PII. S0161171203007944 http://ijmms.hindawi.com © Hindawi Publishing Corp.
LATTICE SEPARATION AND MEASURES
JAMES PONNLEY
Received 26 July 2001
The effects of lattice separation such as normality, almost normal, slightly normal on various lattice-derived measure are investigated and generalizations of earlier work on 0-1 valued measures are obtained.
2000 Mathematics Subject Classification: 28C15, 28A12.
1. Introduction. In an earlier paper [6], we considered a variety of special 0-1 valued measures and studied some associated outer measures and their measurable sets. A portion of this paper was then extended in [7, 8] to the more general case where the measures involved need not be just 0-1 valued. However, there were still sections of [6] that were not generalized, especially those related to separation; namely, where the lattice of subsets involved was slightly normal or almost normal. The extension of these results involves a number of different concepts that did not arise in the 0-1 valued case, and the arguments involved are considerably different from the 0-1 valued case.
We pursue these matters in Sections3 and 4. In Section 5, we give some further related type theorems. Again, the major concern is how certain lattice separation properties affect various measures defined on the algebra generated by the lattice. In many cases regularity is implied, in other cases equality of certain associated outer measures is assured on various sets.
We begin inSection 2with a brief review of some terminology and notation used throughout the paper. Also, a number of basic results are stated which are used throughout the paper. More specific facts are given in the sections to which they are most closely related.
2. Background and basic notation. We briefly review here some standard notation and terminology which are consistent with our previous usage in [6,8]. The set X denotes a nonempty arbitrary set, and ᏸ a lattice of subsets of
X. All lattices considered throughout the paper will contain ∅and X. The algebraᏭ(ᏸ)denotes the algebra generated by ᏸ, andM(ᏸ)denotes those nontrivial, finite, nonnegative, and finitely additive measures on Ꮽ(ᏸ). The set MR(ᏸ)denotes those elements ofµ∈M(ᏸ) that areᏸ-regular. The set
The set Mσ(ᏸ)stands for those elements ofM(ᏸ)which are σ-smooth on Ꮽ(ᏸ), and, consequently countably additive. The setMσ
R(ᏸ)stands for those elementsµ∈M(ᏸ)that are common to both setsMR(ᏸ)andMσ(ᏸ), that is,
Mσ
R(ᏸ)=MR(ᏸ)∩Mσ(ᏸ), and it is not difficult to see that ifµ∈MRσ(ᏸ), then
µ∈Mσ(ᏸ).
To aµ∈M(ᏸ), we associate a number of outer measures. Let E⊂X and define
µ(E)=infµL:E⊂L, L∈ᏸ, (2.1)
whereL=X−L;
µ(E)=inf
∞
i=1
µL
i
:E⊂
∞
1
L
i, Li∈ᏸ
. (2.2)
Similarly, we defined ˜µ(E)and ˜µ(E)˜ where in the above definitions we re-place theLandL
ibyLandLi, respectively, where theL,Li∈ᏸ;µ, ˜µare finitely subadditive outer measures whileµ, ˜µ˜are countably subadditive outer
mea-sures.
In general, ifν1, ν2are two set functions defined on a latticeᏸ, we write
ν1≤ν2(ᏸ)ifν1(L)≤ν2(L), for allL∈ᏸ.
We recall some simple relations involving these outer measures.
Theorem2.1. (a)µ≤µ,µ˜˜≤µ˜.
(b)Ifµ∈Mσ(ᏸ), thenµ(X)=µ(X)andµ≤µ(ᏸ).
(c)Ifµ∈Mσ(ᏸ), thenµ(X)˜˜ =µ(X)andµ≤µ(˜˜ ᏸ). (See[2]for details).
Next, we recall that ifνis a regular countably subadditive outer measure, and ifEnis monotonically increasing toE,(En↑E),En⊂X, then
νlim n→∞En
=lim n→∞ν
En see [5]. (2.3)
In this connection, we note that ifµ∈M(ᏸ)and ifµis a regular outer
mea-sure such thatµ(X)=µ(X), thenµ∈Mσ(ᏸ)(see [12]). A similar statement
holds when ˜µ˜is a regular outer measure.
We next recall that a latticeᏸis normal if wheneverA,B∈ᏸandA∩B= ∅, there existC,D∈ᏸsuch thatA⊂C,B⊂D, andC∩D= ∅. Ifᏸ1andᏸ2are
two lattices of subsets ofX, then ᏸ1is said to semiseparate ᏸ2if whenever
A∈ᏸ1,B∈ᏸ2, andA∩B= ∅, there exists aC∈ᏸ1withB⊂CandA∩C= ∅. The latticeᏸ1, is said to separateᏸ2if forA,B∈ᏸ2,A∩B= ∅, there exist
C,D∈ᏸ1, such thatA⊂C,B⊂D, andC∩D= ∅. Finally,ᏸ1coseparatesᏸ2 if, forA,B∈ᏸ2,A∩B= ∅, there existC,D∈ᏸ1such thatA⊂C,B⊂D, and C∩D= ∅.
countable intersections. We also denote byᏸthe set{L:L∈ᏸ}, and ifν is
an outer measure either finitely or countably subadditive,ν designates the
ν-measurable sets.
3. The general case (a). In this section, we generalize a number of theo-rems established in [6] for the special case of 0-1 valued measures to the more general case. We denote forµ∈M(ᏸ), and
E⊂X, µi(E)=supµ(L):L⊂E, L∈ᏸ, (3.1)
whereµiis an inner measure, and
µi(E)=µ(X)−µE (3.2)
(for details onµ,µ
i, and related matters of measurability, see [2]). Also, forµ∈Mσ(ᏸ)andE⊂X,
µk(E)=µ(X)˜˜ −µ˜˜E=µ(X)−µ˜˜E (3.3)
byTheorem 2.1(c).µkis not, in general, an inner measure; it is, if ˜µ˜is submod-ular (see [4]).
Theorem3.1. Letᏸbe a lattice of subsets ofX, and letµ≤ν(ᏸ),µ(X)=
ν(X), whereµ∈Mσ(ᏸ)andν∈MR(ᏸ). Ifµ˜˜is a regular outer measure and if
δ(ᏸ)separatesᏸ, then
ν(L)=sup µk
∞
j=1
Bj
: Bj⊂L, Bj∈ᏸ
, L∈ᏸ. (3.4)
Proof. Sinceµ≤ν(ᏸ)andµ(X)=ν(X),ν≤µ(ᏸ), soν∈M
σ(ᏸ), and, therefore,µ≤µ(˜˜ ᏸ)and ν ≤ν(˜˜ ᏸ). Also, sinceν ∈M
R(ᏸ), there exists an
A⊂L, A∈ᏸ such thatν(L)−ν(A) < , where is an arbitrary positive
number. Sinceδ(ᏸ)separatesᏸ, there existA
i,Bj∈ᏸsuch that
A⊂A
i⊂ Bj⊂L (3.5)
and where we may assume that theA
iis monotonically decreasing toA, (Ai↓A) and theBjis monotonically increasing toL, (Bj↑L).
Clearly,ν(A)≤νi(∩Ai). Now,
˜ ˜
µ Ai
=lim ˜µ˜Ai (since ˜µ˜is regular) ≤limνA
i≤ν Ai
. (3.6)
Hence,
µk
A
i
=µ(X)−µ˜˜ Ai≥ν(X)−ν A
i
=νi
A
i
Therefore,
ν(A)≤νi
A
i
≤µk
A
i
≤µk Bj
≤µ˜˜ Bj=lim ˜µ˜Bj≤limµBj ≤limνBj≤ν Bj
≤νL.
(3.8)
From which the result follows immediately.
Corollary3.2. Letᏸbe a lattice of subsets ofX, and letµ∈Mσ(ᏸ)such thatµ˜˜is a regular outer measure. Ifν∈MR(ᏸ)is such thatµ≤ν(ᏸ),µ(X)=
ν(X), and ifδ(ᏸ)separatesᏸ, thenνis unique.
Proof. The proof that such aν∈MR(ᏸ)exists is well known (see [3,10]). The uniqueness follows immediately from the theorem, since ifν1,ν2∈MR(ᏸ) both satisfy the conditions, thenν1=ν2(ᏸ), and, therefore,ν1=ν2. We de-note byI(ᏸ)the 0-1 valued measures ofM(ᏸ), and similarly, for the other subsets ofM(ᏸ); for example,Iσ(ᏸ)denotes those elements ofI(ᏸ)that are
σ-smooth onᏸ. We note that any 0-1 valued outer measure is trivially regular. Also recall the following definition.
Definition3.3. The latticeᏸis slightly normal if forµ∈Iσ(ᏸ)andµ≤
ν1(ᏸ),µ≤ν2(ᏸ), whereν1,ν2∈IR(ᏸ)impliesν1=ν2.
Hence, as a special case ofCorollary 3.2, we get the following result of [6].
Corollary3.4. Ifδ(ᏸ)separatesᏸ, then the latticeᏸis slightly normal.
We note that ifᏸ itself separatesᏸ, that is, ifᏸ is normal, then the set inclusions in the proof ofTheorem 3.1become simplyA⊂A
1⊂B⊂L, and in this case, it is easy to see that
ν(A)≤νA
1
≤µA
1
≤µ(B)≤ν(B)≤νL (3.9)
without any need forµ to belong toMσ(ᏸ), or for ˜µ˜to be regular, (3.9) of course implies thatν=µi(ᏸ), or, equivalently,ν=µ(ᏸ). Thus, we have the following corollary.
Corollary3.5. Letᏸbe a lattice of subsets ofX, and letµ≤ν(ᏸ),µ(X)=
ν(X)whereµ∈M(ᏸ)andν∈MR(ᏸ). Ifᏸis normal, then (a) ν=µ(ᏸ),ν=µ
i(ᏸ); (b) νis unique.
We can use the result to obtain a simple proof of the following corollary.
Corollary3.6. Letᏸbe a lattice of subsets ofX, and letµ≤ν(ᏸ),µ(X)=
Proof. LetA
n↓ ∅,An∈ᏸ, then byCorollary 3.5(a), there existsBn⊂An,
Bn∈ᏸ, which we may assume↓such that
νA
n
< µBn+, (3.10)
where >0 is arbitrary.
Since∩Bn= ∅,µ(Bn)→0; whenceν∈Mσ(ᏸ).
The last two corollaries are known, but shown in a different manner (see [2]).
4. The general case (b). In this section, we extend the results of [6] pertain-ing to almost normal lattices.
Recall the following definition.
Definition4.1. The latticeᏸis almost normal if, forA,B∈ᏸandA∩B= ∅, there existA
i↑,Ai∈ᏸ such thatA⊂ ∪∞1Ai, and there existBi∈ᏸ with
A
i⊂Bi, for alliandBi∩B= ∅, for alli.
It is not difficult to show that ifᏸis a delta lattice, and ifᏸis almost normal, thenᏸis normal.
We now have the following theorem.
Theorem 4.2. Let ᏸ be a lattice of subsets ofX which is almost normal. Supposeµ≤ν(ᏸ),µ(X)=ν(X), whereµ∈M(ᏸ)andν∈Mσ
R(ᏸ)and where
µis a regular outer measure. Thenµ=ν(ᏸ).
Proof. We note that sinceν∈Mσ
R(ᏸ), µ∈Mσ(ᏸ). Also since µ≤ν(ᏸ) andµ(X)=ν(X),ν≤µ and in particularν≤µ(ᏸ). Suppose that there
exists anA∈ᏸsuch thatν(A) < µ(A). We note thatν=ν=ν(ᏸ)since ν∈Mσ
R(ᏸ). Hence,
ν(A)≤ν(A) < µ(A). (4.1)
Then there exists aB∈ᏸsuch thatB⊃A, and
ν(A)≤νB< µ(A). (4.2)
SinceA∩B= ∅, there existA
i↑,Ai∈ᏸ, such thatA⊂ ∪Ai, and there exist
Bi∈ᏸwithAi⊂Bi,Bi∩B= ∅, for alli. Hence,
µ(A)≤µ A
i
=limµA
i
≤limµA
i
≤limµBi≤limνBi≤ν Bi
≤νB< µ(A),
(4.3)
We note thatTheorem 4.2can be generalized. For this purpose, recall the following definition (see [9]).
Definition 4.3. Let µ ∈ Mσ(ᏸ), µ is called vaguely regular if µ(A) = sup{µ(B):B⊂A, B∈ᏸ}forA∈ᏸ.
The set of vaguely regular measures is denoted byMv(ᏸ). Forµ∈Mσ(ᏸ),
E⊂X, let
µj(E)=µ(X)−µ(E)=µ(X)−µ(E). (4.4)
Then we haveµ∈Mv(ᏸ)if and only if
µ(A)=infµjB:A⊂B, B∈ᏸ, forA∈ᏸ. (4.5)
It is, of course, clear thatMσ
R(ᏸ)⊂Mv(ᏸ), and we can now establish the fol-lowing generalization ofTheorem 4.2.
Theorem 4.4. Let ᏸ be a lattice of subsets ofX which is almost normal. Suppose that µ≤ν(ᏸ), µ(X)=ν(X)whereµ ∈M(ᏸ) andν ∈Mv(ᏸ)and whereµis a regular outer measure. Thenµ=ν(ᏸ).
Proof. As in the proof ofTheorem 4.2, we suppose that there exists an
A∈ᏸsuch thatν(A) < µ(A)and will arrive at a contradiction. There exists
aB∈ᏸ, such thatB⊃Aand
ν(A)≤νjB< µ(A). (4.6)
Then, there existsA
i↑,Ai∈ᏸ, such thatA⊂ ∪Ai, and there existsBi∈ᏸwith
A
i⊂Bi,Bi∩B= ∅, for alli. Finally, we note that sinceν∈Mv(ᏸ),ν=ν=
ν(ᏸ). Thus, we have
µ(A)≤µ A
i
=limµA
i
≤limµA
i
≤limµBi≤limνBi=limνjBi ≤νj Bi
≤νjB< µ(A),
(4.7)
a contradiction, and we are done.
Again, recalling that any 0-1 valued outer measure is regular, we get as spe-cial cases of the preceding theorems, the following result of [6].
Corollary4.5. Letᏸbe a lattice of subsets of X which is almost normal. If µ≤ν(ᏸ)whereµ∈I(ᏸ), and whereµ∈Iσ
R(ᏸ)or, more generally,µ∈Iv(ᏸ), thenν=µ(ᏸ).
Corollary4.6. Letᏸbe a lattice of subsets ofXwhich is almost normal. If µ≤ν1(ᏸ),µ≤ν2(ᏸ)whereµ(X)=ν1(X)=ν2(X),µ∈M(ᏸ),ν1,ν2∈MRσ(ᏸ) and ifµis a regular outer measure, thenν1=ν2.
Proof. ByTheorem 4.2,
ν1=ν1=µ=ν2=ν2(ᏸ). (4.8)
Hence,ν1=ν2.
5. Further extensions. We begin by recalling the following definition.
Definition5.1. Letµ∈M(ᏸ). Thenµ∈Mw(ᏸ)(the weakly regular mea-sures) if, forL∈ᏸ,
µL=supµ˜L: ˜L⊂L,L˜∈ᏸ. (5.1)
Clearly,MR(ᏸ)⊂Mw(ᏸ), andMv(ᏸ)⊂Mw(ᏸ).
The following result in [6] has been generalized in [12].
Theorem5.2. Ifᏸis a lattice of subsets ofXsuch thatδ(ᏸ)separatesᏸ, thenµ∈Iσ(ᏸ)∩Iw(ᏸ)implies thatµ∈IR(ᏸ).
For completeness we state and prove the generalization, correcting a refer-ence which appears in [11].
Theorem5.3. Ifᏸis a lattice of subsets ofXsuch thatδ(ᏸ)separatesᏸ, thenµ∈Mσ(ᏸ)∩Mw(ᏸ)implies thatµ∈MR(ᏸ).
Proof. We recall (see [2]) that
µ∩ᏸ=L∈ᏸ:µ(L)=µ(L). (5.2)
Hence, if we can show thatᏸ⊂µ, thenµ=µ(ᏸ), andµ∈M
R(ᏸ). To this end, let >0 andA∈ᏸ. Since µ∈Mw(ᏸ), there exists a B∈ᏸ such that
B⊂A, andµ(A)−/2< µ(B)≤µ(A).
Now, there exists∩∞
1An,∩∞1Bn ∈δ(ᏸ)withAn ↓,Bn ↓, andA⊂ ∩∞1An,B⊂ ∩∞
1Bn, and∩∞1(An∩Bn)= ∅.
Nowµ∈Mσ(ᏸ); henceµ(An∩Bn)→0, soµ(An∩Bn) < /2 forn≥N. But
µA
n∪Bn
=µA
n
+µB
n
−µA
n∩Bn
≥µA
n
+µB
n
−2. (5.3)
Hence,
µA
n∪Bn
≥µ(A)+µ(B)−
2. (5.4)
Then
µA
n∪Bn
or
µA
n∪Bn
≥µ(A)+µA− (5.6)
and therefore,
µ(X)=µ(X)≥µ(A)+µA. (5.7)
This implies (see [2]) thatA∈µ, and, sinceA∈ᏸ is arbitrary, this com-pletes the proof.
A related result is the following.
Theorem5.4. Letᏸbe a lattice of subsets ofXand letµ∈Mw(ᏸ)∩Mσ(ᏸ). Ifµi(L)=sup{µ(˜L): ˜L⊂L,˜L∈ᏸ}, forL∈ᏸ, and ifᏸsemiseparatesδ(ᏸ), thenµ∈Mσ
R(ᏸ).
Proof. ForL∈ᏸ, we have
µiL=µjL=supµL˜: ˜L⊂L,˜L∈ᏸ (5.8)
(see [7] for details). Hence,µi=µj(ᏸ)and, therefore,
µ=µ(ᏸ). (5.9)
Thus,
µiL=supµ˜L: ˜L⊂L,˜L∈ᏸ=µL sinceµ∈Mw(ᏸ). (5.10)
Hence,µi=µ(ᏸ), and this implies thatµ∈MR(ᏸ), soµ∈MR(ᏸ)∩Mσ(ᏸ)=
Mσ R(ᏸ).
In a slightly different direction, we give one more result which has significant applications.
Theorem5.5. Letᏸbe a lattice of subsets ofX, and letµ≤ν(ᏸ)whereµ∈
M(ᏸ),ν∈Mσ(ᏸ), andµ(X)=ν(X). Ifᏸis normal and ifµ(L)=sup{µ(A):
A⊂L, A∈ᏸ}, forL∈ᏸ; thenµ=ν(ᏸ).
Proof. Clearly, µ ∈ Mσ(ᏸ) and ν ≤ µ. Suppose that there exists an
L,L∈ᏸ, such thatν(L) < µ(L). By hypothesis, there then exists anA⊂L, A∈ᏸsuch thatν(L) < µ(A). Now, there existsB,C∈ᏸsuch thatA⊂B⊂ C⊂L. Then,
νL< µ(A)≤µB≤µB≤µ(C)
≤ν(C)≤ν(C)≤νL, (5.11)
The result as mentioned has many applications; in particular, we note that in the case of a delta lattice and a measureν∈Mσ(ᏸ),νwill be submodular if it is submodular onᏸ.Theorem 5.5assures us, under the stated hypothesis,
thatµ will be submodular if and only ifν is submodular. These facts are
useful since submodularity of aν∈Mσ(ᏸ)assures us that the set{E⊂X:
ν(E)=ν
j(E)}is aσ-algebra, and thatνrestricted to this set is a countably additive measure (see [4]). We will not pursue these matters here.
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