International Journal of Theoretical Physics

, Volume 41, Issue 4, pp 791–810 | Cite as

On Fuzzy Probability Theory

  • E. D. Habil
  • T. Z. Nasr
Article

Abstract

Our main aim from this work is to see which theorems in classical probability theory are still valid in fuzzy probability theory. Following Gudder's approach [Demonestratio Mathematica 31(3), 1998, 235–254; Foundations of Physics, 30, 1663–1678] to fuzzy probability theory, the basic concepts of the theory, that is of fuzzy probability measures and fuzzy random variables (observables), are presented. We show that fuzzy random variables extend the usual ones. Moreover, we prove that for any separable metrizable space, the crisp observables coincide with random variables. Then we prove the existence of a joint observable for any collection of observables, and we prove the weak law of large numbers and the central limit theorem in the fuzzy context. We construct a new definition of almost everywhere convergence. After proving that Gudder's definition implies ours and presenting an example that indicates that the converse is not true, we prove the strong law of large numbers according to this definition.

fuzzy sets fuzzy probability theory observables quantum mechanics σ-morphisms 

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Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • E. D. Habil
    • 1
  • T. Z. Nasr
    • 1
  1. 1.Department of MathematicsIslamic University of GazaGazaPalestine

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