When fuzzy measures are upper envelopes of probability measures

Fuzzy Sets and Systems 138 (2003) 455–468www.elsevier.com/locate/fss

When fuzzy measures are upper envelopes ofprobability measures

Volker Kr)atschmer∗

Statistik und Okonometrie, Rechts- und Wirtschaftswissenschaftliche Fakultat der Universitat des Saarlandes, Bau 31,Postfach 151150, D-66041 Saarbrucken, Germany

Received 29 June 1999; received in revised form 24 May 2002; accepted 24 September 2002

Abstract

Fuzzy measures which are upper envelopes of probability measures may play an important role to develop ageneral theory of Bayesian statistics. Especially from a technical point of view, a widely accepted generalizedBayes rule would be applicable for those kind of fuzzy measures.

We give su3cient general conditions to ensure that fuzzy measures are upper envelopes of probabilitymeasures. They are applied to some special classes of important types of fuzzy measures, namely Sugenofuzzy, plausibility and possibility measures. The proof of the main result is based on recently systemized innerextension procedures within abstract measure theory.c© 2002 Elsevier B.V. All rights reserved.

Keywords: Fuzzy measures; Coherent upper probability; Inner regularity; Semiregularity; Inner extensions; Inner premeasures

0. Introduction

Let (X;A; W�) denote a statistical space, depending on an unknown parameter # in a param-eter space �. The knowledge of # might be represented by some fuzzy measure on a �-algebraon �. Unfortunately, a general theory of Bayesian statistics on the basis of fuzzy measures is notestablished yet.

But there is a link with robust Bayesian statistics, which deals with situations of the so-calledincomplete information about the parameter #. Incomplete information about # means that there issome true probability measure which represents the information about # but is only reliable in theform of a family of probability measures. The upper and lower envelopes of those families have beenused to establish statistical tests in the context of incomplete information (e.g. [8]). Since an upper

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456 V. Kratschmer / Fuzzy Sets and Systems 138 (2003) 455–468

envelope � of a family of probability measures is conjugated with the respective lower envelope �by �(A) := 1− �(�\A), it su3ces to treat either the upper or the lower envelopes. Upper envelopescan be proved as fuzzy measures, which implies that the prior knowledge of incomplete informationabout the parameter # is in fact represented by a fuzzy measure.

Conversely, if the fuzzy measure that summarizes the prior knowledge of the parameter # isan upper envelope of probability measures, we have the situation of incomplete information, andwe can utilize methods of robust Bayesian statistics. So in order to develop a Bayesian theory onthe basis of fuzzy measures, it might be worthy to deal with the problem under which conditionsfuzzy measures can be approximated by probability measures as upper envelopes. Especially, we areprovided with a very natural procedure to update the prior fuzzy measure. It is has been advocated,e.g. by Dubois and Prade [6], using the name Bayesian conditioning. Bayesian conditioning relieson taking the upper envelope of the conditional probability measures that can be constructed withthe probability measures that are dominated by the fuzzy measure.

The very general result of the approximation problem has been established by Adamski [1] andis not well known in fuzzy literature. We shall generalize it, following a diFerent line of reasoningwhich emphasizes the contribution of the inner extension procedures within abstract measure theory.Further application leads to plausibility and possibility measures as upper envelopes of probabilitymeasures, even in the context of inGnite universes of discourse. Especially, this supports the se-mantical point of view to regard possibility degrees as upper bounds of probabilities, an argumentproposed by Dubois and Prade (cf. [5,6]).

The paper is organized as follows. Section 1 recalls some notions from the abstract measuretheory, and links them with Walley’s theory of imprecise probabilities [20]. Section 2 is devoted tothe exposition of the issue of this paper, namely the approximation of fuzzy measures by probabilitymeasures. We shall state our main results, especially our main theorem which generalizes the above-mentioned result by Adamski. In Sections 3–5 the main theorem is speciGed for fuzzy measures onBorel subsets of HausdorF, metrizable and discrete topological spaces. Afterwards we shall pointout su3cient conditions for some special classes of important types of fuzzy measures to be upperenvelopes of probability measures. In particular, downward continuous plausibility measures on powersets of arbitrary universes and all possibility measures are upper envelopes of probability measures.This extends the known results from Gnite to inGnite universes of discourse.

The remainder of the paper is dedicated to the proof of the main theorem. In Section 7, themain tools will be presented in particular, K)onig’s procedures to extend set functions on lattices tocontents on algebras, or measures on �-algebras [11–14]. The main theorem is then an examinationof these extension procedures as Section 8 will show.

1. Notations and preliminaries

Let (;F) be a measurable space, let P() denote the power set of , and let � :F→ [0;∞[be a fuzzy measure, i.e. �(A)6�(B) for A⊆B, and �(∅) = 0; �() = 1. Throughout this paper wewant to investigate the following approximation problem:

� = sup P�;

where P� consists of all probability measures P on F with P6�. The supremum is taken pointwise.Our aim is to evaluate necessary and su3cient conditions.

V. Kratschmer / Fuzzy Sets and Systems 138 (2003) 455–468 457

For preparation, we recall some notions from abstract measure theory ([11], partly [4]) and intro-duce the concept of coherent upper probabilities [20].

Let us consider a set function � on a lattice S in , i.e. a function � :S→ [0;∞] with∅∈S; �(∅) = 0. � will be called isotone if �(A)6�(B) holds for A⊆B. In this view, a fuzzymeasure is an isotone real-valued set function � on a �-algebra with normalizing condition �() = 1.

A set function � on a lattice S is said to be submodular (supermodular) if the inequality�(A∪B) + �(A∩B)6(¿)�(A) + �(B) holds for all A; B∈S, and modular if it is super- and sub-modular. Every isotone, modular set function on the lattice S can be described as the restriction ofan isotone, modular set function on the power set P() [14, Theorem 1.3].

Lemma 1.1. A set function � on a lattice S in is isotone and modular if and only if thereexists some isotone and modular set function �̂ on the power set P() such that �= �̂|S holds.

A set function � on a lattice S in is deGned to be upward (downward) � continuous if for

an arbitrary isotone (antitone) sequence (Ak)k in S, with∞∪k=1

Ak∈S (∞∩k=1

Ak∈S),

�

( ∞⋃k=1

Ak

)= lim

k→∞�(Ak) resp: �

( ∞⋂k=1

Ak

)= lim

k→∞�(Ak)

holds. We shall call � downward � continuous at A (A∈S) if �(A) = limk→∞

�(Ak) for every antitone

sequence (Ak)k in S with∞∩k=1

Ak =A.

Introducing Walley’s concept of coherence, we mention that a set function � on a lattice S in is a coherent upper probability if it is real-valued, and if

sup!∈

[n∑

i=1

(�(Ai) − 1Ai(!)) − m(�(A0) − 1A0(!))

]

is nonnegative, whenever m; n are nonnegative integers, and A0; A1; : : : ; An are in S with correspond-ing characteristic functions 1A0 ; : : : ; 1An . Within Walley’s framework of rational behaviour, coherenceis a minimal requirement to behave consistently. The reader is referred to Walley’s textbook [20]for further discussion. From the point of view of the measure theory, coherent upper probabilitiesare exactly those set functions which can be described as upper envelopes of isotone modular setfunctions (cf. [20, proof of (3.3.3, 2.8.9)]).

Lemma 1.2. Let � :S→ [0;∞[ be a set function on a lattice S in .Then � is a coherent upper probability if and only if there exists a family �� of isotone modular

set functions � on P() such that the following properties are satis:ed:

(1) �|S6� for every �∈��.(2) �(S) = sup

�∈��

�(S) for every S∈S, where the upper bound is attained in each case.

(3) �() = 1.


Next, we shall introduce special isotone submodular set functions which are prominent examplesfor coherent upper probabilities.

Lemma 1.3. Let � :S→ [0;∞[ be a set function on a lattice S in . Then we can state that

(1) � is isotone and submodular if and only if for every nonvoid subset C⊆S, totally orderedunder set inclusion, there exists an isotone modular set function � on S with �6� and�|C=�|C.

(2) Assume that � is isotone and submodular. If additionally � is bounded by 1, and �() = 1holds in the case of ∈S, then � is a coherent upper probability.

Proof. Statement (1) is a result by Kindler [9, Examples 1,3]. Statement (2) follows from statement(1) and the Lemmas 1.1 and 1.2.

Another important concept within the measure theory is the regularity (cf. [11,19]).Setting inf ∅ :=∞; sup ∅ := 0 a set function � on a lattice S in is said to be inner (outer)

regular T if � is isotone, T⊆S, and

�(A) = supA⊇T∈T

�(T )(�(A) = inf

A⊆T ∈T�(T )

)

for all A∈S. Combining the notions of inner and outer regularity, we call � semiregular T ifT⊆P() and

�(A) = infA⊆T∈T

supT⊇A′∈S

�(A′)

for all A∈S.The most di3cult part of this paper is to Gnd su3cient conditions for our approximation

problem. To state the main result, we need the following notions and notations. For a pairS;T of set systems in , we deGne the set system S⊥ := {\A |A∈S}, and the transporterS�T := {B⊆ | ∀A∈S:B∩A∈T}.T separates S is deGned to mean that for every disjoint pair A; B∈S, there exists some disjoint

pair U; V ∈T with A⊆U; B⊆V .Within the frame of pairs of lattices S;T we obtain an interesting relationship between downward

� continuity and semiregularity.

Lemma 1.4. Let S;T be lattices in with ∅∈S∩T; T⊆ (S�S)⊥; T separates S. Thenevery coherent upper probability � on S is downward � continuous if it is downward � continuousat ∅ and semiregular T.

Remark. The statement of the lemma generalizes a result by Adamski [1, Lemma 3.1]. Adamskihas dealt with isotone submodular set functions, and additionally has assumed the lattice S to bestable under countable intersections with S⊆ (T�T)⊥.

Proof. We shall adopt Adamski’s proof of his result.


Let (Si)i denote an antitone sequence in S with∞∩i=1

Si∈S. The coherent upper probability � is

obviously isotone, hence

�

( ∞⋂i=1

Si

)6 inf

i�(Si) = lim

i�(Si):

Fix a positive real number �. Since � is semiregular T, there exists some T ∈T with∞∩i=1

Si ⊆T

such that

�

( ∞⋂i=1

Si

)+

�2

¿ supT⊇S∈S

�(S):

(Observe that the set {T ∈T | ∞∩i=1

Si ⊆T} is nonvoid because T separates S.)

T⊆ (S�S)⊥ implies Si ∩\T ∈S for every i. Furthermore∞∩i=1

Si; S1\T are disjoint, and T

separates S. Therefore, we can Gnd disjoint members T1; T2 from T with S1\T ⊆T1 and∞∩i=1

Si ⊆T2.

From T⊆ (S�S)⊥ we obtain that S1\T1∈S holds and that (Si ∩\T2)i is an antitone sequence

in S with∞∩i=1

(Si ∩\T2) = ∅. � is downward � continuous at ∅, thus there exists some i0 such that

�(Si ∩ \T2)¡�=2 holds for every i¿i0.Since � is a coherent upper probability, we can Gnd, according to Lemma 1.2, an isotone modular

set function � on S with �6� and �(Si0) =�(Si0). Now Si0 ⊆ (S1\T1 ∪ Si0\T2) and S1\T1 ⊆T implyfor i¿i0

�(Si) 6 �(Si0) = �(Si0)

6 �(S1\T1 ∪ Si0\T2)

6 �(S1\T1) + �(Si0\T2)

6�(S1\T1) + �(Si0\T2)

6 supT⊇S∈S

�(S) + �(Si0\T2)

¡�( ∞∩

i=1Si

)+ �:

This completes the proof.

We are now provided to turn over to the announced conditions for our approximation problem.

2. Statement of the main results

First, we shall give necessary conditions which fuzzy measures have to fulGl if they are upperenvelopes of probability measures.


Proposition 2.1. Let � be a fuzzy measure on F.If � = supP�, then � is an upward � continuous coherent upper probability.

The continuity property follows from routine procedures. The part of the statement related to thecoherence property can be found in [20, (3.3.3, 2.8.9)].

In the context of a Gnite universe , the approximation problem is solved (cf. Lemma 1.2), notethat the continuity property in Proposition 2.1 is trivial.

Proposition 2.2. Let be :nite, and � be a fuzzy measure on F. Then � = supP� if and only if� is a coherent upper probability.

But we are not able to give a solution for our approximation problem generally. OFering su3cientconditions, we can obtain immediately the following Grst result.

Proposition 2.3. If � denotes a coherent upper probability on F which is downward � continuousat ∅, then � = supP�, and the upper bound is attained for each subset in F.

Proof. Applying Lemma 1.2, we can Gnd for every A∈F some isotone modular fuzzy measure Pon F which satisGes P6� and P(A) = �(A). Then we can conclude immediately that P is downward� continuous at ∅, since � has this property by assumption, and 06P6�. Hence P is a probabilitymeasure (cf. [3, Proposition 1.2.4]), which completes the proof.

Proposition 2.3 is not satisfactory since it does not cover simple examples like the following.

Example 2.4. Assume that F contains all the singletons of , and consider the fuzzy measure �on F, deGned by

�(A) :={

1; A �= ∅;0; A = ∅:

For each A∈F\{∅}, every Dirac measure Pa in a∈A is in P� with Pa(A) = �(A).

The example gives rise to look for other su3cient conditions, as general as possible. Our mainresult is an attempt to give an answer.

Theorem 2.5. Let S;T be lattices in with ∅∈S∩T; T⊆ (S�S)⊥, separates S; S isclosed under countable intersections. Additionally, � denotes some fuzzy measure on F such that

1. S⊆F⊆ �(S�S), i.e. the �-algebra generated by S�S,2. � is inner regular S,3. �|S is semiregular T,4. �|S is downward � continuous at ∅.

Then we can state:(1) If ∈S and �|S is a coherent upper probability, then � = supP�.(2) If �|S is submodular, then � = supP�, and the upper bound is attained for each subset in F.


Remark 2.6. Let � denote the fuzzy measure as deGned in Example 2.4. Then we can apply Theo-rem 2.5(2) directly to � with T :=F and S := {A⊆ |A Gnite}.

The statement of the theorem is a generalization of a result by Adamski [1, Theorem 3.9]. OnesimpliGcation is that Adamski additionally has assumed S⊆ (T�T)⊥. Furthermore, statement (a)of Theorem 2.5 deals with restrictions �|S as coherent upper probabilities instead of submodularrestrictions that have been considered by Adamski. These new results of the approximation problemcan be obtained by working consequently within the framework of inner extension theory related toset functions on lattices.

The idea to base the extension of set functions on inner regularity is a rather new developmentstarted with Kisynski [10] and TopsHe [17–19]. Recently, K)onig succeeded in systemizing [11–13].So his work will be the main source we refer to.

As a consequence of inner extension theory, we shall mention—besides the main result—somegeneral result about approximating set functions by modular ones. It is of own interest.

But Grst, we shall give some applications of the main theorem in the following four sections.

3. Fuzzy measures on Borel subsets of Hausdor* spaces

As a Grst application of our main theorem we shall investigate our approximation problem forfuzzy measures on Borel subsets of HausdorF spaces.

Let (; �) be a HausdorF space, let Borel() be the set of all Borel subsets of , and let Comp()be the set of all compact subsets.

Then �⊆ (Comp()�Comp())⊥; � separates Comp(), and Borel() is contained in�(Comp()�Comp()), i.e. the �-algebra generated by Comp()�Comp(). Moreover, Comp()is closed under countable intersections. We are now prepared to apply Theorem 2.5 to S :=Comp()and T := �.

Proposition 3.1. Let � be some fuzzy measure on Borel(), that is inner regular Comp(), suchthat �|Comp() is submodular and semiregular �. Then � = supP�, and the upper bound is attainedfor each Borel subset.

Observe that every isotone set function on Comp() is obviously downward � continuous at ∅.

4. Fuzzy measures on Borel subsets of metrizable spaces

Within this section we shall treat the approximation problem for fuzzy measures on Borel subsetsof metrizable spaces.

Let (; �) be a metrizable space with metric d, let Borel() be the set consisting of all Borelsubsets of , and let Cl() be the set of all closed subsets.

Then � separates Cl(); �⊆ (Cl()�Cl())⊥; Borel()⊆ �(Cl()�Cl()), i.e. the �-algebragenerated by Cl()�Cl(), and Cl() is closed under countable intersections.

Every probability measure on Borel() is inner regular Cl() [16, p. 27]. Therefore, each fuzzymeasure on Borel(), which can be approximated by probability measures as an upper envelope, is


inner regular Cl(). Completing Proposition 2.1 this regularity property is an additional necessarycondition for our approximation problem. Analogously, each fuzzy measure which is an upper en-velope of probability measures has to be inner regular Comp() in the case that (; �) is a Polishspace, because then every probability measure on Borel() is inner regular Comp() [16, p. 29].

Hence, we can summarize the following necessary conditions for fuzzy measures on Borel()which are upper envelopes of probability measures.

Proposition 4.1. Let � be a fuzzy measure on Borel() with � = supP�. Then

(1) � is a coherent upper probability which is also upward � continuous and inner regular Cl(),(2) additionally, � is inner regular Comp() if (; �) is a Polish space.

Now, we turn to the su3cient conditions for approximating fuzzy measures by probability mea-sures. We shall proGt from the particular topological structure of metrizable spaces to substitute thesemiregularity by downward continuity.

Lemma 4.2. Let � :Cl()→R be some cohorent upper probability.Then � is downward � continuous if and only if it is downward � continuous at ∅ and semireg-

ular �.

Proof. The ‘if’ part of the statement is shown in Lemma 1.4, so the ‘only if’ part remains to beproved. Clearly, inf

A⊆G∈�sup

G⊇A′∈Cl()�(A′)¿�(A) for arbitrary closed subset A, and equality holds for

∅. Fix nonvoid A∈Cl(), and deGne for the positive integer k the subsets

Ak :={! ∈

∣∣∣∣ inf!′∈A

d(!;!′)61k

};

Gk :={! ∈

∣∣∣∣ inf!′∈A

d(!;!′) ¡1k

};

where Ak is closed, Gk is open, A⊆Gk ⊆Ak , and A=∞∩k=1

Gk =∞∩k=1

Ak . Since � is downward �

continuous and an isotone, we get

�(A) = limk→∞

�(Ak) ¿ limk→∞

supGk⊇A′∈Cl()

�(A′)

¿ infA⊆G∈�

supG⊇A′∈Cl()

�(A′):

The proof is complete.

Next we shall combine Lemma 4.2 with Theorem 2.5.

Proposition 4.3. Let � be a fuzzy measure on Borel(), which is inner regular Cl(), such that�|Cl() is a downward � continuous coherent upper probability.

Then � = supP�, and the upper bound is attained for each Borel subset.


5. Fuzzy measures on power sets

This section deals with fuzzy measures on the power set of . As will be shown, our approximationproblem can be solved for submodular fuzzy measures. This is due to the fact that no atomlessprobability measure on P() can be constructed, a result that emerges from the following Lemmaof Ulam [15, Satz 5.6].

Lemma 5.1 (Ulam). Let be of cardinality as P(N), the power set of the positive integers.If � is a :nite measure on P() with �({!}) = 0 for !∈, then �(A) = 0 for A ⊆ .

Corollary 5.2. Let be of cardinality as P(N), the power set of the positive integers. P denotessome probability measure on P(). Then there is some subset A of , at most countable, withP(A) = 1; P({!})¿0 (!∈A).

Proof. According to the result of Ulam, the subset A := {!∈ |P({!})¿0} is nonvoid. Moreover,A is at most countable since P(A)61 and therefore every {!∈A |P({!})¿1=n} (n positive integer)contains at most n members. Thus P(\A) has the same cardinality as P(N). Application of Ulam’sresult leads to P(B) = 0 for every subset B of \A, which completes the proof.

We consider P() as a topology on , and we want to apply Proposition 3.1 to P() andComp(), the set of compact subsets with respect to P():Comp() consists of all Gnite subsets.Therefore, under the continuum hypothesis, each probability measure on P() is inner regularComp() due to Corollary 5.2. So every fuzzy measure on P(), which can be approximated byprobability measures as an upper envelope, is inner regular Comp().

On the other hand, every isotone set function � on Comp() is semiregular P(). Hence everysubmodular fuzzy measure on P(), inner regular Comp(), is an upper envelope of probabilitymeasures according to Proposition 3.1.

Thus, we have the following solution of our approximation problem for submodular fuzzy measureson P().

Proposition 5.3. Let � be a submodular fuzzy measure on P(). Then � = supP�, where the upperbound is attained for every subset, if and only if � is inner regular Comp().

Remark. The ‘if’ part of the statement is independent of the continuum hypothesis.

6. Applications to some known fuzzy measures

Next, we want to point out some special classes of important types of fuzzy measures as upperenvelopes of probability measures.

A fuzzy measure � on F is called a Sugeno fuzzy measure if it is upward and downward �continuous [21, p. 43].

For Sugeno fuzzy measures which are also coherent upper probabilities, we shall draw on Propo-sition 4.3 (note that P(), as a topology, is metrizable).


Proposition 6.1. Let F be the set of Borel subsets with respect to a metrizable topology � on and let � denote some Sugeno fuzzy measure on F, which is also a coherent upper probability.Additionally, let us de:ne the set system Cl() := {A⊆ |\A∈�}. Then,

(1) � = supP�, where the upper bound is attained for every subset in F, if � is inner regularCl(),

(2) � = supP�, where the upper bound is attained for every subset, if �=P().

A fuzzy measure � on F is deGned to be a plausibility measure if

∑J⊆{1;:::;n}

(−1)]J �

B0 ∪

⋃j∈J

Bj

6 0

holds for every natural number n and B0; B1; : : : ; Bn∈F (]J denotes the cardinality of J ) (cf. [7,p. 321]). Every plausibility measure is submodular. We shall treat plausibility measures on Borelsubsets of metrizable spaces, and on power sets (cf. Propositions 4.3 and 5.3).

Proposition 6.2. Let F be the set of Borel subsets with respect to a metrizable topology � on .Assume that � is a plausibility measure on F, and let Cl() := {A ⊆ |\A∈�}. Then,

(1) � = supP�, where the upper bound is attained for every subset in F, if � is inner regularCl(), and �|Cl() is downward � continuous,

(2) � = supP�, where the upper bound is attained for every subset, if �=P() holds, and � isinner regular {A⊆ |A Gnite}.

A fuzzy measure � will be called supremum preserving if

�

⋃

j∈J

Bj

= sup

j∈J�(Bj);

whenever J is a nonvoid index set and Bj∈F for j∈J . Supremum preserving fuzzy measures onF=P() are genuinely known as possibility measures (cf. [21, p. 63]).

In the following, we restrict ourselves to �-algebras F in which contain all singletons. Everysupremum preserving fuzzy measure is submodular and extendable to the power set. This allows usto apply Proposition 5.3.

Proposition 6.3. Let F be a �-algebra in containing all singletons, and � be some supremumpreserving fuzzy measure on F. Then we have � = supP�, and the upper bound is attained forevery subset in F.

Proof. Fixing sup ∅ := 0, the set function �̂ on P(), deGned by �̂(A) := sup!∈A

�({!}), is a possibility

measure with �̂|F= �. In particular, it is submodular and inner regular {B⊆ |B Gnite}. Thestatement follows from Proposition 5.3, applied to �̂.


7. Inner extension theory

For the purpose of proving the main theorem we shall draw on inner extension procedures withinthe abstract measure theory. The subject is to construct contents or measures by extending setfunctions on lattices to inner regular contents or measures on algebras resp. �-algebras. This meansto change the point of view to treat extension procedures. Classically, going back to Caratheodory,set functions have been extended to outer regular contents or measures (cf. [3, Section 1.3; 2, SectionI.5]). As mentioned above, this rather new development started with Kisynski [10] and TopsHe [17–19]. The general inner extension theorem has been established by K)onig [11–13]. It will be the issueof this section.

Extension theories come in three simultaneous versions. They are marked •=?; �; �, where ? isto be read as Gnite, � as countable, and � as arbitrary. For a nonvoid set system S in we deGneS• to consist of its respective intersections. For any subset A of a nonvoid set we shall call a setsystem M in to be nonvoid • with M ↓⊆A if M is nonvoid and has cardinality of at most •,the intersection of its members is contained in A, and for every M;M ′∈M, there is some M ′′∈Mwith M ′′⊆M ∩M ′.

In the following, only lattices S in a Gxed nonvoid set with ∅∈S are considered.Generalizing the notion of downward continuity, we call an isotone set function � on S downward

• continuous if �( ∩M∈M

M) = infM∈M

�(M) holds for every M nonvoid • with M ↓⊆; ∩M∈M

M

in S, and �(∅) = 0. An isotone set function � on S with �(∅) = 0 will be called downward •continuous at A (A∈S) if it satisGes �(∩M∈MM) = infM∈M �(M) for every M nonvoid • withM ↓⊆A; ∩

M∈MM =A.

A content + :A→R on a ring is said to be an inner • extension of a set function � on S with�(∅) = 0 if S•⊆A; +|S=�, and

1. + is inner regular S•,2. +|S• is downward • continuous.

An isotone set function � on S with �(∅) = 0 will be called an inner • premeasure if it admits aninner • extension.

The general inner extension theorem is expressed in terms of the inner • envelopes�• :P()→ [0;∞] of a set function � on S, and their satellites �B• :P()→ [0;∞](B∈S) givenin an element A of P() by

�•(A) = sup{

infM∈M

�(M) |M ⊆ S nonvoid • with M ↓⊆ A}

and

�B•(A) = sup

{inf

M∈M�(M) |M ⊆ SB nonvoid • with M ↓⊆ A

};

where SB := {S∈S | S ⊆B}.We can now state the general extension theorem (cf. [11, Theorem 6.22; 12, 1.3; 13, p. 83]).


Theorem 7.1. Let S be a lattice in containing ∅: � denotes some isotone, supermodular setfunction on S with �(∅) = 0. For the inner • envelope �• we de:ne L(�•) as the set consistingof all subsets B of satisfying for every A⊆

�•(A) = �•(A ∩ B) + �•(A\B):

Then

(1) The following assertions are equivalent:(i) � is an inner • premeasure,

(ii) �•|L(�•) is an inner • extension of �,(iii) � is downward • continuous, and

�(B) 6 �(A) + �•(B\A) for A; B ∈ S; A ⊆ B;

(iv) � is downward • continuous at ∅, and

�(B) 6 �(A) + �B•(B\A) for A; B ∈ S; A ⊆ B:

If one of the equivalent conditions in (1) is satis:ed, the following statements hold:

(2) �•|L(�•) is a measure on a �-algebra for •= �; �,(3) every inner • extension of � is a restriction of �•|L(�•),(4) S�S⊆S�L(�•)⊆L(�•).

As a direct consequence of the theorem ((1)(iv)), and �B?6�B

� for B∈S, an inner ? premeasureis even an inner � premeasure if it is downward � continuous at ∅.

Corollary 7.2. Let S be a lattice in containing ∅, and let � denote some inner ? premeasureon S. Then � is an inner � premeasure if it is downward � continuous at ∅.

Extension Theorem 7.1 also provides us with the tool to approximate set functions by inner ?premeasures [14, Theorem 4.5].

Theorem 7.3. Let S;T be a pair of lattices in . Assume that T separates S; T⊆ (S�S)⊥;∅∈S∩T. Additionally, , denotes some isotone, submodular set function on S, semiregular T.Then for a :xed nonvoid subset C of P(), totally ordered under set inclusion, there is someinner ? premeasure � on S with �6, and �?(A) = ,?(A) for every A∈C.

Theorem 7.3 will be one crucial argument to prove the main result of our approximationproblem.

8. Proof of the main theorem

The proof of Theorem 2.5 can be sketched in the following way. First, under the assumptions ofTheorem 2.5, the restriction of the fuzzy measure on the lattice S can be described as an upper


envelope of isotone, modular set functions. Then approximating inner ? premeasures emerges fromthese approximating isotone and modular set functions. The inner ? premeasures can be extended toprobability measures using the results from inner extension theory. What is still missing in order tofollow the line of reasoning is to develop inner ? premeasures from isotone modular set functions.This can be done by the following result [14, Proposition 3.3].

Proposition 8.1. Let S;T be a pair of lattices in . Assume that T separates S; T⊆ (S�S)⊥;∅∈S∩T: � denotes some bounded isotone, modular set function on S. Then � :S→ [0;∞[,de:ned by

�(S) := infS⊆T∈T

supT⊇S′∈S

�(S ′)

is an inner ? premeasure with �6�.

Proof of Theorem 2.5. Retake the notations and assumptions made in Theorem 2.5. Let B∈F.

Proof of (1). Let � be some Gxed positive real number. � is inner regular S. Therefore thereexists some S0∈S with S0 ⊆B and �(B) − �¡�(S0).

Since �|S is assumed to be a coherent upper probability, we can Gnd by Lemma 1.2 someisotone and modular set function � on P() such that �() = 1, �|S6�|S and �(S0) = �(S0) hold.Applying Proposition 8.1, the set function � : S→ [0;∞[, deGned by

�(S) := infS⊆T∈T

supT⊇S′∈S

�(S ′);

is an inner ? premeasure with �|S6�.Furthermore, �6�|S because �|S is semiregular T. Hence �(S0) = �(S0), and, noticing ∈S;

�() = 1. Moreover, � is downward � continuous at ∅ since �|S has this property, and 06�6�|S.Then, according to Corollary 7.2, � is an inner � premeasure. Thus we can extend � to a measure� on some �-algebra F′⊇ �(S�S) such that � is inner regular S� =S.

The set function P := �|F is a probability measure with

P(A) = supS∈S

�(S) 6 supS∈S

�(S) = �(A)

for all A∈F. Additionally,

P(B) ¿ P(S0) = �(S0) ¿ �(B) − �

holds.Proof of (2). If �|S is submodular, then, according to Theorem 7.3, there is some inner ?

premeasure � on S with �6�|S; �?(B) = (�|S)?(B), and �?() = (�|S)?(). Since � is innerregular S, we have

�?(B) = �(B); �?() = �() = 1:

Moreover, � is downward � continuous at ∅ because �|S has this property, and 06�6�|S. Hence� is an inner � premeasure due to Corollary 7.2. Applying the inner extension Theorem 7.1 this


means that there is some measure � on a �-algebra F′⊇ �(S�S) with �|S=�, and that � isinner regular S� =S. Thus P := �|F is a measure, inner regular S, with P|S=�. P is as required.

This completes the proof.

Acknowledgements

The author is grateful to Professor K)onig whose lecture about nonadditive measure and integrationtheory inspired the line of reasoning of the article, and who accompanied the development of thepaper with critical remarks. The author also would like to thank the referees for their comments andsuggestions to improve the Grst draft.

References

[1] W. Adamski, Capacitylike set functions and upper envelopes of measures, Math. Ann. 229 (1977) 237–244.[2] H. Bauer, MaR- und Integrationstheorie, 2nd Edition, de Gruyter, Berlin, New York, 1992.[3] D.L. Cohn, Measure Theory, Birkh)auser, Boston, 1997.[4] D. Denneberg, Non-additive Measure and Integral, Kluwer Academic Publishers, Dordrecht, 1994.[5] D. Dubois, H. Prade, When upper probabilities are possibility measures, Fuzzy Sets and Systems 49 (1992) 65–74.[6] D. Dubois, H. Prade, Bayesian conditioning in possibility theory, Fuzzy Sets and Systems 92 (1997) 223–240.[7] I.R. Goodman, H.T. Nguyen, Uncertainty Models for Knowledge-based Systems, North-Holland, Amsterdam, 1985.[8] P.J. Huber, V. Strassen, Minimax tests and the Neyman–Pearson lemma for capacities, Ann. Statist. 1 (1973) 252–

263.[9] J. Kindler, A Mazur–Orlicz type theorem for submodular set functions, J. Math. Anal. Appl. 120 (1986) 533–546.

[10] J. Kisynski, On the generation of tight measures, Studia Math. 30 (1968) 141–151.[11] H. K)onig, Measure and Integration, Springer, Berlin, 1997.[12] H. K)onig, Measure and integration: mutual generation of outer and inner premeasures, Ann. Univ. Sarav. 9 (1998)

99–122.[13] H. K)onig, Ausgew)ahlte Kapitel aus Mass und Integral: Nicht-additive Integration, Lecture at University of Saarland,

Winter Term 1998/99.[14] H. K)onig, Upper envelopes of inner premeasures, Ann. Inst. Fourier, Grenoble 50 (2) (2000) 401–422.[15] J.C. Oxtoby, MaR und Kategorie, Springer, Berlin, Heidelberg, New York, 1971.[16] K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, London, 1967.[17] F. TopsHe, Compactness in spaces of measures, Studia Math. 36 (1970) 195–212.[18] F. TopsHe, Topology and Measure, in: Lecture Notes in Mathematics, Vol. 133, Springer, Berlin, Heidelberg, New

York, 1970.[19] F. TopsHe, On construction of measures, in: Proceedings of the Conference on Topology and Measure I (Zinnowitz

1974) Part 2, Ernst–Moritz–Arndt Universit)at Greifswald, 1978, pp. 343–381.[20] P. Walley, Statistical Reasoning with Imprecise Probabilities, Chapman & Hall, London, 1991.[21] Z. Wang, G.J. Klir, Fuzzy Measure Theory, Plenum, New York, London, 1992.

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When fuzzy measures are upper envelopes of probability measures