Interat. J. Hath. & Hath.
VOL. 16 NO.
(1993) 23-32
23
LATTICE NORMALITY AND OUTER
MEASURES
PANTAGIOTIS D. STRATIGOS
Department of Mathematics
Long Island University
Brooklyn, NY 11201
(Received August 20, 1991 and in revised form March 26, 1992)
ABSTRACT.
A lattice space is defined to be an ordered pair whose first component
is an arbitrary set X and whose second component is an arbitrary lattice [ of sub-
sets of X.
A lattice space is a generalization of a topological space. The concept
of lattice normality plays an important role in the study of lattice spaces.
The present work establishes various relationships between normality of
lattices of subsets of X and certain "outer measures" induced by measures associ-
ated with the algebras of subsets of X generated by these lattices.
KEY WORDS AND PHRASES.
Lattice space, lattice normality, lattice regularity and
-smoothness f a measure, weak lattice regularity of a measure, finitely subadditive outer measure; countably subadditive outer measure.
1991 AMS SUBJECT CLASSIFICATION CODES.
i.
28A12, 28A60, 28C15.
INTRODUCTION.
It is our aim in this paper to establish various relationships between normality of lattices on an arbitrary set and certain "outer measures" induced by meas-
ures on the algebras generated by these lattices.
More specifically, consider any set X and any lattice
on’X,
[.
The algebra
on X generated by [ is denoted by A(i).
In Section 3, a finitely subadditive "outer measure" is associated with an
arbitrary (0-1)-valued measure on
A([).
The behavior of "outer
measures" of this
type on i can be used effectively to characterize normal lattices, and this is
investigated in Section 3.
Also in Section 3, a notion of weak regularity of
measures is introduced in terms of the associated finitely subadditive "outer
measures" and it is shown that if [ is normal, then this notion coincides with
regularity.
In Section 4, a countably subadditive "outer measure" is associated with an
arbitrary (0-1)-valued measure on A(i)o
The relationship between this "outer
measure" and the associated (0-1)-valued measure is considered in the presence of
smoothness.
Also in Section 4, the equality of certain of these measures and
"outer measures" on [ or [’
the complementary lattice of i
is considered,
in particular in the case where [ is normal and countably paracompact or [ is
24
normal and
P.D. STP.ATIGOS
.
6 are given, including normality, which
In addition, conditions on
automatically imply regularity of certain smooth measures.
Typical applications are given throughout the paper in the case of topological
It is also
lattices; it is clear that many more such application could be given.
clear that certain of our results extend to arbitrary measures, but we will pursue
this matter elsewhere.
We adhere to standard by now lattice terminology and notation, which can be
found, for example, in [1,2,6,7] and, for convenience, we review some of the more
important terminology and notation used throughout the paper.
2o
TERMINOLOGY AND NOTATION.
We shall assume that @, X E
(a) Consider any set X and any lattice on X, 6.
6,
without loss of generality for our purposes.
U,
Now, consider any topological space X and denote the class of open sets by
the class of closed sets by
sets by
Z.
Recall
F,
the class of clopen sets by
Note each of the classes
U, F, C, Z
is also referred to as the topology on X
defined to be
(X,O).
Thus
(X,6)
this reason, we shall refer to
C,
and the class of zero
is a lattice of the prescribed type.
and the topological space X is
is a generalization of a topological space.
X,6
theory, it is convenient to regard
as a lattice space.
For
In topological measure
as the topology on X and
(X,)
as the topo-
logical space.
6 is said to be complement generated iff for every element of 6, L, there
exists a sequence
6,
in
(.n),
such that L
every two elements of 6, A, B, if A N B
6, C, D,
,
_%’n"
6 is said to be normal iff for
then there exist two elements of
D’ and C’ N D’
@. 6 is said to be countably
paracompact iff for every sequence in 6, (An), if (An) is decreasing and
limnAn=’
then there exists a sequence in 6, (B), such that for every n, A C B’ and (B’)
n
n
n
n
such that A C C’ and B C
B’=.
is decreasing and lim
n n
(b) The algebra on X generated by 6 is denoted by
on X,
A.
A measure on
A
finitely additive and bounded.
is a measure on
A(6)
A(6).
is defined to be a function U from
(See [I], p. 567.)
is denoted by M(6).
Consider any algebra
A
to R such that
An element of
M(6),
,
6-regular iff for every element of A(6), E, for every positive
exists an element of
6, L,
such that L C E and
I(E)-
< e.
(L)
is said to be
number
M(6),
,
is said to be
(A),
if (A) is decreasing
n
n
,
there
The set whose
general element is an element of M(6) which is 6-regular is denoted by
element of
is
The szt whose general element
MR(6).
An
6-(o-smooth) iff for every sequence in A(6),
,
and lim A
then lim (A
0o The set whose
n n
n
n
general element is an element of M(6) which is 6-(o-smooth) is denoted by
M(6).
The set whose general element is an element of M(6) which is 6-(o-smooth) just for
(A) in 6 is denoted by M (6).
n
o
The set whose general element is an element of
M(6),
,
such that
(A (6) )={0,1}
that is, the set whose general element is a (0-1)-valued measure on A(6) is denoted
by I (6).
NOTE
Since every element of M(6) is expressible as the difference of nonneg-
ative elements of
M(6),
loss of generality.
we shall work with nonnegative elements of
M(6),
witho.
(Related matters can be found, for example, in [3,4,5,8].)
LATTICE NORMALITY AND OUTER MEASURES
25
(c) A finitely subadditive outer measure is defined to be a function
P(X) to R such that ()
from
is nonnegative, increasing, and finitely sub-
0 and
additive.
from P(x)
A countably subadditive outer measure is defined to be a function
to R such that (}
3.
is nonnegative, increasing, and countably subadditiveo
0 and
LATTICE NORMALITY AND FINITELY SUBADDITIVE OUTER MEASURE.
In this section, we work with an arbitrary set X and an arbitrary lattice on
t.
X,
L
it to obtain conditions for
,
M(t),
’
and the function
L’ DA}.
PROPOSITION 3.2.
(ii)
<
(iii)
If
(iv)
’
,’
’
(i)
i’.
on
’
and
e I(i),
to be normal.
Consider any lattice space
DEFINITION 3.1.
of
P(x) and use
We introduce a certain finitely subadditive outer measure on
then
(Proof omitted.
P(X}
(X,L)o Nqw,
determined by
’
consider any element
inf{(L’)IL e L
(A)
and
is a finitely subadditive outer measure.
is
iff
’
on
i-regular.
(P(X))
{O,1}.
The following theorem gives "a characterization of normality of
L
in terms of
finitely subadditive outer measures of the type introduced by the preceding
definition.
L
LEMMA 3.3.
is normal iff for every element of I (L),
IR(L), Ul’ u2"
elements of
<
such that
I’ u2
on
L,
,
for every two
(Known.)
2"
i
THEOREM 3.4. The following statements are equivalent:
(a) [ is normal.
< v on
L,
’ ,
v’ on
(c) For every
and
IR(L’),
L.
,,
for every element of
v,
as in (b), for every element of
L,
(a) implies (b).
and any element of
by assumption, v’ < ’.
every element of
element of
,
v
L,
L,
I(l),
L, A,V’(A)
IR(/),
v,
such that
Hence to show
/
’
(A).
A, such that v’ (A) <
v’ (A)
0 and
’
(A}
’. Consequently V(A)
0.
L, such that L’ D A and V(L’)
’
’
’
i.
Consequently
’
(A}
assumption is wrong.
0.
Then
<
L,
on
Consoquontly
’
L’
’
v’ on
L’ D A} and
because
it suffices to show for
(A).
Consider any such A.
Now, note since v is
IR(L),
Then since
L-regular
by assumption,
there exists an element of
Consider any such L. Then since
L,
IJ(D) < V(D) < V(L’}
u’ on
Assume (b) and show (a).
L
is normal
C, D, such that A C C’ C D C
O.
Thus a contradiction has been reached.
was not needed in the proof.
(b) implies (a).
IJ(C’)<
i.
_< v on L and show
inf{(L’)ILEL and
v’ on
Hence since v e
0.
A, such that
Assume the contrary. Then there exists an
by assumption, there exist two elements of
Consider any such C, D.
such that
Consider any element of
(A)
Note since for every element of P(X), A,
inf{V(L’)IL E L and L’ D A} by definition and
L
L,
B, such that B C A’ and (B)
Assume (a) and show (b).
u’ (A)
on
,
IR(L),
L.
there exists an element of
PROOF.
,
IR(L’)
(b) For every element of
L.
L’.
llcncc li(C’)
O.
Therefore, the
(Note the L’-regularity of
For this, use Lemma 3.3, namely,
26
P.D. STRATIGOS
IR(6},
u I, u 2, such that
(= I(6’)) by assumption,
consider any element of I (6), p, and any two elements of
P <
on 6 and show u
Ul’ u2
I
6.
iR(6’)and
6
IR(6)
6
Ul,U 2
’
u2"
’
I’
and
6
on
2
Consequently,
u2
2.
,
<
on
2
Ul’ 92
are [-regular, u
2
Ul
I
Then by Lemma 3.3, 6 is normal.
2"
(b) is equivalent to (c).
.
Consider any such
on 6. Thus
i by assumption, <u l, u
2
on 6. Hence since (b) is true by
Further, note since u I,
i
6’.
P, such that D < p on
Hence since p < u I, u
assumption,
and
Note since D 6 I([)
IR(6’),
there exists an element of
Then B < P on
u
[Proof omitted. (Again it should be noted that the
6’-regularity of p is not needed in the proof of (b) implies (c).)]
on
o
(c)
,
IR(U),
for every element of
For every , and ,
Z
u’ on
Z.
such that u(Z’)
element of
IR(Z),
PROF.
IR(i), ,
.,
Z,
i.
_<
u, such that
Asse [ is noal.
[,
on
.,
C
Io(i’),
To show u 6
by the relevt definition, there exists a sequence in
Z’
and
I0(6),
Io([’ ).
6
,
Io([),
Consider y element of
u on i.
such at
Then
T3.
u
u, such that
such that
If 6 is normal, then for every element of
IR(L),
(F)
Z,
and u, as in (b), for every element of
(c) For every
i, there exists an element of
THEOREM 3.7.
of
C U a,,d
I."
Theorem 3.4, the following statements are true:
is normal, according to
’
Z,
U, U such that
of
element
P, such that
For every element of I(Z), p, for every element of
(b).
on
[,
F, B’
u on
Consider any topological space X such that X is
APPLICATION 3.6.
since
IR(F),
u, such that
as in (b), for every
i, there exists an element of
(U)
(b) For every
(a) X is normal.
3.4, the following statements are equivalent:
element of
Then according to Theorem
Consider any topological space X.
APPLICATION 3.5.
(.)=i.
for every
and any elent
asse the contrary.
[,
(L
Z,
Then
(L’)n
such that
is
Then since
decreasing d lim L’
# 0. Consider any such (L).
@ and limn
n n
n
u 6 I([) by assption, for every n, V(L’)
i. Hence for every n, since 6 I([’)
n
u on i and [ is noal by assption, by Theorem 3.4, there
and
d u 6
U(L:)
IR([)
L, {.
exists an element of
Now, for
eve
().
=i k;
consider
n
{.
r;ucl, tl,at
n
note
L’
C
I,(
a,,d
=I k
I; co,nid,r any sucl
n
1%
i; set
n
.
Further
C L’ and (i)
n
n
n
n
Hence
and
(L’)
L’
is
and
decreasing
lim
i
since
is decreasing
n
n n
n n
0. Thus a contradiction has been
by assumption, lim
since M 6
n
nsider
() is in [
n
Note
and for every n,
Consider any tological
APPLICATION 3.8.
en
v e
element of
PROF.
,,
I(),
If
for every element of
I(L),
Asse
L
I(i’).
X such that X is noal.
IR(),
u, such that
u on
is countably paracompact and noal, then for every
P, for every element of
L
p, and any element of
IR(L),
L-regular,
u,
Theorem 3.7,
countably paracompact by assumption,
is
IR(L),
u,
u on L, uI (L).
such that
R
is countably paracompact and noal. Consider any element of
normal by assumption, by
since
sacc
I(U).
COLY 3.9.
Io(L)
Consequently u
is noal, according to Theorem 3.7, the following statement is true:
since
For every element of
F,
,
I and
M(n
lu(/)
Therefore the assption is wrong.
reached.
,(in)
e I
R(L)
such that p <
E
I(L’)
Iu(L’).
C
I(L).
on
L.
L
Then since
Further, note since
Consequently
E
L
is
is
I(L).
Then
LATTICE NORMALITY AND OUTER MEASURES
27
If [ is countably paracompact and normal, then for every
COROLLARY 3.10.
Is(t), i’
I<2
APPLICATION 3.11.
element of
for every element of
v,
< v on
such that
[,
2
I(/),
Is(1).
6
space’X
Consider any topological
F
Then since.
paracompact and normal.
for every element of
2’
such that X is countably
is countabiy paracom[act and nozTnal,
For every element
according to Corollary 3.9, the following statement is true:
of
.
9 on F, u6 I
u, such that
R
Then
Consider any topological space X such that X is T 3
is countably paracompact and normal, according to Corollary 3.9, the
Is(F),
IR(F),
V, for every element of
<
APPLICATION 3.12.
since
Z
For every element of
following statement is true:
of
u,
IR(Z),
_<
such that
u on
Z,
I(Z).
v 6
,
I([),
Consider any element of
DEFINITION 3.13.
,
Is(Z),
for every element
has the follow-
such that
ing property:
For every element of
B, such that B C
A’ and
IR(/-),
6
Note if
C
(B)
(*)
i.
I([)
PROOF
IW([),
.
t
Assume
u.
Note to show
0 and U(A)
t,
Iw(t).
.
on [.
Hence (A’)
i.
[,
Now, note since
Consider any such B.
E
IR(t),
9’.
9
’
Consequently
REMARK.
6
IR(t).
IW(I)
Thus
X.
t-ultrafilters
G1
G2
F
t.
Hence since
0.
and
Further, note
’
(B)
i,
Thus a contradiction
on
Consequently
t.
,
P(X),
F
IR(t)
of
Consequently to show
Accordingly, note since (A
B’ is
’
IW(i)
and
Consequently
B)
t)
()
C
.
described by
and denote it by
Note I([)
respectively.
IW(L).
by Theorem 3.4.
IR(t)
IR(t).
C
F
{X),
then consider the
Further, consider the two
{,A,A U B,X} and G2= {,B,A B,X),
determined by G
G 2 and denote them by
1
described by
then consider the elements
A’
’
I(/’) and
Hence u(A)
i.
Next, consider the prime t-filter
subset of
6
by assumption,
I, by the definition
(B)
and B
AB=
A, B, such that A
[,A,B,AL)B,X).
Further, consider the lattice [ described by [
element of I([) determined by
V
on
Then
Consider any set X such that X has at least three elements.
Now, consider any two elements of
IR(t).
9
Further,
The converse is false.
COUNTEREXAMPLE.
and A U B
IW(t)
Therefore the assumption is wrong.
has been reached.
Vl u2
’
Hence since B C A’, U(A’)
I.
t
v’ on
Consequently
on /.
Consider any such A
Hence since
i.
element
Then there
Assume the contrary.
B, such that B C A’ and
on t and t is normal by assumption,
(B)
Thus
C
A, such that (A) < U(A).
there exists an element of
9
Iw(/)o
IR([
IW(t) @ and consider any
<
element of I R(t), , such that
6
IR(t), it suffices to show
Iw(t)
note for this, it suffices to show
since
by
Now, assume
is nornl
Then there exists an
exists an element of
of
For this reason, U is said to be
has Property (*)} is denoted
If [ is normal, then
Consider any such
(A)
/,
lw(/).
THEOREM 3.14.
of
’
i, there exists an element of
A, such thau (A’)
then U has Property (*).
weakly regular and { 6
’IR(/)
[,
G1
{,u I,2 )"
Iw(t)
C
IR([),
0, (A’
0.
IR(L).
B’)
Show I
w(t)
C I
R(/).
Note
it suffices to show
i.
IW(I)-
Further, note the only
Hence by the definition of
IW(I),
28
P.D. STRATIGOS
Finally, note [ is not normal.
Thus I
W([)
C I
R([)
and [ is not normal.
An alternative proof of the equivalnce of parts ,a) and (c) of Theorem 3.4
will be given, which does not involve an outer measu’e.
based
o**
Consider any lattice space
Further, consider {L E i
M }
This proof will be
and denote it by
(X,[).
Now, consider any element of I(1),
[,
for every element of
I, L O A
A, such that (A)
%.
,G
i is normal iff for evezy element of I([),
LEMMA 3.15.
(ii) Show for every two elements of
%,
two elements of
LI, L
there exists an element of
any such A.
Then A C (L
2
L.
C
1
L
L2)’
1
t)
A 2,
1
L
G,
L
.
2)
6 A
G,
Consider
1
Hence since i is nokai by assumption,
2.
such that A
A
t)
1
G,
A
and A
2
and L
i
(
U
A
1
,L
C
and
by the
i
,(A)
Thus a
A2,
1
Therefore the assumption is wrong. Consequently,
i)
6
2
(iii) Note for every element of
G..
(8) Show G
%
A
0.
element of
%,
H,
G, L,
for every element of
such that
is an i-ultrafilter.
H
and H M
G
G.
Consider any such
such that
H, such that H
G
H
D
H.
there exists an element of [, A, such that (A)
der any such A. Then by the definition of
and A q
[, S,
is an [-filter.
Consider any [-filtez
H
1 and (L
%.
L
Consequently
H E
Consider any
Then by the definition of
Consider any such A I, A
2. Note since L i
"(A
0 (i= 1,2). Hence since A
LC S, S
of
G.
L
2.
2
1
i, A, such that (A)
contradiction has been reached.
L
G, L I, L 2, L1 O L2 E G.
Assume L
there exist two elements of i, A
definition of
is
an [-filter
%.
9
(i Note
A
,%
(a) Assume [ is norl and show for every element of I(1),
i-ultrafilter.
(s) Show G is
an
k-ultra-
is an
filter.
PROOF.
.
i-ultrafilters.
a characterization of normality of i in terms of certain
H
and
H
is a filter.
%,
Hence H
A
A
M
.
.
Then by the definition
1 and H
A
Consequently
Consi-
H.
G. Thus a contradiction hasThus
Therefore the assumption is wrong.
been reached.
Then there exists an
Consequently
A 6
is an i-ultra-
filter.
OBSERVATION.
u. Note
,
<
9
Consider the element of
on [.
(b) Assume for every element of
normal.
I([),
IR([)
determined by
, Gis
G
and denote it by
an [-ultrafilter and show i is
For this, use Lemma 3.3, .namely, consider any element of I([),
two elements of
IR(1), Ul’ 2’
such that
I’ 2
on i and show
,
I 2"
and any
Note
Hence since
GvI, Gu2 G
and
.are i-ultrafilters and
is an k-filter by the assumption,
G
GVl,GV2
G92
i Hence
D {A
I} and
i
Further, note since
Ui(A)
%.
Gu2"
%1
Gvi
{A
i
ui (A) I} is an i-ultrafilter since ui IR(i) and Gui is an [-filter,
{A i
ui(A) I} Gui (i= 1,2). Hence since i Gu2 {A i uI(A) I}
since
Ul,U2
on i,
c
by the relevant definition.
LATTICE NORMALITY AND OUTER MEASURES
{A
6
t
i}.
V2(A)
Thus V
U
1
on
2
t.
Hence u
29
Consequently
u2"
I
t
is
normal.
The alternative proof of the equivalence of parts
,a)
and (c) of Theorem 3.4
is now given.
(Recall statements (a) and (c) of Theorem 3.4:
(a) i is normal.
IR([), ,
_<
such that
[,
on
there exists an element of
[,
of
u,
IR(1),
_<
such that
A’
B, such that B C
Consider any element of
[,
on
(A)
G.
Consider
G
tion, by (Lemma 3.15, (a)),
IR()
,
6
determined by
IR([)
U(A’)
<
and
G,
I.
i, D(A’)
@.
G,
[, C, D,
e
[’
.
IR([’), ,
of generality).
A 6 i and 9(A’)
element of
0.
Thus U 6 I([) and
Then since
D.
(Property of D.) Hence
1 and
[ and B
Then B
For this, assume the contrary.
.
B
and for
evez
Then there
two elements of
and A ,q B
@, (A)
Then U(A’)
I.
I.
0.
IR([’),
Consider any such
on [.
such that U <
0 or 9(B)
6
Hence there exists an element
L’, (L’)
IR([), ,
Thus
i.
,
Now,
Assume U(A)
6
IR([)
Now, note
0 (without loss
<_
and
on
[,
and
Hence since (c) is true by assumption, there exists an
C
A’
an@
(C)
Consider any such C.
i.
Thus a contradiction has been reached.
Ther6fore
Then
C’
the assumption
Consequently [ is normal.
is wrong.
4.
G
Therefore A
0.
A, B, such that A
i, C, such that C
and (C’)
A, such that
Consider the element of
< p on [.
Note
such that for every element of
I([)
9
i.
Note
i.
[,
there exists an element of [, B, such that (B)
Further, consider any element of
since
and (B)
has the Finite Intersection Property.
Consequently
of
any element
C’ D A and D’ D B, C’
Consider any such A, B.
D’
L’
A or L’ D B} and denote it by
such that
consider {L’
A’
Hence since i is normal, u
Consider any such B.
[,
,
IR([’)
for every element of
is an [-ultrafilter.
(ii) Assume (c) and show (a).
exist two elements of
6 [
and denote it by p.
D on [.
l.)
i,
Further, note since [ is normal by assump-
Hence p(A)
by the definition of
A N B
{L
} by definition.
i, L N A
such that U(A’)
and any element of [, A, such that u(A’)
G
Note
for every element of
and (B)
Show there exists an element of [, B, such that B C
e I(1).
[, A,
for every element of
(i) Assume (a) and show (c).
,
IR([’),
(c) For every element of
LATTICE NORMALITY AND COUNTABLY SUBADDITIVE OUTER MEASURE.
In this section we work with an arbitrary set X and an arbitrary lattice on X,
[.
We introduce a certain countably subadditive outer measure on F(X) and use it
to obtain conditions for [ to be normal.
DEFINITION 4.1.
of M([),
,
and the function
"
6 [ for every k and t} L D
k k
PROPOSITION 4.2.
(ii)
"
<
(iii) If
(X,[).
Consider any lattice space
’.
e I(i),
(i)
"
then
on
(X) determined
by
A}.
Now, consider any element
"
(ii) If
I([) and (X)
inf{Xk= 1 (L)
is a countably subadditive outer measure.
"(P(X)) C
(Proof omitted.
PROPOSITION 4.3. (i) If
(A)
6 I
(1), then
"(X), then
"
e
<
on [.
I ([).
o
30
P.D. STRATIGOS
, E Iu (L).
(i) Assume
PROOF.
[, A,
Then there exists an element of
by the definition of
7. ,(L)
"
there exists a sequence in
)
,(A N
i.
I;
k)
L
is in
,
since
t,
k
(Lk),
t)k
such that
I(L), ,(A}
D A
1
(A N L
k)
and for every k,
n
Note
(A
Now, for every natural number n, consider
(A N L
(’n)
.
assume the cntrary.
Then since ,
().
< (A). Consider any such
k
0. Consequently
and for every k, ,(L)
and
L,
on
,"(A) < P(A). Consider any
D A}
E [ .for every k and
such that
inf{k=l P()
,"(A)
Then since
such A.
, _< ,"
To show
and
nk=l
set
(.n)
(A N
)
note
n
is decreasing and
limn
,
n
=I
, (.n)
i.
n ’n
).
Consider
%
n
)
(A
.
().
Note
Hence
(.)
by assumption, lim
0. Thus a contradiction has been
n
n
< ," on L.
Therefore the assumption is wrong. Consequently
IO(L)
reached.
,
(ii) (Proof omitted.)
Consider any lattice space
NOTATION.
,
I([),
,
such that
For every sequence in
e N}.
n
Now, consider any element of
L, (L),
n
such that
L
n
q
n
L,
inf{,(L );
,(N L
n n
n
(**)
I(L), then , has Property (**) and if , has Property (**),
Io(L). Thus I(/) C {, 6 I(/) l, has Property (**)} c Io(/). set {,
J{i). Thus I (L) C J(L) C Io([).
has Property (**)}
,
Note if
,
(X,L).
has the following property:
6
then
6
" PROPOSITION 4.4. If ,EI(L),
, q I(i).
PROOF.
Assume
,"
(i) Assume P
,
then
on i’ and show
,
," on i’ iff
,
J(L).
e l(t)
J(i).
Assume the contrary. Then by the
L, (L),
such that
L
i and
n
n n
inf{,(Ln); n E N}. Consider any such (Ln
is in
Now, note since (L)
n
relevant definition, there exists a sequence in
P(% n)
L
L
and
since
Un L’n DUn L’,n
, e I(L),
,(_ L
n
1.
Further, note since
0.
n
E i, ,(t L’)
n
n n
,
Hence fo every n, P(L_’)
,"
.
element of
Assume
L
is
.,
6omplement generated.
J([),
To show e
.
,.
L,
such that
C
,
I,
L’
Consider any such
ff
Iw([),
(.)
C
’k [k
<
"’’ (k)"
Consider any such m.
L’ and ,’
(m)
THEOREM 4.6.
I.
If
L
,
Then
I.
,"
L’
Iw().
use the relevant definition,
L, such that ,(L’)
and ,’
().
by assumption, by Proposition 4.4,
m
by assumption and
Note since I (L) C j(L), J(i)
1 and show there exists an
Note since L 6
ment generated by assumption, there exists a sequence in
.
L’
If i is complement generated, then J(i) C
namely, consider any element of
(Ok
on
O.
J(L).
Consequently
Consider any element of
P"
Now, note
N}. Hence since
(Proof omitted.)
PROOF.
i.
n ,(L._’}.
," (t)
L
PROPOSITION 4.5.
L
--<
L’). Consequently ,(t) L’)
0.
n
n n
n n
i. Thus a contradiction has been reached. Therefore the assump-
Hence ,( L
n n
tion is wrong.
(ii)
"(n L)
ihf{P(L ); n
n
<
n
i’ because
t)
NI, ,(n Ln
n
0 and for every n, ,(L
n
,"(U L’)
Consequently
L’
n n
by the definition of ,"
inf{"(Ln};
,(% n) M
L
t)
k
on i’.
[,
L
(%),
and
L
is comple-
such that
%.Consequently 1
Further, note since
,(L’)
Hence there exists a value of k, m, such that
Then since
Consequently
(.m)
, e" Iw(i). ’Thus
J(t) C IW(L)
<_ ,’,
i.
Thus L
m
is normal and complement generated, then J(L)
" (.m
i and
I (L)
R
LATTICE NORMALITY AND OUTER MEASURES
31
Assume 6 is normal and complement generated.
o
() Show j(6) C
Note since 6 is complement generated, by Proposition
PROOF.
IR(6).
J(6)CIw(6).
4.5,
Further, note since 6 is normal, by’Theorem 3.14,
Consequently J(6) C I R(6).
(8) Show
I(6)
C J(6).
APPLICATION 4.7.
46
I(6)
I(6).
J(6).
CI([
C
J([) C
I(6).
I O().
R
APPLICATION 4.8.
is normal and complement generated by definition, by Theorem
J(F)
Z
since
Consider any to[ological space X suc]l that X is
is normal and complement generated, by Theorem 4.6, J(Z)
" ’
T3.
R(z)I(6),
E
Assume 6 is normal and countably paracompact and
Io(6).
6
Note since
M 6 I(6) by assumption, there exists an element of
IR(6), ,
such that M
IR(6)
and M < u
on" 6.
Consider any such
.
6 I(6’) and 9 6
Thus
is normal by assumption, by Theorem 3.4,
M’
u’ on 6.
(Proposition 4.3, (ii), v < u"
v"
on
on
Then since
6. Also, note since M
<
M" < M’, v’ <
Z
’. < u’ and since
6,
< u on
" ’
APPLICATION 4.10.
since
on 6] and
on 6.
If
6
Io(Z),
then
" ’
C
"
If
L
is normal and 6 and’
Assume 6 is normal and 6 and
PROOF.
<
Then
there exists a sequence in
.k
der any such
% e 6.
and
6
Set
I(6)
k
If
6
’k U’ B.
to show
Io(),
APPLICATION 4.14.
since
If
6
is normal and
IO(Z),
then
"
(A)
,6Is(L),
6I(6).
0 and
’
(A)
such that
^’
Uk Lk
(
Then A C
B’o
(A)
,
" ’
,
is normal and
then
L,
Note A C
APPLICATION 4.13.
since
6 I
Then
T3.
4.9, the following
’
’
"
" ’
’
U"
(C), then N"
s
then
on
.k
i.
^’
Uk Lk
6, A,
U"(L) /
on
"(A)
Now, note since
[k
)’ and since 6 is
()=
L,
(L).
(A)<
such that
D A and
C.
on
L.
Note to Show
Then there exists an element of
Consider any such Ao
"
6, D"’= M’
on
Z.
’, it suffices to show for every element of 6, L,
Assume the contrary.
%
’ ’
.
and
1
is normal and countably paracompact, according to
Theorem 4.9, the following statement is true: If
since
Consequently 9’ <
u" < M"
Consider any top61ogical space X such that X is T
Then since
THEOREM 4.12.
IR(1) and
Io(6), by
is 6-regular, u’=
Hence since
on
6
E
Consider any topological space X such that X is
APPLICATION 4.11.
0-dimensional.
Hence since
6
is normal and countably paracompact, according to Theorem
statement is true:
u on i.
Now, note since i is
cuntably paracompact and normal by assumption and M 6 Io(6) and
6 I(6). Further, note [since
M _< u on 6, by Corollary 3.9,
6.
6.
then
6.
on
PROOF
Hence 9’
T|en
I
If 6 is normal and countably paracompact and
THEOREM 4.9.
on
IR([)-
C
Consider any topological space X such that X is perfectly
F
Then since
O(6),
Note
(7) Consequently J(6)
normal.
Hence since J(6) C I
IW(6)
’
(A).
0,
0. Cons[-
by assumption
Now, use the assumption that 6 is normal
0, thus reaching a contradiction.
Consider any topological space X such that X is normal. Then
according to Theorem 4.12, the following statement is true:
on
o
Consider any tollogical space X such that X is
T3.
Then
according to Theorem 412, the following statement is true:
" ’
on
Z.
32
P.D. STRATIGOS
ACKNOWLEDGEMENT.
The author wishes to cxpross his
a|preciation
to Long Island
University for partial support of the present work through a grant of released
time from teaching duties.
REFERENCES
I.
ALEXANDROFF, A. D., Additive set-functions in abstract spaces,
(N.S.) 9 (51)
2.
BACHMAN, G.
Matt
Sb_._.
(1941), 563-628
and SULTAN, A., Regular lattice measures: mappings and spaces,
Pacific J. Math. 67, no. 2, (1976), 291-326.
3.
..,
On regular extensions of measures, Pacific J. Math. 86, no. 2,
(1980), 389-395.
4.
CAMACHO, J., On maximal measures with respect to a lattice,
Internat.’. Math.
& Math. Sci., 14, No. 1 (1991), 93-98.
5.
EID, G., On normal lattices and Wallman spaces, Internat. J. Math. & Math.
Sci. 13(i) (1990), 31-38.
6.
FROLIK, Zo, Pime filters with the C.I.P., Comm. Math. Univ. Carolinae, 13,
7.
NBELING,
8.
SZETO, M., Measure repeletness and mapping preservations, J. India
(1972), 553-575.
G.,
Grundlaen
der
analtischen Topoloie,
Springer, Berlin, 1954.
n
Math. Soc. 43 (1979), 35-52.