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On Choquet Theorem for Random Upper Semicontinuous Functions CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector International Journal of Approximate Reasoning 46 (2007) 3–16 www.elsevier.com/locate/ijar On Choquet theorem for random upper semicontinuous functions Hung T. Nguyen a, Yangeng Wang b,1, Guo Wei c,* a Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, USA b Department of Mathematics, Northwest University, Xian, Shaanxi 710069, PR China c Department of Mathematics and Computer Science, University of North Carolina at Pembroke, Pembroke, NC 28372, USA Received 16 May 2006; received in revised form 17 October 2006; accepted 1 December 2006 Available online 29 December 2006 Abstract Motivated by the problem of modeling of coarse data in statistics, we investigate in this paper topologies of the space of upper semicontinuous functions with values in the unit interval [0,1] to define rigorously the concept of random fuzzy closed sets. Unlike the case of the extended real line by Salinetti and Wets, we obtain a topological embedding of random fuzzy closed sets into the space of closed sets of a product space, from which Choquet theorem can be investigated rigorously. Pre- cise topological and metric considerations as well as results of probability measures and special prop- erties of capacity functionals for random u.s.c. functions are also given. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Choquet theorem; Random sets; Upper semicontinuous functions 1. Introduction Random sets are mathematical models for coarse data in statistics (see e.g. [12]). A the- ory of random closed sets on Hausdorff, locally compact and second countable (i.e., a * Corresponding author. Tel.: +1 910 521 6582; fax: +1 910 522 5755. E-mail addresses: [email protected] (H.T. Nguyen), [email protected] (Y. Wang), [email protected] (G. Wei). 1 Supported in Part by NSF of Shaanxi (98SL06), PR China. 0888-613X/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.ijar.2006.12.004 4 H.T. Nguyen et al. / Internat. J. Approx. Reason. 46 (2007) 3–16 topological space that admits a countable base) spaces (HLCSC) as bona fide random ele- ments is established in Matheron [10] (see also [11]) in which the Choquet theorem is cru- cial for statistical applications. In the past two decades or so interests were on enlarging the domain of applicability of standard statistical inference to imprecise data such as per- ception based information which can be modeled as fuzzy sets (see e.g. [9]). To extend Matheron’s theory of random closed sets to random fuzzy sets, we consider fuzzy sets whose membership functions are upper semicontinuous (u.s.c.), generalizing indicator functions of closed sets. Thus, we are concerned with random elements taking values in the space UðEÞ of u.s.c. functions defined on a HLCSC space E (e.g. Rd) with values in the unit interval [0,1]. Previous works in this direction were restricted to compact random fuzzy sets (e.g. [9]), i.e. the largest class of fuzzy values considered so far in the literature is the class of u.s.c. functions with compact supports (see [8]), although the misleading term ‘‘fuzzy random variables’’ might give the impression that all fuzzy sets can be taken as values of these ran- dom elements! In view of Matheron’s theory of random closed sets (dually, open sets), it should be possible to consider random fuzzy closed sets in their full generality (without the restriction on compact support), and that is the purpose of this paper. We will mention that dual results are also obtained for continuous and lower semicontinuous (l.s.c.) func- tions as well. Note however that our approach can be also applied to u.s.c. functions with values in the extended real line which are useful in stochastic processes (e.g. [4]), and var- iational problems (e.g. [14]). Essentially, we will use a standard method of identifying functions with sets in a product space via the concepts of hypographs (and epigraphs) as in [1–3,15,16]. For a D½0; 1 represen- tation, see [5]. The idea is to embed the space of u.s.c. functions into some appropriate spaces. Note that it was mentioned in [14] that embedding via countable families of a-level sets fails. It turns out that embedding via hypographs or epigraphs leads successfully to Matheron’s framework. We establish also the Choquet theorem for random u.s.c. functions and related results. In the concluding remarks, we mention another radically different approach which will lead to a full generalization of Matheron’s theory of random closed sets. 2. Random closed sets and upper semicontinuous functions Throughout this paper, E denotes a Hausdorff, locally compact and second countable (HLCSC) topological space, such as Rd. Since singleton sets are closed, an extension to closed sets of E is a generalization of standard multivariate statistical analysis. Using Matheron’s notation [10], let FðEÞ or sim- ply F denote the class of all closed sets of E. To define random elements with values in F, we need to equip F with a r-field. A standard way for doing so is to consider a topology s on F and take the Borel r-field rðsÞ. Following Matheron, we take s to be the so-called hit-or-miss topology which is generated by the base B below: B ¼f K : K 2 ; G 2 ; n P 0g; FG1;...;Gn K i G where K ¼ K \ \ÁÁÁ\ , ; denote the classes of compact sets and FG1;...;Gn F FG1 FGn K G open sets of E, respectively, and for A E, A F ¼fF 2 F : F \ A ¼;g; FA ¼fF 2 F : F \ A 6¼;g: Note that, in B, when n ¼ 0, K means K . FG1;...;Gn F H.T. Nguyen et al. / Internat. J. Approx. Reason. 46 (2007) 3–16 5 The topological space ðF; sÞ is Hausdorff, compact and second countable [9], and hence metrizable the Urysohn’s metrization theorem [6]. As such the convergence in ðF; sÞ is sequential. More specifically, a sequence Fn of closed subsets of E is convergent to a closed subset F of E if and only if the following two conditions are satisfied [10]: (1) For every x 2 F , there exists for each integer n (except, at the most, for a finite num- ber of Fn’s) a point xn in Fn such that limn!1xn ¼ x in E. (2) If F nk is a subsequence of Fn,andxnk (xnk 2 F nk ) is an arbitrary convergent sequence, then limn!1xnk is in F. As a separable and complete metric space (Second countability implies separability and they are equivalent for metrizable spaces; compactness implies completeness and totally boundedness), it is a good candidate for defining random closed sets. Specifically, a ran- dom closed seton E is a measurable map X, defined on some probability space ðX; A; PÞ with values in ðF; rðsÞÞ. The probability law of the random closed set X is the probability À1 measure P X ¼ PX on rðsÞ. Like distribution functions of random vectors, PX is charac- terized by a capacity functional TX or simply T by Choquet theorem. Specifically, T : K ! R is called a capacity functional if it satisfies: (i) 0 6 T ðÞ 6 1, T ð;Þ ¼ 0, (ii) If Kn & K in K then T ðKnÞ&T ðKÞ, (iii) T is alternating of infinite order, i.e., T is monotone increasing and for K1; ...; Kn, n P 2, X n 6 jIjþ1 T ð\i¼1KiÞ ð1Þ T ð[i2I KiÞ ;6¼If1;2;...;ng where jIj denotes the cardinality of the set I. Choquet theorem. Let T : K ! R. There exists uniquely a probability measure Q on rðsÞ satisfying QðFK Þ¼T ðKÞ for K 2 K if and only if T is a capacity functional. Note that Choquet theorem can be also formulated in terms of open sets (see [10]). Now the indicator function of a closed set F,1F :!f0; 1g,1F ðxÞ¼0 or 1 according to x 62 F or x 2 F , is an upper semicontinuous (u.s.c.) function. Thus, the class UðEÞ or sim- ply U of u.s.c. functions defined on E with values in [0,1] generalizes F when we identify sets with their indicator functions. These are random fuzzy closed sets. Recall that a function f : E !½0; 1 (or to the extended real line) is said to be upper semicontinuous at a point x 2 E if for any a 2½0; 1 such that f ðxÞ < a, there exists dðx; aÞ > 0 such that f ðyÞ < a for all y 2 Bðx; dÞ, an open ball centered at x with radius d (noting here that E is metrizable). The interpretation is that each such a is an upper bound for f on some neighborhood of x and hence the adjective ‘‘upper’’. The function f is upper semicontinuous on E if it is upper semicontinuous at every point of E. Dually, f is said to be lower semicontinuous (l.s.c.) on E if 1 À f is u.s.c. on E (for u.s.c. and l.s.c. functions with values in the extended real line, f is l.s.c. if and only if Àf is u.s.c.). Clearly, f is continuous if f is both u.s.c. and l.s.c., hence the term ‘‘semi’’ in semicontinuous. 6 H.T. Nguyen et al. / Internat.
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