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    STUDIA MATHEMATICA - Issue no. 3 / 2007  
         
  Article:   COMPLETENESS WITH RESPECT TO THE PROBABILISTIC POMPEIU-HAUSDORFF METRIC.

Authors:  ŞTEFAN COBZAŞ. 		 
       
         
  Abstract:  The aim of the present paper is to prove that the family of all closed nonempty subsets of a complete probabilistic metric space L is complete with respect to the probabilistic Pompeiu-Hausdorff metric H. The same is true for the families of all closed bounded, respectively compact, nonempty subsets of L. If L is a complete random normed space in the sense of Šerstnev, then the family of all nonempty closed convex subsets of L is also complete with respect to H. The probabilistic Pompeiu-Hausdorff metric was defined and studied by R.J. Egbert, Pacific J. Math. 24 (1968), 437-455, in the case of Menger probabilistic metric spaces, and by R.M. Tardiff, Pacific J. Math. 65 (1976), 233-251, in general probabilistic metric spaces. The completeness with respect to probabilistic Pompeiu-Hausdorff metric of the space of all closed bounded nonempty subsets of some Menger probabilistic metric spaces was proved by J. Kolumbán and A. Soós, Studia Univ. Babes-Bolyai, Mathematica, 43 (1998), no. 2, 39-48, and 46 (2001), no. 3, 49-66.  
         
     
         
         
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