NY.: Ithaca, Cornell University Library, 2016. — 29 p. English. (OCR-слой).{Free Published: Cornell University Library (arXiv:1509.00447v2 [math.GM] 6 Mar 2016)}.[Huan Huang. Department of Mathematics, Jimei University, Xiamen. China.
Congxin Wu. Department of Mathematics, Harbin Institute of Technology, Harbin. China].Abstract.
Diamond gave compact criteria in fuzzy numbers space endowed with Lp metric and compact criteria in the space of fuzzy star-shaped numbers with respect to the origin endowed with Lp metric. However, these compact criteria are wrong. Wu and Zhao proposed right characterizations in these two spaces. Based on this result, Zhao and Wu further gave compact criteria in the space of fuzzy star-shaped numbers with Lp metric. However, compare the existing compactness characterizations of fuzzy sets spaces endowed with Lp metric with Arzel`a–Ascoli theorem, it finds that the latter gives the compact criteria by characterizing the totally bounded sets while the former does not seem to characterize the totally bounded sets. Since, in metric spaces, totally boundedness is a key feature of compactness. We present characterizations of totally bounded sets, relatively compact sets and compact sets in the fuzzy sets spaces FB(Rm) and FB(Rm)p equipped with Lp metric, where FB(Rm) and FB(Rm)p are two kinds of fuzzy sets on Rm which do not have any assumptions of convexity or star-shapedness. All fuzzy sets spaces considered in this paper are subspaces of FB(Rm)p endowed with Lp metric. Based on these characterizations and the discussions on convexity and star-shapedness of fuzzy sets, we construct the completions of every fuzzy sets space mentioned in this paper.
Then we clarify relation among all the ten fuzzy sets spaces discussed in this paper including the general fuzzy star-shaped numbers space introduced by Qiu et al.
At last, it gives characterizations of totally bounded sets, relatively compact sets and compact sets in all the fuzzy sets spaces mentioned in this paper.Introduction.
The Hausdorff metric.
The spaces of fuzzy sets.
Characterizations of relatively compact sets, totally bounded sets and compact sets in (FB(Rm)p, dp).
Lebesgue’s Dominated Convergence Theorem.
Fatou’s Lemma.
Absolute continuity of Lebesgue integral.
Minkowski’s inequality.
Relationship between (FB(Rm), dp) and (FB(Rm)p, dp) and properties of (FB(Rm), dp).
Subspaces of (FB(Rm)p, dp).
Acknowledgement.
References (32 publ).
