Vol. 38, No. 2, 2008, 15-23
EXTENSION OF NONADDITIVE MEASURES ON LOCALLY COMPLETE σ-CONTINUOUS LATTICES
Mona Khare1, Soni Gupta2
Abstract. We introduce the concept ofMσ-approachability for a semi- continuous function (i.e. a nonadditive measure) on aσ-complete sublat- tice of a locally completeσ-continuous latticeLand using it we extend a nonadditive measure from a sublattice ofLto aσ-complete sublattice of L.
AMS Mathematics Subject Classification (2000): 28A12, 28C15, 28B10 Key words and phrases: measure, locally complete, σ-continuous, σ-complete, absolute continuity,Mσ-approachability
1. Introduction
In measure theory, a basic procedure is that of extending the notion of a measure on a given class of sets to a larger class of sets. Kelley, Nayak and Srinivasan [12] proved that a nonnegative real-valued function µ defined on a latticeLof sets is a premeasure (meaning that it extends to a countably additive measure on aδ-ring containingL) providedµis tight and continuous at∅.The extension of this theorem to the class of real-valued (not necessarily nonnegative real-valued) function is dealt in [19]. In 1981, Morales [18] established a quite general extension theorem for a uniform semigroup valued tight set functionλ on a lattice L of subsets of a setX, the domain of extension being theσ-ring generated by the latticeL. He also discussed the extension ofλon theσ-algebra of a locallyL-measurable sets.
Rieˇcan [22] proved an extension theorem for a positive real-valued modular functions defined on a suborthomodular lattice of a σ-continuous, σ-complete orthomodular lattice. An extension theorem has been proved for measures on MV-algebra in fuzzy measure theory ([4, 5, 23]; see also [10, 13, 25]). In [2], Avallone and Simone proved an extension theorem for nonnegative real-valued modular functions defined on suborthomodular lattices of a σ-continuous, σ- complete orthomodular lattice, using topological approach. They used the the- ory of lattice uniformities (i.e. a uniformity which makes the lattice operations
∨and∧uniformly continuous). They further extended the theory in context of lattice ordered effect algebras in [3].
1Department of Mathematics, University of Allahabad, Allahabad 211 001, India.
Mailing address: 10, C.S.P. Singh Marg, Allahabad-211 001; Phone: 0532-2623080; Fax:
0532-2623553; e-mail: [email protected];
2Allahabad Mathematical Society, 10 C.S.P. Singh Marg, Allahabad 211 001, India; e-mail:
A variety of structural characteristics of nonadditive set functions are intro- duced and discussed by Dobrakov [7, 8], Drewnowski [9], Wang [24], Wang and Klir [26], Pap [20] and Denneberg [6], and the relevant theories are developed by them separately. Nonadditive measures appear today in many branches of pure mathematics with many important applications ([21, 24], see also [14, 15, 16]).
The aim of the present paper is to study an extension problem for nonaddi- tive measures defined on a sublattice of a locally completeσ-continuous lattice L.Some basic definitions are collected in Section 2. In Section 3, we introduce notions of absolute continuity and Mσ-approachability and we extend a semi- continuous function (i.e. a nonadditive measure, or simply a measure) µ on a sublatticeM ofLto a uniqueMσ-approachable measureµbon aσ-complete sub- latticeN containingM under suitable conditions. This goal has been achieved in three steps: firstly we extend any lsc-measure on M to an lsc-measure on Mσ.To obtain extension of a measureµonM to a measureeµonMσ,we need a sufficient condition involving the notion of absolute continuity; µe preserves absolute continuity. Finally, in the third step we prove that µ can uniquely be extended to µb onN containing Mσ, using Mσ-approachability. Extensions obtained at each step are uniquely determined. Some basic results on modular functions are also obtained and we prove that if µ is submodular, then µe is submodular and consequentlyµbis submodular.
2. Preliminaries and Basic Results
Let (P,≤) be a poset. An elementx∈P is called anupper bound ofA⊆P ifa≤xfor everya∈A;xis called alower bound ofA, ifx≤afor everya∈A.
An element x∈ L is called the join (or the least upper bound, or the sup) of A⊆L, denoted by W
A, if (i)xis an upper bound ofA,
(ii) ify is an upper bound ofA, thenx≤y.
If A is finite, we call W
A (if it exists) afinite join. If A contains only two elementsaandb, then we sometimes also writea∨b instead ofW
{a, b}for our convenience; similarlyW
A=a1∨a2∨. . .∨anwhereA={a1, a2, . . . , an}(n∈N;
Ndenotes the set of all natural numbers). A lattice is a poset (L,≤) in which both join and meet for every finite subset of L exist. For a lattice L, for all a, b∈L, a≤bif and only ifa∨b=bif and only ifa∧b=a.A latticeL= (L,≤) is calledcomplete if every (possibly empty) subset ofLadmits an infimum (or equivalently if every subset ofLadmits a supremum). The symbol 1 denotes the top element (or supremum) ofLand 0 denotes the bottom element (or infimum) ofL(cf. [17]). A latticeLis said to beσ-complete, if every countable subset of Lhas a supremum and an infimum [3].
2.1 [1]. A lattice L is locally complete if it satisfies one of the following equivalent conditions:
(i) Every nonempty lower bounded subset of Ladmits an infimum.
(ii) Every nonempty upper bounded subset of Ladmits a supremum.
(iii) There exists a complete lattice, denoted byL,with the bottom element 0 and top element 1, such that Lis a sublattice ofL,L=L∪ {0,1}, infL= 0
and supL= 1.
It can be observed that every complete lattice is locally complete.
2.2. Let{an} be a sequence in a latticeL= (L,≤).We call an ↑a,(a∈L) if and only if a1≤a2 ≤. . .≤an ≤. . .,W
an exists andW
an =a. In this case we also write a= limn→∞an. If an ↑ a, bn ↑ b and an ≤bn,for alln, then we may deduce thata≤b.
2.3 [2]. A latticeLis said to beσ-continuous ifan↑aimpliesan∧b↑a∧b (or equivalently,an↓aimpliesan∨b↓a∨b) for everyb∈L.IfLisσ-continuous then, for the sequences{an}and{bn}inLsuch thatan↑aandbn↑b, we have an∧bn ↑a∧b(or equivalently,an↓a, bn↓bimpliesan∨bn↓a∨b).
Every infinitely distributive lattice [17]Lisσ-continuous.
For any setX, (P(X),⊆), (LX,≤) (whereLis locally completeσ-continuous lattice) and (I,≤) where (Iis the closed unit interval [0, 1] of the real lineR) are locally complete σ-continuous lattices. For more examples of locally complete lattices, we refer to [1].
2.4. A functionµ:L→[0,∞) is said to be modular if, for every a, b∈L, µ(a∨b) +µ(a∧b) = µ(a) +µ(b). We say that µ is submodular if for every a, b∈L, we haveµ(a) +µ(b)≥µ(a∨b) +µ(a∧b).
3. M
σ-Approachability and Extension of Nonadditive Mea- sures
LetL be a locally complete σ-continuous lattice and let C be a nonempty subset of L, M be a sublattice of L, and N be a σ-complete sublattice of L containing M. We denote Mσ = {b ∈ L : there exists a sequence{an}in Msuch thatan ↑ b}. We may also describe Mσ as the family of all countable joins of elements fromM. ThenMσ is a sublattice ofLcontainingM.
Definition 3.1. A function µ : C → [0,∞) is called a semi-continuous measure(or nonadditive measure, or simply a measure)onC, if it satisfies the following conditions:
(i)µ(0) = 0, whenever0∈C,
(ii) (monotone)ifa≤b, a, b∈C, thenµ(a)≤µ(b),
(iii) (semi-continuous from below) if an ↑ a, a ∈C, an ∈C(n ∈N), then limn→∞µ(an) =µ(a),
(iv) (semi-continuous from above) if an ↓ a, a∈ C, an ∈ C (n ∈N), then limn→∞µ(an) =µ(a).
The functionµis said to be a lower semi-continuous measure(or lsc-measure) if it satisfies (i),(ii)and (iii), while µ is said to be an upper semi-continuous measure(or usc-measure)if it satisfies (i),(ii) and(iv).
Definition 3.2. A nondecreasing function µ : C → [0,∞) is said to be lower(respectively, upper)consistent onC, if for everyb∈C,an∈C(n∈N), an ↑a withb≤a we havelimn→∞µ(an)≥µ(b) (respectively,an↓aand a≤b we have limn→∞µ(an)≤µ(b)).
We obtain the following:
Proposition 3.1. Let µ : C → [0,∞) be a monotone function. If C is closed under the formation of finite meet (respectively, finite join), then µ is lower (respectively, upper) consistent if and only if µ is semi-continuous from below (respectively, semi-continuous from above).
Lemma 3.1. Let µ : C → [0,∞) be a monotone and semi-continuous from below function, whereC is closed under the formation of finite meet. For a, b∈L, let an ↑a, bn ↑b, an, bn∈C(n∈N). If a≤b,thenlimn→∞µ(an)≤ limn→∞µ(bn).
Theorem 3.1. Ifµis an lsc-measure onM,thenµcan be extended uniquely to an lsc-measure on Mσ.
Proof. Forb∈Mσ, defineµ(b) = lime n→∞µ(an) when{an}is a sequence inM andan↑b. In view of Lemma 3.1,µeis well defined.
For monotonicity, suppose that a, b ∈ Mσ and a ≤ b. Then there exist sequences {an} and {bn} in M such that an ↑ a and bn ↑ b. Since L is σ- continuous, soan∧bn ↑a∧b=a.Now
e
µ(b) = lim
n→∞µ(bn)≥ lim
n→∞µ(an∧bn) =µ(a).e
Next, suppose that{an}is a sequence inMσandan↑a, a∈Mσ.Then there exists a sequence{ani}∞i=1inM such thatani↑anandµ(ae n) = limi→∞µ(ani), (n∈N).Fori∈N,setbi =a1i∨a2i. . .∨aii.Thenbi∈M,{bi}is an increasing sequence and bi ≤ ai ≤ a for all i, which yield that b = limi→∞bi = ∨bi ≤
∨ai=a.It may be noted thatb∈Mσ.
Also, aki≤bi for 1≤k≤i.Therefore, ak = limi→∞aki≤limn→∞bn =b.
It follows thata=∨ak ≤b.Thusa=b.Now, from the monotonicity ofµ,e e
µ(a) = lim
n→∞µ(bn) = lim
n→∞µ(be n)≤ lim
n→∞µ(ae n),
and the result follows. Obviously, the extensionµeis unique. 2
Definition 3.3. Let µ and ν be two measures on C. We say that µ is absolutely continuous with respect to ν (denoted by µ¿ν), if for every ε >0, there exists δ > 0 such that |µ(c)−µ(b)| < ε, whenever b, c∈ C and |ν(c)− ν(b)|< δ.(cf. [11])
Theorem 3.2. Let µ be a measure on M. Then µ can be extended to a measureµeonMσ,provided there exists a measureν onMσ such thatµ¿ν on M. The extension is unique andeµ¿ν onMσ.
Proof. In view of Theorem 3.1, we need only to prove thatµeis semi-continuous from above. For this, let {an} be a sequence in Mσ and an ↓ a0, a0 ∈ Mσ. Then there exists a sequence{ani}∞i=1 in M such thatani↑ an (n∈N∪ {0}).
Letε >0.Sinceµ¿ν onM there exists δ >0 such that|µ(c)−µ(b)|< ε/2, wheneverb, c∈M and|ν(c)−ν(b)|< δ.Sinceν is a measure onMσandan ↓a0
there existsm∈Nsuch that
ν(an)< ν(a0) +δ/2,for alln≥m.
Since a0i ↑ a0 and ani ↑ an (n ∈ N) and ν is a measure on Mσ, there exists k∈Nsuch that, for alli≥k,
ν(a0)< ν(a0i) +δ/2, ν(an)< ν(ani) +δ and µ(ae n)< µ(ani) +ε/2, (by definition ofµ). So, fore n≥m, i≥kwe have
ν(ani)≤ν(an)< ν(a0) +δ/2< ν(a0i) +δ, and also
ν(a0i)≤ν(a0)≤ν(an)< ν(ani) +δ,
which yield that|ν(ani)−ν(a0i)|< δ.Sinceµ¿ν, we have|µ(ani)−µ(a0i)|<
ε/2.Therefore, we get (fori≥k) e
µ(an)< µ(ani) +ε/2< µ(a0i) +ε≤eµ(a0) +ε,for alln≥m.
Since a0 ≤ an and eµis monotone, the result follows. Using similar argument we have µe¿ν onMσ.The uniqueness is proved in Theorem 3.1. ¤
Now to extend a measure from a sublattice M (containing 1, the largest element ofL) to aσ-complete sublatticeN,containingM,of a locally complete σ-continuous lattice L,we introduce a new concept ofMσ-approachability of a measure onN.
Definition 3.4. A measure µ on N is said to be Mσ-approachable if for a∈N and forε >0 there existsb∈Mσ such thata≤b andµ(b)< µ(a) +ε.
Theorem 3.3. A measureµonM can be extended to anMσ-approachable measure onN, provided there exists anMσ-approachable measureν onN such that µ¿ν on M. The extension is unique and it preserves the absolute conti- nuity with respect to ν.
Proof. From Theorem 3.2, we haveµe¿ν onMσ and the extensionµeis unique.
We define, fora∈N, b
µ(a) = inf{eµ(b) :a≤b, b∈Mσ}.
Clearly,µ(a) =e bµ(a) fora∈Mσ,µ(0) = 0 if 0b ∈M, andµbis monotone.
Now, to prove µb is semi-continuous from below, suppose that {an} is a sequence N such that an ↑ a0, a0 ∈ N. Let ε > 0. Since µe ¿ ν on Mσ, there exists δ > 0 such that |eµ(c)−eµ(b)| < ε/2 whenever b, c ∈ Mσ and
|ν(c)−ν(b)|< δ.Sinceν is a measure on N,there existsm∈Nsuch that ν(a0)< ν(an) +δ/2, for alln≥m,
and consequently, we obtainbm∈Mσ witham≤bmandeµ(bm)<µ(ab m) +ε/2.
Since ν is Mσ-approachable on N, we get b0 ∈ Mσ with a0 ≤ b0 such that ν(b0)< ν(a0) +δ/2.SinceMσ is closed under the formation of finite meet, we may assume thatbm≤b0 (replacebmby (bm∧b0)). Thus
ν(b0)< ν(am) +δ≤ν(bm) +δ,
which yields|ν(b0)−ν(bm)|< δ.Sinceµe¿ν onMσ, we have|eµ(b0)−µ(be m)|<
ε/2.Now, we have b
µ(a0)≤µ(bb 0) =µ(be 0)<µ(be m) +ε/2<bµ(am) +ε.
Since an ≤ a0 and µb is monotone, it follows that µb is semi-continuous from below. Similarly we may prove thatµb is semi-continuous from above, and also that µb¿ν onN, andµb is the unique extension. Thatµb isMσ-approachable follows from its definition.
Proposition 3.2. A functionµ:M →[0,∞) is modular if and only if µ(a1) +µ(b1) =µ(a2) +µ(b2) (1) wherea1, b1, a2, b2∈M with a1∧b1=a2∧b2 anda1∨b1=a2∨b2.
Proof. Letµbe modular. For a1, b1, a2, b2∈M such thata1∧b1=a2∧b2and a1∨b1=a2∨b2, we haveµ(a1∨b1) =µ(a2∨b2), and so,
µ(a1) +µ(b1)−µ(a1∧b1) =µ(a2) +µ(b2)−µ(a2∧b2).
Sinceµ(a1∧b1) =µ(a2∧b2) we have equation (1). Conversely, let (1) hold. Let a, b∈M. Since ((a∨b)∨(a∧b)) =a∨b and ((a∨b)∧(a∧b)) =a∧b.Then from (1), we have
µ(a∨b) +µ(a∧b) =µ(a) +µ(b).
Henceµis modular.
Theorem 3.4. Let µbe an lsc-measure onM (containing1). Then(i)⇒ (ii)⇒(iii)
(i) µis submodular.
(ii)eµis submodular.
(iii)µbis submodular.
Proof. (i)⇒(ii). Letµbe submodular. Leta, b∈Mσ.Then there exist sequences {an}and {bn}in M such thatan↑a, bn ↑b,µ(a) = lime n→∞µ(an) andµ(b) =e limn→∞µ(bn). Since L is σ-continuous, so an ∧bn ↑ a∧b and also we have an∨bn ↑a∨b.Therefore
e
µ(a) +µ(b) = lime
n→∞µ(an) + lim
n→∞µ(bn)
= lim
n→∞(µ(an) +µ(bn))
≥ lim
n→∞(µ(an∨bn) +µ(an∧bn))
= lim
n→∞µ(an∨bn) + lim
n→∞µ(an∧bn)
=eµ(a∨b) +µ(ae ∧b).
(ii)⇒(iii). Leta, b∈N.Forε >0, we havec, d∈Mσsuch thata≤c, b≤d, b
µ(a)>µ(c)e −ε/2 andµ(b)b >µ(d)e −ε/2.Thus, b
µ(a) +bµ(b) +ε >eµ(c) +µ(d)e
≥µ(ce ∨d) +µ(ce ∧d)
≥µ(ab ∨b) +µ(ab ∧b).
Sinceεis arbitrary, we have b
µ(a) +bµ(b)≥bµ(a∨b) +µ(ab ∧b). 2
Concluding Remark
If we takeM to be a sublattice ofL(Lbeing locally completeσ-continuous lattice) and consider a nonnegative extended real-valued µ, then Lemma 3.1, Theorem 3.1, Theorem 3.2 remain valid providedνis a finite measure (obviously then µ(a1) < ∞ would be needed in Definition 3.1(iv)). However, for the extension of µonM toN we need to take the top element 1 inM. In view of Theorem 3.4, ifµis submodular thenµeis submodular.
Since (P(X),⊆), (I,≤) (whereIis the closed unit interval [0, 1] of the real line R), and (LX,≤) (provided L is a locally complete σ-continuous lattice) are locally completeσ-continuous lattices, the present study provides a unified approach for classical theory, popular fuzzy theory and theory ofL-fuzzy sets.
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Received by the editors July 6, 2007
