MONA KHARE AND BHAWNA SINGH
Received 2 September 2004 and in revised form 30 December 2004
The present paper deals with the study of a weakly tight function and its relation to tight functions. We obtain a Jordan-decomposition-type theorem for a locally bounded weakly tight real-valued function defined on a sublattice ofIX, followed by the notion of a total variation.
1. Introduction
The notion of a signed measure arises if a measure is allowed to take on both positive and negative values. A set that is both positive and negative with respect to a signed measure is termed as a null set. Some concepts in measure theory can be generalized by means of classes of null sets. An abstract formulation and proof of the Lebesgue decomposition theorem using the concept of null sets is given by Ficker [5]. A real-valued function satis-fying certain properties that can be expressed as a difference of two nonnegative functions possessing the same properties is called “decomposable.” Several Jordan-decomposition-type theorems are exhibited in [3]. Faires and Morrison [4] exposed conditions on a vector-valued measure that ensure vector-valued Jordan-decomposition-type theorem to hold. For a signed null-additive fuzzy measure, a Jordan-decomposition-type theorem is investigated by Pap in [11].
The problem of generation of measures by tight functions defined on a lattice of sets has been taken up by several authors [1,2,6,8,9]. Nayak and Srinivasan [10] initiated a weaker form of tightness for a real-valued function µdefined on a lattice of sets to decomposeµas a differenceµ+−µ−and then extended it to a countably additive measure.
InSection 2, we have defined and studied the notions of measuring envelopes, modu-lar functions, and additive functions. The notions of superadditive and subadditive func-tions are also given with the help of pointwise addition of elements in IX. The lower envelopeβ∗of a superadditive functionβdefined on a sublatticeKofIXturns out to be
superadditive. InSection 3, we introduce the notion of a weakly tight functionβ:K→R, whereKis a sublattice ofIXcontaining 0 and 1 (cf. [10]). The condition imposed on the [0, 1]-valued functionβto be a weakly tight function is less restrictive than that for being a tight function. It is proved that a superadditive, monotone, and weakly tight function
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βis tight (thereK is taken to be closed under addition). InSection 4, our main result states that a locally bounded, weakly tight real-valued functionβdefined onKhas a rep-resentation of the formβ+−β−; bothβ+andβ−are nonnegative monotone (and hence,
locally bounded) functions defined onK. If, in addition,βis additive and modular, then the decomposed partsβ+andβ−preserve superadditive and supermodular properties.
The total variation|β|ofβis defined as the sum ofβ+andβ−, following the terminology
in the classical measure theory (cf. [12]); few properties of|β|are noted.
Notations. Throughout this paper,Xis a nonempty set andI≡[0, 1] is the unit interval of the real lineR;Cdenotes a subfamily ofIXof all functions fromXtoI;Kstands for a sublattice ofIXcontaining the least element 0 and the greatest element 1, where 0 and 1 are constant functions sending eachx∈Xto 0 and 1, respectively. We will denote byβa function fromKtoIsatisfyingβ(0)=0.
2. Measuring envelopes
LetCbe a sublattice ofIXand letξ:C→Ibe a function. Thenξis calledmonotoneif f, g∈C,g≤f ⇒ξ(g)≤ξ(f). The mappingξis calledsupermodular(submodular, resp.) if for f,g∈C,ξ(f) +ξ(g)≤ξ(f ∨g) +ξ(f ∧g)(ξ(f) +ξ(g)≥ξ(f ∨g) +ξ(f ∧g), resp.); ξ is said to be modularif it is both supermodular and submodular. The mappingξ is said to besuperadditive(subadditive, resp.) if for f1,f2∈Csuch that f1+f2∈C, ξ(f1+
f2)≥ξ(f1) +ξ(f2) (ξ(f1+f2)≤ξ(f1) +ξ(f2), resp.);ξ is said to beadditiveif it is both
superadditive and subadditive. If we restrict ourselves to disjoint crisp sets inC, then the condition of being additive forξcoincides with that of [10].
The familyC⊆IX is said to beclosed under additionif for f,g∈Cwith f +g∈IX, we have f +g∈C, and is said to beclosed under addition modulo1 (orclosed under⊕) if
f,g∈Cimplies that f⊕g∈C, where f ⊕g=(f+g)∧1.
A functionξ:C→Iis calledsubadditive modulo1 if forf1,f2∈Cwithf1⊕f2∈C, we
haveξ(f1⊕f2)≤ξ(f1) +ξ(f2); hereC⊆IX.
If f,g∈IX and f +g∈IX, then f ⊕g= f +g. Thus ifCis closed under addition modulo 1, thenCis closed under addition. Also, ifξ is subadditive modulo 1, thenξ is subadditive.
The definition for a functionξ:C→Ito be supermodular (submodular, superaddi-tive, subaddisuperaddi-tive, resp.) continues to hold for a real-valued functionξdefined onC. Definition 2.1. Let β:K→I be a function satisfying β(0)=0. Defineβ∗:IX→I and
β∗:IX→Iby
β∗(f)=supβ(g) :g≤ f,g∈K,
β∗(f)=infβ(g) :g≥f,g∈K, f ∈IX. (2.1)
β∗andβ∗are called thelower envelopeand theupper envelopeofβ, respectively.
We obtain
(i)β∗(0)=0=β∗(0);
(iii)β∗|K≤β≤β∗|K;
(iv)βis monotone if and only ifβ∗|K=β=β∗|K.
Proposition2.2. (i)Ifβis supermodular, thenβ∗is supermodular.
(ii) Ifβis submodular, thenβ∗is submodular.
Proof. (i) Let f1,f2∈IX. Letε >0. Then there existg1,g2∈K, f1≥g1, f2≥g2such that
β∗f1
−ε 2< β
g1
,
β∗f2
−ε 2< β
g2
. (2.2)
It follows that
β∗f1
+β∗f2
−ε < βg1∨g2
+βg1∧g2
. (2.3)
Consequently,
β∗f1
+β∗f2
−ε < β∗f1∨f2
+β∗f1∧f2
. (2.4)
Sinceεis arbitrary, we get
β∗f1
+β∗f2
≤β∗f1∨f2
+β∗f1∧f2
. (2.5)
Proof of (ii) follows analogously.
Proposition2.3. (i)IfKis closed under addition andβis superadditive, thenβ∗is
super-additive.
(ii) IfK is closed under addition modulo1andβis subadditive modulo 1, thenβ∗is subadditive modulo1, and henceβ∗is subadditive.
Proof. (i) Letf1and f2be inIXsuch that f1+ f2∈IX. Letε >0. Then there existg1,g2∈K
withg1≤ f1andg2≤ f2such thatβ(g1)> β∗(f1)−ε/2 andβ(g2)> β∗(f2)−ε/2. Hence,
β∗(f1) +β∗(f2)−ε < β(g1) +β(g2). Since, for anyx∈X, 0≤(g1+g2)(x)≤(f1+f2)(x)≤
1, we getg1+g2∈IX and sog1+g2∈K. Sinceβis superadditive,β∗(f1) +β∗(f2)−ε <
β(g1+g2)≤β∗(f1+f2) and hence, we obtainβ∗(f1) +β∗(f2)≤β∗(f1+f2).
(ii) Let f1,f2∈IX. Let ε >0. Then there exist g1,g2∈K with f1≤g1 and f2≤g2
such that
β∗f 1
+β∗f2
+ε > βg1
+βg2
. (2.6)
Sinceg1,g2∈KandKis closed under⊕, we getg1⊕g2∈K. Also,f1⊕f2≤g1⊕g2. Hence
(2.6) gives
β∗f 1
+β∗f2
+ε > βg1⊕g2
≥β∗f 1⊕f2
. (2.7)
Definitions 2.4. Letβ1andβ2be real-valued functions defined onK. Define (β1+β2)(f)=
β1(f) +β2(f), f ∈K, and (β1−β2)(f)=β1(f)−β2(f), f ∈K. Likewise, forβ:K→R
andλ∈R, define (λβ)(f)=λβ(f), f ∈K.
Proposition2.5. (i)Ifβ1 andβ2are supermodular (submodular, resp.), thenβ1+β2 is supermodular (submodular, resp.).
If bothβ1andβ2are modular, then so areβ1+β2andβ1−β2.
(ii) Ifβ is supermodular (submodular, resp.) then λβ is supermodular (submodular, resp.), whereλis a nonnegative real number.
(iii) IfK is closed under addition, andβ1,β2are superadditive (subadditive, resp.), then
β1+β2is superadditive (subadditive, resp.). If bothβ1andβ2are additive, then so areβ1+β2 andβ1−β2.
(iv) IfKis closed under addition, andβis superadditive (subadditive, resp.), thenλβis superadditive (subadditive, resp.), whereλis a nonnegative real number.
Proof. We will prove only (i) and (iii).
(i) Letβ1andβ2be supermodular. Letf,g∈K. Then,
β1+β2
(f) +β1+β2
(g)=β1(f) +β2(f) +β1(g) +β2(g)
≤β1(f∨g) +β1(f ∧g) +β2(f ∨g) +β2(f∧g)
=β1+β2(f∨g) +β1+β2(f ∧g).
(2.8)
Ifβ1andβ2are submodular, then, by similar arguments,β1+β2is submodular.
(iii) LetK be closed under addition. Letβ1 and β2 be superadditive. Let f,g∈K.
Then,
β1+β2
(f) +β1+β2
(g)=β1(f) +β2(f) +β1(g) +β2(g)
≤β1(f+g) +β2(f +g)=β1+β2(f +g).
(2.9)
3. Weakly tight functions
Definition 3.1. Letβ:K→Iwithβ(0)=0. Thenβis calledtight(cotight, resp.) if
βf2
=βf1
+β∗f2−f1
, f1,f2∈K, f1≤ f2,
βf2
=βf1
+β∗f2−f1
, f1,f2∈K, f1≤f2, resp.
. (3.1)
Ifβis tight (or cotight), then βis modular and monotone. Furthermore, ifβis tight (cotight, resp.), then β∗ (β∗, resp.) is an extension ofβ. A detailed study of tight and
cotight functions is made in [7,13].
Definition 3.2. Letβ:K→Rbe a function withβ(0)=0. Thenβis calledweakly tight if for every pair f1,f2∈K with f1≤ f2 and for anyε >0, there exists f ∈K such that
f ≤f2−f1and
β
f2
−βf1
Proposition3.3. Letβ:K→I be a function satisfyingβ(0)=0. Ifβis tight, thenβis weakly tight.
Proof. Let f1,f2∈Kwith f1≤f2. Letε >0. Sinceβis tight, there exists f ∈K with f ≤
f2−f1such that
β∗f2−f1
−ε=βf2
−βf1
−ε < β(f) (3.3)
or
βf2
−βf1
−β(f)< ε. (3.4)
Sinceβis monotone and f ≤ f2−f1, we obtain
β(f)=β∗(f)≤β∗f2−f1
=βf2
−βf1
, (3.5)
and so
βf2
−βf1
−β(f)≥0. (3.6)
Thus,|β(f2)−β(f1)−β(f)|< ε.
Proposition3.4. LetKbe closed under addition. Letβ:K→Ibe superadditive, monotone, and weakly tight. Thenβis tight.
Proof. Let f1,f2∈K with f1≤ f2. Letε >0. Sinceβis weakly tight, there exists f ∈K
such that f ≤ f2−f1and
β
f2
−βf1
−β(f)< ε. (3.7)
Consequently,
βf2
−βf1
< β(f) +ε≤β∗f2−f1
+ε. (3.8)
Sinceεis arbitrary, we getβ(f2)−β(f1)≤β∗(f2−f1).
Since, byProposition 2.3(i),β∗is superadditive, we getβ(f2)=β∗(f2)≥β∗(f2−f1) +
β∗(f1), which yields thatβ∗(f2−f1)≤β∗(f2)−β∗(f1)=β(f2)−β(f1).
Thusβ∗(f2−f1)=β(f2)−β(f1), that is,βis tight.
4. A Jordan-decomposition-type theorem
In this section,βis a real-valued function defined on a sublatticeK ofIX containing 0 and 1. Also, it is assumed throughout this section thatβislocally bounded, that is, for any f inK, sup{β(g) :g≤ f,g∈K}exists. For a locally bounded real-valued functionβ, the definitions of lower and upper envelopes,β∗andβ∗, ofβmay be given in the same way.
Definition 4.1. For f ∈K, defineβ+(f) andβ−(f) as follows:
β+(f)=supβ(g) :g≤f,g∈K,
Remarks 4.2. (i)β+=β ∗|K.
(ii) β−=(−β)+;β+=(−β)−.
(iii) Bothβ+andβ−are nonnegative, monotone (and hence, locally bounded).
(iv) −β−≤β≤β+.
Theorem4.3. Letβbe a weakly tight real-valued function defined onK. Thenβ=β+−β−.
Proof. Letε >0. Let f ∈K. Then there exists f1∈Ksuch that f1≤ f and
β+(f)< βf 1
+ε
2. (4.2)
Sinceβis weakly tight, there exists f2∈Ksuch thatf2≤f −f1and β(f)−
βf1
+βf2<ε
2. (4.3)
This implies that
βf1
+βf2
< β(f) +ε
2. (4.4)
Also, f2≤ f − f1≤ f yields that −β−(f)≤β(f2). Hence, using (4.2) and (4.4), we get
β+(f)−β−(f)< β(f1) +ε/2 +β(f2)< β(f) +ε. Sinceεis arbitrary, we get
β+−β−≤β. (4.5)
Replacingβin (4.5) by−β, we get
(−β)+−(−β)−≤ −β, (4.6)
or
β−−β+≤ −β, (4.7)
or
β+−β−≥β. (4.8)
Thus,β+−β−=β.
Proposition4.4. LetKbe closed under addition. Ifβis additive, then bothβ+andβ−are
superadditive.
Proof. The proof follows fromProposition 2.3(i).
Proposition4.5. Ifβis modular, then bothβ+andβ−are supermodular.
Proof. The proof follows fromProposition 2.2(i).
Theorem4.6 (Jordan-decomposition-type theorem). LetKbe a sublattice ofIX contain-ing0and1. Ifβ:K→Ris locally bounded and weakly tight, thenβcan be written as
β=β+−β−, (4.9)
where bothβ+andβ−are nonnegative and monotone (and hence, locally bounded) functions
defined onK. Furthermore, ifβis modular (βis additive andK is closed under addition), then the decomposed partsβ+andβ−are supermodular (superadditive).
Definition 4.7. For a functionβ:K→R, define thetotal variationofβ, written as|β|, by
|β| =β++β−. (4.10)
Theorem4.8. Letβ:K→Rbe a locally bounded function. (i) Ifβis weakly tight, thenβ=0⇔ |β| =0.
(ii) For eachf ∈K,|β(f)| ≤ |β|(f). (iii) Ifβis modular, then|β|is supermodular.
(iv) LetKbe closed under addition. Thenβbeing additive implies that|β|is superaddi-tive.
Proof. (i) Ifβ=0, thenβ+=0=β−and so|β| =0. Conversely, if|β| =0, then bothβ+
andβ−vanish. Sinceβis weakly tight, byTheorem 4.3,β=β+−β−. Henceβ=0.
(ii) Let f ∈K. Then byRemark 4.2(iv),β(f)≤β+(f), and−β(f)≤β−(f).
Ifβ(f)>0, then
|β|(f)=β+(f) +β−(f)≥β(f)=β(f). (4.11)
Ifβ(f)<0, then
β(f)= −β(f)≤β−(f)≤ |β|(f). (4.12)
(iii) Follows fromProposition 4.5andProposition 2.5(i).
(iv) Follows fromProposition 4.4andProposition 2.5(iii). Remark 4.9. If we restrict ourselves to{0, 1}-valued functions inK, thenKmay be viewed as a sublattice ofP(X). For a [0,∞]-valued functionβdefined on this restrictedK over P(X), the definitions of lower and upper envelopes,β∗andβ∗, reduce to the
correspond-ing definitions in classical theory given by Adamski [1]. In this manner, the present study generalizes the theory in [1].
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Mona Khare: Department of Mathematics and Statistics, Faculty of Science, University of Allah-abad, Allahabad 211002, Uttar Pradesh, India
E-mail address:raghav9802@rediffmail.com
Bhawna Singh: Allahabad Mathematical Society, 10 CSP Singh Marg, Allahabad 211001, Uttar Pradesh, India
