Canonical sequences of monotone measures☆
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 . The set function g on is called a monotone measure [10] or capacity [9] if 1) , (norming); 2) if (monotonicity). Let be monotone measures. We use the notation iff for all , and also iff and . A monotone measure g is (see [10])
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continuous from below if for any sequence , , ;
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continuous from above if for any sequence , ,
Canonical sequences of monotone measures (finite case)
In this section we will consider linear operators on the set of all monotone measures . The mapping is called a linear operator if for every and such that .
Lemma 2 Let and . Then the set function is in , and the corresponding mapping defined by is a linear operator on . 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 of a space X. In addition, and g is continuous. Then μ is also continuous.
Proof It is sufficient to show that for any sequence in such that and it is valid . Under the condition, g is continuous and , therefore, , i.e. , and μ is continuous. □
Corollary 3 Let g and μ be superadditive measures on a σ-algebra of a space X. In addition, 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 be a submodular measure on a finite algebra of a space X, 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 of X that contains singletons , where ( may be the powerset of X, i.e. ). A monotone measure Π is called a possibility measure if there is a function called the possibility distribution and Since , , and every function satisfying 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)
- et al.
Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion
Math. Soc. Sci.
(1989) Representation of the Choquet integral with the σ-additive Möbius transform
Fuzzy Sets Syst.
(1997)- et al.
Interaction transform of set functions over a finite set
Inf. Sci.
(1999) - et al.
Aggregation functions and capacities
Fuzzy Sets Syst.
(2018) When fuzzy measures are upper envelopes of probability measures
Fuzzy Sets Syst.
(2003)Cores of exact games, I
J. Math. Anal. Appl.
(1972)- et al.
Theory of Charges. A Study of Finitely Additive Measures
(1983) - et al.
Event-based transformations of capacities
- et al.
Statistical classes and fuzzy set theoretical classification of possibility distributions
Canonical sequences of fuzzy measures
Approximation of coherent lower probabilities by 2-monotone measures
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