Completeness in Probabilistic Metric Spaces
Subject Areas : Statistics
D.
Varasteh Tafti
1
(Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.)
M.
Azhini
2
(Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.)
Keywords: توپولوژی قوی, تابع مثلثی, قضیه اشتراکی کانتور, قضیه بئر,
Abstract :
The idea of probabilistic metric space was introduced by Menger and he showed that probabilistic metric spaces are generalizations of metric spaces. Thus, in this paper, we prove some of the important features and theorems and conclusions that are found in metric spaces. At the beginning of this paper, the distance distribution functions are proposed. These functions are essential in defining probabilistic metric spaces. Then the Dirac function is presented as an important example of distributions functions. We also, introduced Sibley’s metric or Levy metric on the set of distance distribution functions and so this space becomes a metric space. In the following, probabilistic metric spaces are defined from the Serstnev’s view, and some examples such as Menger probabilistic metric spaces, are introduced. In this paper, the strong topology induced by distance distribution functions is introduced, and then probabilistic diameter, probabilistic bounded, probabilistic semi-bounded, probabilistic unbounded and probabilistic totally bounded sets are introduced. Also, we prove that in every probabilistic metric, every probabilistic totally bounded set is probabilistic bounded set. We also present the cantor intersection theorem and a formulation of Bair’s Theorem in complete probabilistic metric spaces. In addition, we prove that the Heine-Borel property and the Bolzano-Wierestrass property are equivalent.
[1] K. Menger, Statistical metrics, Proc, Nat, Acad, of sci. U.S.A. 28 (1942), 535-537.
[2] K. Menger, Probabilitstic geometry, Pro,Nat, Acad, of sci, U.S.A. 37(1951), 226-229.
[3] K. Menger, Gemetric generale (chap.VII), Memorial des Sciences Mathematiques, No, 124, Paris 1954.
[4] V. Istratescu and I. Vaduva, Products of statistical metric spaces, (Romanian)Acad.R.P.Romine Stud.Cere.Mat.12(1961),567-574.
[5] A. N. Serstnev, On the concept of random normed spaces, (Russian), Dokl. Akad, Nauk
SSSR 149(1963) 280-283.
[6] E. Thorp, Generalized topologies for statistical metric spaces, Fundamenta Mathematicae
51 (1962), 9-21.
[7] A. Wald, On a statistical generalization of metric spaces, Proc, Nat. Acad. of Sci U. S. A.29 (1943), 196-197.
[8] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 ( 1960), 314-334.
[9] B. Schweizer and A. Sklar, Probabilistic metric spaces, North-Holland, New York(1983).
[10] O. Hadzic and E. Pap, Fixed point theory in probabilistic metric spaces, Institue of Mathematics, University of Novi Sad, Yugoslavia (2001).
[11] Engelking, R; General Topology, Warszawa (1977).
[12] Zhang Shi-Sheng, Basic theory and application of probabilistic metric space (I), Applied Mathematics and Mechanics, 9,2 (1988); 123-133.
[13] S. M. Vaezpour and M. Shams, Topological properties of probabilistic metric space Int. Journal of Math. Analysis, Vol. 1,no. 20 ,( 2007), 957-964.
