Elsevier

Fuzzy Sets and Systems

Volume 379, 15 January 2020, Pages 1-19
Fuzzy Sets and Systems

Canonical sequences of monotone measures

https://doi.org/10.1016/j.fss.2018.10.017Get rights and content

Abstract

It is an important feature of a monotone measure that it is not additive in general. In the paper we propose the mathematical tool, based on canonical sequences of monotone measures, for analyzing additivity of monotone measures on subalgebras and give a way of generating such monotone measures. It turns out that the generating rule can be considered as an effect of a linear operator defined on the set of monotone measures. We also investigate in what cases the sequence of such operators behaves commutatively and preserve continuity properties from the generating monotone measure.

Introduction

This paper is devoted to the detailed investigation of a new mathematical construction for analyzing monotone measures. First results concerning canonical sequences of monotone measures were announced in [4]. Then we showed, how these results can be used in finding Pareto-optimal 2-monotone approximations of coherent lower probabilities [5], in finding the least favorable pairs in Huber–Strassen theory [6], and for characterizing extreme 2-monotone measures [7].

Canonical sequences of monotone measures are generated by simple linear operators defined on the whole set of monotone measures. They allow us to produce measures which are additive on subalgebras and give us a new way to prove known and new results concerning 2-monotone and 2-alternating measures.

One can find recently published several papers [2], [16], [19] devoted to so called event based transformations of capacities (or monotone measures). We see that this notion has been not elaborated yet, but authors of [16] study four linear operators on the set of monotone measures, and one of them is investigated in our paper. They show that these operators form a semigroup and describe convex closure of these operators defined on monotone measures on the powerset of a finite set with cardinality 2 and 3. Some generalizations of event based transformations of monotone measures using aggregation functions can be found in [2].

In the paper we use the terminology adopted in the book on non-additive measures by Dieter Denneberg [10], so 2-monotone measures are called supermodular measures, their duals submodular measures, etc. The paper has the following structure. In Section 2 we remind some notions and results from the theory of non-additive measures. In Section 3 we define canonical sequences of monotone measures and investigate their properties for a finite case. In particular, we show that any sequence of monotone measures can be produced by the successive mappings of linear operators, associated with a sequence of sets. We show that this sequence of sets can be always restricted to a chain in the corresponding algebra. In Section 4 we analyze such linear operators associated with countable chains of sets, in particular, we find sufficient conditions under which the limit measure preserves continuity properties. In Section 5 we show how introduced linear operators can be generalized for arbitrary chains in the corresponding σ-algebra, and show how canonical sequences of monotone measures give us new ways for characterizing cores of submodular measures. In Section 6 we show how canonical sequences of monotone measures can be used for analyzing possibility measures, and we give examples of finitely additive measures, produced by canonical sequences.

Section snippets

The main notions and constructions

Let X be a measurable space with a σ-algebra A. The set function g on A is called a monotone measure [10] or capacity [9] if 1) g()=0, g(X)=1 (norming); 2) g(A)g(B) if AB (monotonicity). Let g1,g2 be monotone measures. We use the notation g1g2 iff g1(A)g2(A) for all AA, and also g1<g2 iff g1g2 and g1g2. A monotone measure g is (see [10])

  • continuous from below if for any sequence {Ai}i=1A, A1A2..., g(n=1An)=limng(An);

  • continuous from above if for any sequence {Ai}i=1A, A1A2...,

Canonical sequences of monotone measures (finite case)

In this section we will consider linear operators on the set of all monotone measures Mmon. The mapping L:MmonMmon is called a linear operator if L[αμ1+βμ2]=αL[μ1]+βL[μ2] for every μ1,μ2Mmon and α,β0 such that α+β=1.

Lemma 2

Let μMmon and BA. Then the set functionμB(A)=μ(AB)μ(B)+μ(AB),AA is in Mmon, and the corresponding mapping L{B}:MmonMmon defined byL{B}[μ](A)=μ(AB)μ(B)+μ(AB),AA, is a linear operator on Mmon.

Proof

The properties of the mapping described in the lemma can be easily checked. 

Canonical sequences of monotone measures (countable case)

Lemma 11

Let g and μ be subadditive measures on a σ-algebra A of a space X. In addition, gμ and g is continuous. Then μ is also continuous.

Proof

It is sufficient to show that for any sequence {An}n=1 in A such that A1A2... and n=1An= it is valid limnμ(An)=0. Under the condition, g is continuous and gμ, therefore, 0limnμ(An)limng(An)=0, i.e. limnμ(An)=0, and μ is continuous. 

Corollary 3

Let g and μ be superadditive measures on a σ-algebra A of a space X. In addition, gμ and g is continuous. Then μ is

Canonical sequences of monotone measures (general case)

In this section we will generalize the notion of canonical sequence or rather exactly the underlying linear operator for an arbitrary uncountable chain of sets. For this purpose we will use the probabilistic interpretation of the generating monotone measure from the canonical sequence (see Lemma 5). Notice that Lemma 5 for submodular measures is formulated as

Lemma 14

Let μ0Γ=g be a submodular measure on a finite algebra A of a space X, {μkΓ}k=0n be a canonical sequence of monotone measures constructed

Canonical sequences and possibility measures

Let X be an arbitrary reference set. In this section we will consider possibility measures on the σ-algebra A of X that contains singletons {x}, where xX (A may be the powerset of X, i.e. A=2X). A monotone measure Π is called a possibility measure if there is a function π:X[0,1] called the possibility distribution andΠ(A)={supxAπ(x),A,0,A=. Since Π(X)=1, supxXπ(x)=1, and every function π:X[0,1] satisfying supxXπ(x)=1 defines the possibility measure by formula (3).

We will use here the

Conclusion

The paper gives us the detailed study of canonical sequences of monotone measures. This mathematical tool shows good results for analyzing supermodular and submodular measures, and may be useful in many problems of discrete optimization [13]. We show that canonical sequences of monotone measures can be described by the special linear operators on the set of all monotone measures. It is worth to mention that many of known transformations of monotone measures or set functions, for example, Möbius

References (22)

  • A.G. Bronevich et al.

    Approximation of coherent lower probabilities by 2-monotone measures

  • Cited by (2)

    • A universal approach to imprecise probabilities in possibility theory

      2021, International Journal of Approximate Reasoning

    The short version of this paper has been presented at IPMU 2004 [4].

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