L),
in
L
is decreasing and
is denoted by
L-(T-smooth) iff for every
(L)
0.
net in
lim
n
M(L)
M(o* L).
L,
gener-
A.
X,
g
S().
A{LI, E
L
M(L) which
L-(o-smooth)
An
’
then
< L
>
A{L
An element of
Ig(E)-g(L)l
E
and
is
M(L),
L-regular
< e.
is denoted by
iff for every sequence in
n)
lim g(A
n
L-(o-smooth)
M(L) which
An element of M(L) g
>
is de-
for every positive num-
which is
if < L
A
is
the support of
The set whose general element is an element of
< A >
n
then lim
R, such that
to
such that
is said to be
whose general element is an element of
M(o,L).
A
and is denoted by
L
there exists an element of
An element of
X
whose general element is a measure on
The set
The set whose general element is an element of
MR(L).
A{LI,
The algebra of subsets
Next, consider any algebra of subsets of
L-regular iff for every element of
g, is said to be
L.
The o-algebra of subsets of
For the general element of
L/Ig l(e)
n{e e
fined to be
ber, e
A{LI.
is defined to be a function, g, from
bounded and finitely additive.
is denoted by
X,
and any lattice of subsets of
is denoted by
0.
The set
is denoted by
is
o-smooth just
is said to be
is decreasing and lim
L =4,
The set whose general element is an element of
is said
L-(-smooth) is denoted by M(,L). An element of M(L),
an
exists
there
number,
g
M(o,L) and for every positive
to be L -tight iff
M(L)
which is
L-compact
set,
,
K, such that
Igl,(K)
<
The set whose general element is an ele-
SUPPORT FOR LATTICE REGULAR MEASURES
M(6)
709
6-tight is denoted by M(t,6). The set whose general
such that (A([))
{0,i}, that is, the set of 0-I
element is an element of (6),
measures is denoted by I([).
6 is said to be replete iff whenever an element of 1(6)
belongs to
6 is said to be prime complete iff whenever an element of
IR(o,6), then S() #
6 is said to be measure replete iff
I(6),
belongs to I(o*,6), then S() #
MR(o,[)
MR(r,6). [ is said to be strongly measure replete iff MR(o,6)
MR(t,6).
ment of
which is
,
.
.
Next, consider any topological space
0,
its collection of open sets by
Z.
tion of zero sets by
X and denote its collection of closed sets by
In case
X
is replete.
X
is said to be s-complete
C
is replete.
X
X
X
M(6),
elements of
II.
,,
and its collec-
is replete.
X
is said to be N-
is said to be measure compact iff
M(6)
C
F
Z
is
is measure replete.
is measure replete.
is expressible as the difference of nonnega-
without loss of generality, we shall work with nonnegative
M(6).
Among the principal tools utilized in the present work are three measures in-
M(6), denoted by
duced by the general element of
and ’)
,,
and certain criteria for
which are expressed in terms of
(these measures are denoted by
o-smoothness,
’.
or
r-smoothness, or tightness,
(See [5].)
,
For the reader’s convenience, in this part we collect thedefinitions of0,
’
Z
is said to be realcompact iff
is said to be Borel measure compact iff
Since every element of
rive elements of
F
iff
Moreover,
is said to be clopen measure replete iff
NOTE.
X
T3,
is
compact iff
measure replete.
C,
its collection of clopen sets by
and
and we summarize (in the form of a theorem) the principal facts pertaining to the
criteria mentioned above.
Preliminaries.
that
<
6
is separating and disjunctive.
IR(6), tW(6)
and [9]).
>
Consider the function
IR(6).
identified with
It is known that the topological space
T
it is
2
iff
6
.
Consequently
Then
is a
IR(6)
<
t6, tW(6)
I}
W(A).
#
Moreover,
X.
Moreover,
2.
For every two
W(A u B)
)
8)
3.
A{6), A, W(A)’
elements of A{61, A, B,
For every element of
B)
W(A
y)
If
A
6)
e)
If
W(A)
A(6)
{W(L); L
A
B
iff
W(B),
W(A)
W(A)
then
is
6}
is denoted by
W(A’).
W(B).
A{w{L)
6
(X)
A.
W(B);
A
B;
(Note all these statements are true, if
and
by
W(A) u W(B);
W(A)
W(B);
B, then
X
In case (X)
Proposition I.I.
I.
is
> -homeomorphlsm.
IR(6).
Denote the general element of
is denoted by
X.
is a compactificatlon of
is said to be embedded in
Definition of
(See e.g., [4]
is normal.
is topologically identifiable with
X, X
{
IR(6)/(A)
w(L).
x
X, 6, such
and any lattice of subsets of
which is such that the domain of
X, x, (x)
For this reason, (X)
(i)
TI;
is compact and
for every element of
dense in
X
Consider any set
is just disjunctive.)
Then
is
G. BACHMAN and P.D. STRATIGOS
710
M(L),
Next, consider any element of
which is such that
and the function
(A).
is A(W(L)) and for every element of A(W([)), W(A), (W(A))
E MR(W([)).
Conversely, consider any
MR(L), then
M(W(L)) and if
is
and he function
element of M(W(L)),
whic is such that the domain of
M(L) and
(W(A)). Note
A{LI and for every element of AILI, A (A)
MR(L).
E MR(W(L)), then
and if
Note since W(L) is compact,
MR(t,W([)).
MR(,W([))
MR(o,W(L))
MR(W([))
MR(o,W([)). Hence
MR(W([))
Then
Next, consider any element of MR(L),
the domain of
Note
E
6W(L)-regular.
(ii)
’.
I}
sets, uniquely, and the extension is
for this extension.
Continue to use
Definition of
IR(o,f)/(A)
{
*-measurable
o-algebra of
is extendible to the
A{LI
Denote the general element of
Wo(A}.
is denoted by
{Wo(L);
Moreover,
A.
by
L}
L
Then
is denoted by
w ILl.
If, in each statement of Proposition I.I,
REMARK.
W o, the
is replaced by
W
resulting statement is true.
M(L),
Next, consider any element of
’
’
A(Wo(L))
Note
M(Wo(L)) and if
any element of M(Wo(L)),
the domain of
(A).
consider
domain of
M(L)
then
is
A{LI
is
’
and
MR(o,L)
e
MR(Wo([)),
’
e
Conversely,
which is such that the
A{LI, A
o(Wo(A)).
(A)
MR(L).
then
’(Wo(A))
(A),
MR(Wo(L)).
and the function
and for every element of
and if
’
MR(L), then
which is such that
Note
MR(L),
e
Moreover, if
MR(o,Wo(L)).
Definition of
(iii)
Consider any set
Lemma I.i.
such that
iff
E
’
A(Wo(L)),
and the function
and for every element of
L
MR(L2)’ 2’
c2"
For every element of
such that
21 A(L I)
i
MR(LI), I’
and if
L
[I’
X,
and any two lattices of subsets of
X
there exists an element of
L 2, then
separates
2
is unique.
(See [2].)
Next, consider any set
disjunctive.
X
.
and any lattice of subsets of
Consider any element of
MR(L),
Lemma I.i, there exists an element of MR(tw(L)),
Then
X,
L
is
Hence, by
IA(w(L))
uch that
L
such that
MR(W(L)).
and
is unique.
tW(L), because W(L) is compact,
tW(L) is compact,
MR(,tW([))
MR(t,tW(L)).
MR(tW(L)) MR(o,tW(L))
is extendible to the u-algebra of
MR(o,tW(L)). Hence
Consequently,
measurable sets, uniquely, and the extension is tW(L)-regular. Continue to use
W(L)
since
separates
Note since
*-
for this extension.
THEOREM 1.1.
t
Consider any set
is.(separating) and disjunctive.
I.
g E
*(IR(o, L))
MR(o,L)
iff
(IR([)).
O*(X)
X
and any lattice of subsets of
For every element of
O(IR(L)); equivalently,
X, L, such that
MR(L), u:
E
MR(o,L)
iff
711
SUPPORT FOR LATTICE REGULAR MEASURES
2.
MR(,L)
3.
’ MR(T,wo(L))
4.
L
If
(IR(L))
iff
*(X)
(IR(L)).
*(IR(o,L))
iff
(IR(L)).
T 2, then
is also separating and normal, or
X
and
L,
iff for every sequence in
"
i
<
>, if
iff
0"
"
W(L i)
X)
or to
is decreasing and
gi
*(X)
MR(o,L)
e
MR(o,L)
IR(L) -g
(For more details refer to [I].)
parts.
3.
< L
e
Similarly, equivalent statements are obtainable for the other
0".
[W(Li))
*(X)
iff
*-measurable.
is
We not for example, that the statement of part I,
(IR(L))" is equivalent to "p e MR(o,L) iff ,(IR(L)
then
MR(t,L)
p
NECESSARY AND SUFFICIENT CONDITIONS.
In this section we work with an arbitrary set
X
L
L
b)
L
X,
sets of
such that
sufficient conditions for
and an arbitrary lattice of sub-
is separating and disjunctive and we give necessary and
to be
a)
LindelSf,
replete, c)
measure replete,
d) strongly measure replete.
a)
LindelSf property.
The following statements are equivalent:
Theorem 2.1.
i.
L
2.
For every subset of
L,
A,
A*,
is LindelSf.
there exists a subset of
{W(L);
n{W(La); e A*}
A}
IR(L)
IR([)
X
Consider any subset of
L,
A}, if
{L;
such that
X, then
A*
and
countable.
{W(L);
that
L
A*
X.
L
Since
is disjunctive,
Consider any such
A*.
Then
a e A}
{L
A}
{L;
A*
A,
is Lindelf, there exists a subset of
is countable.
Consequently
B)
IR(L)
A}
a
Hence, since
and
Assume i, and show 2.
)
Proof.
A*}
such
CA}
n{L
such that
n{W(La);
.
is
IR(L) -X.
2 is true.
(Proof omitted.)
Conversely, assume 2, and show i.
L
Assume
Corollary 2.1.
Then the follow-
is normal and countably paracompact.
ing statements are equivalent:
of
i.
L
2.
For every element of
is
Lindelf.
Z(tW(L)), K0,
a)
Proof.
K
IR([)
K
{W(L);
X.
tW([), K, if K
K
IR(L)-X.
0
Assume 1, and show 2.
A}.
tW([), there
K
Since
a
K
such that
Consider any such
IR(L)-X, then there
tW([), K, such that
Consider any element of
[,
exists a subset of
A}.
{L;
Then
exists an element
{L
a e
{W(L);
A},
A}
such that
IR(L) -X.
such
by Theorem 2.1, there exists a subset of A, A*
such
X and A* is countable Consider any
IR([)
a e
that
Since [ is normal and countably paracompact, by [5], Theorem 2.2, part 2, there
A*} K 0 IR(L) X. Conexists an element of Z(tW(L)), K 0, such that
Hence,
since
L
is LindelDf
A*
A*
o{W(La);
n{W(L);a
sider any such
B)
such that
K0
Then
K
K0
IR([)
X.
Consider any subset of
Conversely, assume 2, and show
o{W(La)
A}
IR(L)
X.
Consequently 2 is true.
Set
{W(Le);
A}
K.
L,
{L
Then
K
e
A},
IR(L)-X.
G. BACHMAN and P.D. STRATIGOS
712
Z(tW([)), K 0, such
Then, since 2 is true, there exists an element of
X.
IR(i)
tW(/), there
Ln
A}
n[W(L ); a
n{W(La);
;
n)
W(L
Ln
/,
hence
>
a
tW(i)-compact and a
is
0
K
such that
E
A}
n{W(Ln)’
W(i)
since
o{W(Ln)’;
0
N}.
n
a
K
0
G6-set
n E
of
N}.
Hence for every
there exists an ele-
is compact
W(Ln)’;.consider any such n"
n’ such that W(L n
A} and {W(L ); n
N}co{W(Ln)’; n N} K0 IR([)
a
A,
ment of
{L
0.
Then
>"
K
Then, since
exists a sequence in
Consider any such
n
K
Consider any such
K
that
{L
Then
a
n
N}
n
X.
Then, by Theorem
n
L
R. 1,
is LindelBf.
(I).
Examples.
Z.
[
X
Consider any topological space
X,
is
T3 and
is LindelDf iff for every closed subset of
Then, by Corollary 2.1, X
there exists a zero set of
X
such that
KO,
K0
K
such that
let
BX-X, K,
(This result is well-
8X-X.
kno).
(2). Consider any topological space
arid let
of
Then, by Corollary 2.1,
K, there exists a zero set of
80X-X
b)
C.
[
such that
X
is LindelDf
BoX
and O-dimensional
iff for every closed subset
K
such that
K0,
T
is
K
BoX-X
0
Repleteness.
Le 2.1.
Consider any lattice of subsets of
,
([),
(,L).
every element of
I.
2.
X
X
For every net in
X, [
[
such that
is
.
For
the following statements are equivalent:
L
L
>
L
>
A},
if
if
*(nL
is decreasing, then
(n).
inf
3.
a)
V (0L
inf
[
{L
a
{L
=
A}
is a filter base, then
(See [8].)
(La).
The following statements are equivalent:
Theorem 2.2.
i.,
L
For every subset of
is replete.
IR()
X, then there exists an eleV
IR([)
-X.
ment of o(W([)), B, such that V
and consider any element
X #
Proof. a) Assume I, and show 2. Assume IR([)
X. Consequently
X. Since [ is replete, IR(o,[)
Then
X,
of IR([)
>
that < L i > Is desuch
L
IR(,[). Hence there exists a sequence in [
i
2.
For every element of
IR(),
V, if
B
.
creasing and
i,
(Li)
’
limi Li
I.
but
Consequently
limi v(Li)
W(L i)
O.
Consider any such
IR([) -X
and
io W(Li)
<L.1 >"
Note for every
o(W([)).
Conse-
.
.
quently 2 is true.
is replete, it suffices to
) Conversely, assume 2, and show I. Note to show
element of IR(o,[) -X,
any
Consider
X
Assume IR(o,[)
X
show IR(o,[)
IR([) -X. Hence, since 2 is true, there exists an element of o(W([)),
Then
X. Consider any such B.
IR([)
B
B, such that
.
(i)
Since
([), the extension of
sets (also denoted by
)
is
6W([)-regular.
to the
*-measurable
sup{(K)/K W([)
o-algebra of
Consequently
(B)
713
SUPPORT FOR LATTICE REGULAR MEASURES
6w(L),
Consider any element of
B}.
K
and
O.
Hence, by Theorem I.i, part I, (K)
n{W(L)/L
e IR(L), {x,}
(ii) Since
-X.
*({})
2.1,
t
COROLLARY 2.2.
L,
sequence in
L and (W(L))
L
,
(B)
({u})
u
X
IR(o,[)
Consequently
ed.
IR(/)
K
Then
O.
(B)
Hence, by Lemma
I}.
I.
Consequently
(iii)
B.
K
K, such that
Consequently
[
and
is replete.
is replete iff whenever
>, such that
i
L
Thus a contradiction has been reach-
0.
IR([)
b
X, then there exists a
oW(Li) IR(L) X.
’is decreasing and
H
(Proof omitted.)
F,
F
i
>
F
such that
>
i
X
such that
T
is
F.
[
and let
X, then there exists a
nF
X, where
X
i
mX
X is a-complete iff whenever
Then, by Corollary 2.2,
sequence in
X
Consider any topological space
Example.
is decreasing and
the closure is taken in X.
COROLLARY 2.3.
[
Assume
Then the follow-
is normal and countably paracomDact.
ing statements are equivalent:
I.
[
2.
For every element of
is replete.
a)
Proof.
IR([) -X,
[,
quence in
.
K
0
Assume I, and show 2.
L.I >.
Consider any such
IR(L)
Assume
L.
such that
i
X.
8)
Assume
[
Then, since
K
Consider any such
Conversely, assume 2,
IR([)
X
in
tW(L),
every
of
< H
K
Z(tW([)),
an
K
i,
show i.
7(tW([)),
[
H.l
i’
Li’
K
0.
Hence
Then
ing.
K
i
K
ni’
>
is a
H’.
0
K
IR([)
0
[
To show
W(L i)
0
and
K0,
G-set
Z.
Then, by Corollary 2.3,
there exists a zero set of
is known.
Consequently 2 is true.
t|at
1
.
< L
i
>
X
BX, K 0
is
X,
Then, since 2 is
e K
IR([)
0
tW([), there
of
tW(L),
separates
Hi W( i)
H
and
Then for
W( i)
W(L i)
e
such that
realcompact iff whenever
e K
0
BX
Con-
exists a sequence
i
Consequently K
X
X.
there exist two elements
is decreasing and
such that
0
is replete, use Corollary 2.2.
X.
iH"i
W(Li)
0
L.
Without loss of generality, assume that
and
K
.W(Li)
Xo
such that
i; then K0W(Li ) W( i) ’H i’.
is in
[5],
such
Consider any such
W(L)
X.
IR([)
W(Li)
e
KO,
IR()
Hence, by Corollary 2.2, [ is replete.
Examples. (I). Consider any topological space
[
2.2, there exists a se-
i
hence, since
oW(Li)’i
0
< L
’
such that
consider any such
K0
Then, since
such that
i
n
0
K
and consider any element of
0.
and consider any element
is normal and countably paracompact, by
Then
O.
true, there exists an element .of
sider any such
X #
is decreasing and
Theorem 2.2, part 2, there exists an element of
IR([)
X, then there exists an ele-
is replete, by Corollary
Then, since
< L
IR([)
IR([) -X.
if
Z(tW([)), K O, such that
ment of
of
IR([),
0W(Li)
i
X
is
e
>
is decreas-
IR([)
T31/2
X.
and let
BX- X, then
(This special case
G. BACHMAN and P.D. STRATIGOS
714
(2). Consider any topological space X such that X
L C. Then, by Corollary 2.3, X is N-compact
BoX KO,
then there exists a zero-set of
X
X
nH
H’
Consider any such
tW(L).
of
G6-set
L.l >.
any such
L
2.2,
L
L
To show
Hao
o(W(L))
IR([)
i
IR(L)
i)
Since
HO
tW([),
L,
B) of Corollary 2.3.)
e 0W(L
.
X,
A, a O, such that
there exists a sequence in
and
A},
is replete, use
F -set of
is an
a e
Li >
<
Consider
Hence, by Theo-
X.
{Ke;
Z(tW([)),
If there exists a subset of
A},
a
such that
e e A}
{K
tW(L).
a,
K’a
is an
Fo-set
A},
a
of
tW([).
oK’.
X
such that
Z(tW(L)),
K
a, since
Note for every
Hence for every
{Ka;
Z(tW([)),
K
is a
Then, by Corollary
is replete.
c)
Measure repleteness.
W ILl).
’
Next
for the general element of
1}.
(W (L))
.
Consider any element of
’, (L)
by the definition of
S(’)
IR(L),
IR(o,W ([)),
IR(,[)
Note for every element of
Observation.
and
A},
a
is replete.
Consider any such
2.4,
A}.
E
H’O,
e
Proof. Assume there exists a subset of
of
{H=;
{Ha;
tW([),
-sets of
(See the proof of part
0W(L i)
X,
X.
tW(L),
-sets of
BoX
E
is replete.
K’, then
G-set
F
i
COROLLARY 2.5.
X
{Ha;
Then, since
0"
Then, since
Then
BoX
0
Consequently there exists an element of
H’
nW(Li)
i
e
such that
X
K
and consider any element of
X #
uH’
IR([)
is a
rein
IR(L)
nH
0
F
and 0-dimensional
iff whenever
is replete.
Consider any such
Assume
Theorem 2.2.
e
L
Assume there exists a collection of
such that
X
then
0H
Proof.
E
If there exists a collection of
COROLLARY 2 4
such that
such that
T
is
and let
Consequently
L
If
Summarizing:
I.
W (L)
o
L, L
Consequently
’
W(L).
’ (Wo(L))
Hence
’
IR(,
n{W (L)/L
S(’)
note
such that
iff
I.
S(’),
Then,
so
is replete.
is disjunctive, then
W (L)
is replete.
We will obtain a necessary and sufficient condition for
W (L)
to be measure re-
plete.
Preliminaries:
’
such that
According to
MR(,W([)).
[5], Theorem 3.2,
Theorem 2.3.
I.
W (L)
2.
M{L}
Proo@.
show
MR(,L)
Consider the set whose general element is an element of
a)
This set is denoted by
part
R([).
MR([),
(See [5], P. 1517.)
I, PR([)= MR(o,[).
The following statements are equivalent:
is measure replete.
Assume 1, and show 2.
R(L).
Note to show
Consider any element of
R(L)
MR(o,L),
.
MR(o,L),
it suffices to
’
(o,W (L)).
Then
SUPPORT FOR LATTICE REGULAR MEASURES
W (L)
Since
p’ e (,W (L)),
o
MR(o,L).
quently
(L)
B)
asume
Conversely,
COROLLARY 2.6.
I.
R(L).
M(L).
MR(o,i)
Hence
MR(T
Wo([)).
Conse-
Consequently,
(Proof omitted.)
2, and show 1.
The following statements are equivalent:
is measure replete.
[
2.
R([)
is replete and
IR(o,[)
X.
L.
o
is measure replete, [
[
SiTce
W ([)
Consequently,
Wo(L)
assumption,
MR(o,[).
Assume 1, and show 2.
)
Proof.
Hence
p e
so
(o W (L))
by assumption
is measure replete
715
M(L)
Then,by Theorem 2.3,
is measure replete.
is replete.
is measure replete, by
[
Hence, since
MR(o,[).
Conse-
quently 2 is true.
Z.
L
MT(Z)
Consider any topological space
(i).
Examples.
let
(Proof omitted).
Conversely, assume 2, and show 1.
B)
X
such that
X
is measure compact iff
X
Then, by Corollary 2.6,
X
and
T31/2
is
is realcompact and
m(o,Z).
(2).
Then, by Corollary 2.6, X
T
F.
[
and let
such that
X
is
is Borel measure compact iff
X
is -complete and
such that
X
is
X
Consider any topological space
R(F)
MR(o,F).
(3).
and let
C.
compact and
R(C)
is N-
X
MR(o,C).
for every element of
MR([),
For every element of
{W(L);
A}
{W(L );e
and
Since
6W([)
Note
6
is
inf
MR(T,tW([)) and
(W(L
{W(L )-
))
Hence
tW([), K
The condition
"[
tW([)
(K).
tW([), K
L and W(L)
Set {W(L)/L
MR(,TW([)) and {W(Le) e
by Lemma 2.1,
inf (W(L )).
e A}
Hence, by eemma 2. I, inf (W(L ))
e
and
is a filter base
IA{W{[II o
Remark.
K}.
L and W(L)
n{W(L)/L
a e A}.
and any element of
MR([),
Consider any element of
tW(L), K
e
0-dimensional
and
=fi(K).
Proof.
K
T
is clopen measure replete iff
Then, by Corollary 2.6, X
Lemma 2.2
fl*(K)
X
Consider any topological space
_and
6*(K)
tW([)
Now, note
{W(L );
Consequently
inf
e A}
*(K)
Since
K}
A}
6W(L)
(W(Le)).
is
and
is a filter base
(K).
is separating and disjunctive" was not needed in the
proof.
IR([)
,
Observation.
For every element of
THEOREM 2.4.
The following statements are equivalent:
i.
[
2.
For every element of
Lemma 2.2,
MR(,[).
p* e
*.
(Proof omitted.)
is measure replete.
X, then
Proof. )
element of
MR([),
*(K)
MR(o,[),
for every element of
tW(), K, if
K
O.
Assume i, and show 2. Consider any element of MR(o,[),
and any
such that K
IR([)
X
Since K
tW([) and p
MR([), by
*IK (K). Since [ is measure replete, by assumption, MR(o,[)
tW(L), K,
Consequently
Theorem i.I, part 2,
(K)
(MR(T,[).
0.
Hence, since
Consequently
*(K)
K
tW([)
and
K
IR([)
0. Thus 2 is true.
X, by
G. BACHMAN and P.D. STRATIGOS
716
)
Conversely, assume 2, and show I.
MR(o,L)
suffices to show
MR(,L).
[
Note to show
.
is measure replete, it
MR(,L),
tW(L), K
Consider any element of
To show
MR(,L), use Theorem I.i, part 2.
*(K) D(K).
E MR(L), by Lemma 2.2,
tW(L) and
IR(L)
X. Since K
O.
IR(L)
K
X and
tW(L) and K
MR(,[), by the assumption, *(K)
MR(,L). Hence MR(o,L)
0. Then, by Theorem i.I, part 2,
quently (K)
(T, L). Consequently L is measure replete.
Consider any element of
E
such that
K
In this connection, we note
Remark.
L
L2
and any element of
.
IA(LII
*
B*
Then
MR(,L2)
LI,
X,
IA{[)
such that
iff
L2
MR(,LI).
such that
Set
L
on
(Proof omitted.
COROLLARY (). For every element of MR(L),
iff
on
Proof. Consider any element of MR(L),
Now, use Proposition 2.1 with
.
L2
W(L),
tW(L),
*
tW(L).
L
D.
COROLLARY (B).
MR(o,L)
Conse-
following useful result:
.the
Consider any two lattices of subsets of
Proposition 2.1.
Since
For every element of
implies
E
MR(L),
(Proof omitted).
COROLLARY 2.7. If for every element of
exists an element of
*
if
MR(T,L).
(W(L)), B
such that
tW(L), K
K
B
K
D
on
tW(L)’, then
IR(L)
X implies there
X, then L is measure
IR(L)
replete.
Assume for every element of
Proof.
exists an element of
tW(L), K
K
(W(L)), B
IR(L)
X implies there
IR(L)
X. To show L is mea-
such that K
B
sure replete, use Theorem 2.4. Consider any element of MR(o,L),
and any element
of tW(L), K
such that K
IR(L)
X. Then, by the assumption, there exists an
element of (W ()), B, such that K
B
IR(L)
X. Consider any such B. Then
*(K) (B). Moreover, since
E MR(,L)
and B
(B) O. (See the proof of Theorem 2.2, part ).
b
L
Theorem 2.4,
REMARK.
(W(L))
and
Consequently
B
IR(L)
*(K)
X,
0.
Then
is measure replete.
Corollary 2.7 is the measure repleteness analog of Theorem 2.2 for re-
pleteness.
Observation.
o(Z(tW(L)))
Z.
set of
o(W(t)), o(Z(tW(t)),c o(W(t)).
IR(L).)
Z(tW(t))
is the class of Baire sets of
(I).
Examples.
L
Since, In general,
Consider any topological space
If for every closed subset of
BX, B
such that
K
B
BX
X
such that
X
is
T31/2
(Note
and let
BX, K, K
BX X implies there exists a Balre
X, then, by Corollary 2.7, X is measure com-
pact.
(2).
Consider any topological space
X
such that
X
is
T
and let
t
X, K K X X implies there exists a Balre set of
X, then, by Corollary 2.7, X s Borel measure compact.
Consider any topological space X such that X is T
and 0-dlmenslonal
If for every closed subset of
X, B
.
such that
(3).
and let
L
a Balre set of
K
B
X
If for every cosed subset of
BoX,
measure replete.
B, such that KCBC
BO
BOX,
K,.K
BoX-X
implies there exists
X_X, then, by Corollary 2.7, X is clopen
SUPPORT FOR LATTICE REGULAR MEASURES
THEOREM 2.5.
and
The following statements are equivalent:
I.
L
2.
For every element of
is measure replete.
,
A}
n{La;
717
(n{L
MR(u,[),
A, A *, such that
then there exists a subset of
A*})
L,
for every subset of
A},
{L;
if
A* is countable
0.
(Proof omitted.)
REMARK.
is separating and disjunctive" is not needed in the
"/
The condition
proof of this theorem.
d)
Strongly measure repleteness.
THEOREM 2.6.
L
If
is normal (or T
2)
and
a(W(/)), then Lis strongly
X
mea-
sure replete.
Assume
Proof.
[
is normal (or T
2)
X
and
a(U(i)).
[
Note to show
is
(t,/). Consider any eleMR(o,[)
(t,i), use Theorem 1.1, part 4. Since X
strongly measure replete, it suffices to show
ment of
MR(o,[), u.
(W(L)), X
To show
*-measurable.
is
U
*(X)
Moreover,
0(X),
since
(X),
since
O*(X),
*(X)
(IR(L))
T2)
and normal (or
i
Consequently
Examples.
[
Z.
If
X
and
X
is
*-measurable.
Then, since
by Theorem I.I, part 4,
MR(t,[).
is strongly measure replete.
(I). Consider any topological space
is a Baire set of
X
Io(W(i))
[
*-measurable,
X is
since
(IR([)),
5 (IR([)).
*-measurable,
X is
since
Consequently
is separating, disjunctive,
Hence
X
such that
8X, then, by Theorem 2.6, X
MR(t,[).
MR(q,[)
is
T31/2
and let
is strongly measure com-
pact.
(2). Consider any topological space
T
and let i
F. If
2
Borel measure compact.
X
L
C.
If
X
s
such that
a
Bare set of
X
X
is
T31/2
and normal or simply
X, then, by Theorem 2.6, X
is a Baire set of
(3). Consider any topological space
and let
X
such that
BoX,
then,
X
is T
and
by Theorem 2.6,
is strongly
O-dimensional
X is strongly
clopen measure replete.
4. REPLETENESS PROPERTIES.
It is advantageous to be able to
characterize various repleteness properties in
terms of support of certain
measures. In this section we pursue this matter
in general
Consider any set X and any lattice
of subsets of X, [
such that L is separating and disjunctive.
) Repleteness and support.
Preliminaries. Consider the set whose general element
is an elemen of MR(1),
such that whenever 0 e IR(i)
IR(o,[), then there exists an element of
tW([)’, O
such that 0
O and (O)
O. This set is denoted by R([). (See
[I],
p. 1519.)
Next, consider the set whose general element
is an element of MR(L),
such that
whenever 0
IR([)
X, then there exists an element of
tW(L)’, O such that O
O
and (0)
O. Denote this setbyMR(i). Note
MR(L)
# and MR(L) R(i).
G. BACHMAN and P.D. STRATIGOS
718
LEMLMA 3.1.
For every element of
(L), u, p e MR(L) iff S()
X.
}(i), p. Now, consider any element of S(),
p. (Note, since tW(L) is compact, S() # .)
Then, since S()
IR(/), p
IR(L).
Assume p
X. Then p
IR(L) -X. Hence, since u E MR(/), by the definition of
{(L), there exists an element of tW([)’, O such that
0 and 9(0)
0. Consequently, p
S(). Thus a contradiction has been reached. Consequently
S() c X.
8) Consider any element of MR(L),
such that S()
X. Assume IR(L)
X#
and consider any element of IR(L)
S(). Consequently there exists
X, p. Then p
an element of tW(/)’, 0
such that p E O and (0)
O. Then, by the definition of
MR(L).
MR(L), u
COROLLARY 3.1. For every element of MR(L), p
p
MR(L) iff S(p)
S().
Proof. ) Consider any element of MR(L), p. Since }[R(L),
exists and
S(p)
S()
X. Since p E R([), by Lemma 3.1, part ), S()
X. Consequently
s()
s().
e)
Proof.
8)
S(p)
Consider any element of
Consider any element of
X, S()
X.
COROLLARY 3.2.
MR(1),
such that
V
Then; by Lemma 3.1, part 8),
S(U)
MR([).
S().
Then, since
MR(,[).
MR([)
To show U
MR(1),
fl(,[), use Theorem
I.I, part 2. Consider any element of tW(1), K such that K IR(/)
X. Since
p
MR(L), by Lemma 3.1, S() X. Consequently K n S()
Moreover, since S()=
I and (W(L))
o{W(L)/L
(IR([))}, by Lemma 2.1, (S())
(IR([)). Consequently
(K)
0. Then, by Theorem i.I, part 2, p
MR(,[). Hence MR([) MR(,[).
Proof.
Consider any element of
.
Proof.
is replete iff R([)
By [5], Theorem 3.5, part 3, I
X.
By Lemma 3.1, for every element of
THEOREM 3.1.
S()
[
sequently
/
i)
L
If
R(L).
R([)
is replete iff
THEOREM 3.2.
R(/).
is replete iff whenever
MR([),
,
p E
R([)
(Recall that, in general,
is replete, then for every element of
R([),
p
iff
S()
X.
then
Con-
R(/)c R(/).)
M(/),
S(’ )=
s().
2)
If for every element of
1)
Proof.
S()
S(p’)
S()
0
IR(o,[).
S(B).
X
Hence, since
S(p’)
Consequently
assume the contrary.
[
IR(o,[), V, S(V’)
{B}
S(p’)
S()
[
is replete.
M[),
S() n X.
/
is replete,
S(p’)
V
Then
Further, note
. .
IR(o,[)
Then
and
S(V), then
Consider any element of
is replete.
Assume for every element of
2)
Then
[
Assume
S(p).
S(’)
S(). To show / is replet
IR()[), B
Consider any element of IR(o,/)
X #
X,
Thus a contradiction has been reached.
Consequently
is rep.lete.
8)
Measure repleteness and support.
The purpose of the following example is to show that the condition "there exists
an element of
MR(o,/),
such that
S(9) #
measure replete.
sider
L
Example.
Assume
e)
IR(/)
X,
and denote it by v. Since
is not compact.
"
Then
Consider any element of
B
+
x
E
.
is not sufficient for
X #
and any element of
IR(/)
IR(/)
and
x
i
to be
X, x. Then, conIR(L) (because L is
719
SUPPORT FOR LATTICE REGULAR MEASURES
Consider any element of
2. Consequently
+
.
(L) +
S(), and S()
MR(,L).
Next, show
x
quently
X.
L.
Conse-
Px’
p
Then, since
is separating and disjunctive,
[
Thus a contradiction has been reached.
X.
Consequently
Since
x
i, and
+x(X)
(X)
(X)
Then (L)
x(L)
MR(,L).
Assume
IR(,/).
p
Consequently
MR(z,L)
IR(,L)
x(L)
Hence
2.
(X)}.
[/(L)
{L
S()
By the definition of support,
(X).
[, L such that (L)
MR([).
disjunctive),
Consequent-
MR(,L).
ly
8)
[
Assume
a)
(as in part
X, x and
Consequently
MR(,L) (see part )).
any element of
X,
and
MR(o,i),
,
e
MR(o,/),
S()
but
Then
,
IR(o,L)
Consider any element of
IR(o,/) # X.
Then
is not replete.
S()
i
is not
,
measure replete.
IR(z,[).
then v
S()
in terms
repleteness
measure
for
condition
sufficient
and
We will give a necessary
IR(o,[), v
For every element of
Observation.
if
of support.
Consider any set
LEMMA 3.2
MR(L),
any element of
call
w(L)-regular and
is
IR (L), H.
an
X, [. Consider
on o(tW(L)). (Reo(W(/)) and
any subset of
consider
tW(L)-regular.) Next,
and
lattice of subsets of
on
is
Then
,
There exists a countably additive measure on
Case I:
0 < O <
O
Case 2:
0 < O <
X
and the measures
,
W(L)-regular, and
is
p(H)
o(IR(L))
There exists a countably additive measure on
tW(L)-regular, and
is
O
o(H)
o(W(L)), p, such that
(H).
o(tW(L)),
such that
(H).
=p(IR([))
(See [5], Lemma 4.1.)
i.
L
2.
For every element of
is measure replete.
t
Sine
Hence
.
is measure replete
{0},
MR(o,L)
Assume i, and show 2.
)
Poof.
MR(o,L)
S()
{0},
HR(o,L)
MR(z,L) {0}.
Consequently
MR(,L).
MR(o,L)
MR(,L).
,
Consider any element of
S() @
) Conversely, assume 2, and show i.
fices to show
.
The following statements are equivalent:
THEOREM 3.3.
Note to show
[
Consider any element of
MR(,L). Then, by Theorem i.I, part 2, there exists an
K
X and (K)
IR(L)
0. Consider any such K.
there exists a countably additive measure on o(tW(L)), p
*(K). Consider any
tW(L)-regular, and o(K)
O (IR(L))
and the element of M(L), v
which is such that
that
O[A(W(L))
OIA(W{LI
MR(W(L)).
Show
v
such tat
<
W(L i)
MR(o,L).
ei
>
o(W(L))
and
Since
0 <
Uniqueness of Extension
W(L i)
(by
01o(W(L}I
OIo(W(L) <- Io(w(L) ).
Consequently
Then, by Lemma 3.2, Case 2,
such that
such
.
0 < p
-<
,
is
O
Now, consider
0.
0[A(W(L))
Consider any sequence in
IN(L)
since
,
element
Assume
{0},
such
of tW(L), K
Note
MR(L).
Use Theorem 1.1, part i.
is decreasing and
i
p(0.w(el)).
Consequently
is measure replete, it suf-
MR(o,L)
OIo(W(L)
X, and show
.
(w(el))
Uniqueness of Extension),
Further, note
-<
lo(W([))
Consequently
L,
< L
0.
,
i
>,
Note
(W(LI))
i
(W(LI))
i
by
G. BACHHAN and P.D. STITIOS
?20
P(nW(Li))i
(nW(Li))’i
-<
(D.W(Li))
Consequently
.
(IR(i))
s() #
Further, note
tw(L),
A(tw(L))
o(IR(L))
and
.
S(O)
S(v)
.
0
X.
MR(o,L).
u
0.
Moreover,
This theorem generalizes
Remark.
L
0
, (K)
X, S(0) o X
IR(L)
K
Consequently
Thus a contradiction has been reached.
Consequently
([) separates
and
X. Moreover, since
Hence, since
S(0)c K.
0(K),
O]A(w(L))
S(0)
S(9)
Then, by the assumption,
# 0.
(K) # 0,
Also, since
Consequently
MR(,L).
MR(o,L)
*(K)
o(IR(i))
0*(K)
S()
sequently
Hence
Then, by Theorem I.I, part I,
0.
(0W(Li))i
MR(o,L), by Theorem I.I, part I,
U
1
u(X)
since
Since
.
Con-
MR(z,[).
U
is measure replete.
[8], Theorem 2.2, where
it is assumed that
L
is
MR(o,7)
every element of
,
{0}
.
S() #
Consider any topological space
(2).
.
X
such that
Then, by Theorem 3.3, (or by [8], Theorem 2.2), X
Consider any topological space
(3).
C. Then,
and let
[
ment of
MR(o,C)
y)
S() #
{0},
MR(o,)
element of
.
by Theorem 3.3, X
{0},
S() #
X
T31/2
is
and let
is measure compact iff for
Then, by Theorem 3.3, (or by [8], Theorem 2.2), X
Z.
[
X
such that
X
Consider any topological space
(I).
Examples.
X
is
F.
[
and let
T
is Borel measure compact iff for every
such that
X
is
and 0-dimensional
T
is clopen measure replete iff for every ele-
Other properties of support.
The following statements are equivalent:
Theorem 3.4.
I.
[
2.
For every two elements of
S()
then
is regular.
M([),
<
if
[
on
and
(X)
(X),
S().
such
9
Assume i, and show 2. Consider any two elements of M([),
(X), S()
<
on [ and (X)
9(X). Since
< 9 on [ and (X)
that
such that
x
of
element
an
exists
there
S(u),
Then
S(9).
S(B)
#
S(). Assume
{L
[/(L)
(X)}, there exx { S(9). Consider any such x. Then, since S()
Proof.
e)
ists an element of
Then, since
and L
B’
[
[, L
such that
9(L)
9(X)
and
L.
x
is regular, there exist two elements of
i, A
Consider any such
L.
B
A’
such that
x
B’ @. Consider any such A, B. Then L B’ A. Conseand A’
[ and
9(L) < 9(B’) < (B’) < (A) < (X). Consequently A
9(X)
(X)
reached.
been
has
a
contradiction
e
(X). Hence, since x
(A)
S(B), x A. Thus
S().
Consequently S()
8) Conversely, assume 2, and show I. Note, by the assumption, for every two ele< 9 on [, then S()
S(). Then, i is regular.
ments of I([),
9, if
c
quently
(Proof known, see, e.g., 6].)
Remark
I.
The condition "i
is separatlmg and
disjunctive" was not needed in the
proof.
Remark 2.
Observation
measures.
This theorem generalizes a known result on 0
I.
Assume
there exists an element of
[
is regular.
MR([),
9
Consider any element of
such that
<
9
on
[
and
M(i),
iX)
.
Then
(X). (See
SUPPORT FOR LATTICE REGULAR MEASURES
.
Consider any such
e.g.,[10].)
S(W)
is regular, by Theorem 3.4,
i
Then, since
721
s().
assume
M(L),
ment of
(X)
p.
v(X).
MR(o,L).
v
M([),
of
is regular, in
it is permissible to
W
MR(L).
p
Observation 2.
and
element
facts about the support of an
checking
[
Given that
The significance of this observation is this:
Assume L is regular and countably compact. Consider any eleS v
on L
such that
Then, consider any element of MR(L), v
MR(o,L). Consequently
Since L is countably compact, MR(L)
L
Moreover, since
is regular,
S()
S(9).
L
Given that
The significance of this observation is this:
is regular and
M(o*,L),w,it
countably compact, in checking facts about the support of an element of
is permissible to assume
MR(o,L).
W
X
Consider any topological space
Example.
is regular and count-
X
such that
Then, by Observation 2, for every element of M(* F)
such that u
there exists an element of MR(,F), thatis, a Borel measure,
S(v).
and p(X)
9(X) and S()
Observation 3. Assume L is regular, normal, and countably paracompact. Con-
on
F
L
ably compact and let
M(o*,/),
sider any element of
.
MR(o,L), v
pact, there exists an element of
(See [8], Theorem 4.1)
9(X).
S()
Theorem 3.4,
L
Then, since
is normal and countably paracom-
s
such that
Consider any such
v.
L
on
9
L
Then, since
(X)
and
is regular, by
S(9).
L
Given that
The significance of this observation is this:
normal,
is regular,
and countably paracompact, in checking facts about the support of an element of
M(o*,L),
it is permissible to assume
p
MR(o,L).
U
Consider any topological space
Example.
X
X
such that
is
Z.
[
and let
T31/2
there exists an element of
Then, by Observation 3, for every element of M(o*,Z), W
v(X) and
s
v on Z and w(X)
such that p
MR(o,Z), that is, a Baire measure, v
s(.)
s().
Consider any element of
v
p
on
MR(0),
F.
Next, show v
< F >, such that
n
< F
any such
n
>
Fn >
Since
such that
v
Assume
p
on
0
and
p.
and
T31/2
is
Note
M(0).
p
v(X).
p(X)
Then
v
M(*,F).
n)
is pseudocompact, there exists a value of
uF’n
X.
Since
nl --’Fk
X.
Consider any such
ing and
v
X
Then there exists a sequence in F,
0. Consider
0, but lim v(F
is decreasing and lim Fn
n
n
0, < Fn > is increasF
< F > is decreasin and lim
n
n
n
M(*,F).
<
M(F),
Now, consider any element of
F.
L
pseudocompact and let
such that
X
Consider any topological space
Observation 4.
X
such
n, n o
n
that
k
increasing.
note since
--’
limn V(Fn)
MR(, there
sider any such
inim v(Fn)
0,
exists an element of
G
n
then
G
n
F
n
o
Since
and for every
X
Fn0
Consequently,
n
Set
> 0.
0, Gn,
hence
G
n
<
F’n
n > no
n, if
limn 9(Fn)
such that
<
is increasing,
>
.
G
Fn
then
F’n
X.
>
is
Now,
Then for every n, since
F
n
n
F ’; consequently
n
and
v(Gn) > e; con-
G’n
G
n
F’n
hence
G. BACHMAN and P.D.. STRATIGOS
722
’
G’n
if
n, if
Consequently for every
n
n e n
"
Gn
then
o
M(o*,L).
F ’= X.
Consequently
is completely regular, it is regular; equivalently, F
X
is
S().
S(p)
Then, by Theorem 3.4,
X
Given that
The significance of this observation is this:
is
M(F),
compact, in checking facts about the support of an element of
and pseudo-
T31/2
p, it is permissi-
M(o*,F).
e
ble to assume
Hence for every n,
n
Thus a contradiction has been reached.
Finally, note since
regular.
G’n
then
nO
n
The purpose of the following discussion is to illustrate the importance of consid-
ering o-smoothness with respect to lattices.
L
L
If
Lemma 3.3
Assume
Proof.
is complement
Note to show
e
MR(L),
{(’)/
L
and
inf
and for every element of
M(o*
generated, then
MR(L)
L)
L, L
it suffices to show for every element of
’
L}
inf{(’
L
(L)
D
which is such that
Consider the function
L, L, p(L)
M(o*,L’),
Consider any element of
is complement generated.
’
and
L}.
Show for every
(L). Consider any element of L, L. Note (L) N o(L). Now,
show o(L)
(L). Since L [ and L is complement generated, there exists a sequence in
L >. Without loss of generality,
<
Consider any such
L, L n >, such that L nL’.
n
n
L, L
element of
L’n
assume
<
and since
L’n
L’n
>
o(L)
N
Consequently
L
L
If
Assume
Proof.
M(o*,L),
To show
is decreasing and
L,
n
>
is countably paracompact.
0.
Assume
M(a*,L’)
Now, show
e)
Since
8)
Since
Lemma 3 4,
and
L
such that for every n,
THEOREM 3.5.
Proof.
n
lim L
n
n
<
Consider any such
lm (Ln)
and show lim (L
L
L
M(o*,L’)
n
>
L,
L
If
L
O.
L’n
L’
inf{(’)/
L,
limn (L
(L).
Since
such that
< L
n
>
L}.
M(o*,L).
M(o*,L’)
>
Conse-
’
L and
M(o*, L’),
Consider any element of
< L
is in
>
L
M(o*,L’),
L’).
L
<
is in
L
L
’
n
n
<’>
n
and
M(o*,L’),
M(o* L’)
Hence
is complement generated, then
is complement generated.
and
is decreasing and
<
Ln
lm ()=
O.
lim
n
n
Consequent-
M(o* L)
M(o*,L’)
Note in general,
MR(o,L).
MR(o,L)
M(o*,L’).
MR(a,L).
is complement generated, by Lemma
is complement generated,
M(o* L)
.
is de-
>
is countably paracompact, there exists a sequence
Then, since
M(o* L)
Thus
n)
(L
(L) + 0
L’)
i (L’n
is countably paracompact, then
,
e
Then, since
L) +
<
L) +
is decreasing and since
>
consider any sequence in
lim L
n
n
creasing and
ly
L’
Note
>
(L). Thus (L)
(L). Then o(L)
MR(L)
MR(L). Hence M(o* L’)
Lemma 3.4.
.
.
L’n
is decreasing,
(L
(L)
n,
<L’n n L’
Next, consider
L.
(L’n L’) (L) L’
Consequently lm (L)
lm (L
quently
in
L
Note for every
L’)
lim(Ln
O.
is decreasing.
L,
L
Since
>
o(L)
L
3.3,
M(o*,[’)
MR([).
is countably paracompact.
Hence, by
?23
SUPPORT FOR LATTICE REGULAR NEASURES
y)
M(o*,[’)
Consequently
M(o*,L)
MR(L) n
M(o*,[’)
M(o,[)
Note
Observation.
MR(o,L).
MR(o,[)
This result generalizes the following well-known result:
X
X
such that
is
[
and let
T31/2
2.
MR(o,[).
MR(o,[).
M(o,[)
Hence
Consider any topological space
MR(o,Z); expressed otherwise:
M(o,2)
Then
M(o*,[’)
Thus
M(o,L).
every Baire measure is regular.
"[
Note the condition
Remark.
is separating and
disjunctive" was not needed in
the proof.
Finally, we will consider the question of when the support of a measure is LindelSf.
,
Proposition 3.1. ()
S()
is
8)
If
[
If
[
MR(t,L),
6, then for every element of
is
y) For every element of
if
MR(t[),
IA{[
of
(compact), then for every element of M([),
is Lindelof
Lindelf (con,act).
,
MR(o,[),
S() is LindelDf, then for every element
if
*(S()).
(S())
then
(X).
(S())
(Proof omitted.)
"[
The condition
Remark.
disjunctive" was not needed in the
is separating and
proof.
Definition.
[
is
paracompact iff
{As;
A},
if
u{A’"
of
[,
e
A}, such
X, x
a e
that for every
B’
a
[
A}
X,
A’
and
[, L
there exists an element of
is regular and for every subset
then there exists a subset of
u{B;
A}
a
such that
X
L’
x e
{B;
[,
and for every element of
A/L’ n B’
{e
and
}
is finite.
THEOREM 3.6.
.
is LindelDf.
S() #
A}o
t[
Assume
Proof.
Then
t[
(t[)’,
Hence, since
t[
{0
A*
A,
a subset of
G
to show that for every n,
[
Since
Observation.
(Proof omitted.)
is disjoint.
{S()
S()
O; 0
0
#
;
Gn
MR(t,t[),
G,
0
S()
Since
0
G
n
0
<
A*}
.
,
if
S(B) #
is discrete,
Therefore to show
Gn
S()
{0},
.
u{Oe;
S(B)
is paracompact.
S() cuG
and
uGn and for every n,
G n >.
Note to show there exists
A*
is countable, it suffices
and
MR(t,[), by [5], Theorem 2.5,
91A{[
and
L’ #
S()
,
then
is unique.
(S()
L’)>O.
without loss of generality assume
Gn
is disjoint; hence
{S(B) O;
is countable, it suffices to show
is countable; consider any element of
consider any element of
t[
S()
such that
S()
To do this, proceed as follows:
[, L
#
such that
Gn >, such that G
such that
9
uG and
S()
Since
Now, consider any n.
n
countable.
For every element of
for every element of
G
s
A}, G
is separating and disjunctive, and
there exists an element of
0
n
is paracompact,
e
u{Oe,
S()
such that
<
{Oe;
A},
e
G and any such
Consider any such
is discrete.
G,
of
there exists a sequence of subsets
MR(t,[)
Consider any element of
is paracompact.
Then there exists an open refinement of
Gn
MR(t,[),
is paracompact, then for every element of
Consider any subset of
S()
Note
t[
If
G
n
O; then, by assumption,
O, x; then, there exists an element of [,
G. BAC and P.D. STRATIGOS
724
O; consider any such
(S() n O) > 0. Therefore
consequently
0;
;
’
x
such that
Gn
(proof omitted’) consequently
S()
Whence
Examples.
(Note
compact.
MR(,Z),
X
(3).
Lindelf.
Consider any topological space
is
T31/2).
Let
is Lindelf.
by Theorem
[
Z.
3.6
Let
C.
X
such that
Then by Theorem
3.6
X
is
T
and para-
for every element of
(This result is known.)
X
such that
for every element of
Consider any topological space
and paracompact.
S ()
countable.
Consider any topological space
F. Then
[
(I).
S()
(2).
is
is
then by the observation (S(p)
Gn} is countable;
{S() n O; O
X
is
T
is
T
MR(,F),
such that
Then by Theorem 3.6,
X
X
S()
and paracompact and
is LindelDf.
and 0-dlmenslonal
for every element of
MR(,C);
is LindelDf.
ACKNOWLEDGEMENT:
The second author wishes to express his appreciation to Long Island
University for partial support of the present work through a grant of released time
from teaching duties.
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I.
2.
3.
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107-114.
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5.
6.
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8.
BACHMAN, G. and SULTAN, A., On Regular Extensions of Measures Pacific J. Math, 86
(1980) 389-395.
BACHMAN, G. and STRATIGOS, P., Criteria for -smoothness, T-smoothness, and tightness of Lattice Regular Measures, with Applications, Can. J. Math., 33(1981)
1498-1525.
FROLIK, Z., Prime Filters with the C.I.P., Comm. Math. Univ. Carolinae, 13 (1972)
553-575.
NOEBELING, G., rundlagen der analytischen topolpKie, Springer-Verlag, Berlin 1954.
SZETO, M. Measure Repleteness and Mapping Preservations, J. Indian Math. Soc. 43
(1979) 35-52.
"On Maximal Measures withRespect to a Lattice," in Measure Theory and
its Applications proceedings of the 1980 conference (edited by G. Goldin
and P.F. Wheeler) Northern Illinois University Press DeKalb, Iii, (1980)
277-282.
I0. WAKMAN,H., Lattices and Topological Spaces, Ann. Math., 39(1938) 112-126.