Convergence, Strong Law of Large Numbers, and Measurement Theory in the Language of Fuzzy Variables

Details Convergence, Strong Law of Large Numbers, and Measurement Theory in the Language of Fuzzy Variables Convergence, Strong Law of Large Numbers, and Measurement Theory in the Language of Fuzzy Variables

2009-03-05

arxiv.org/abs/0903.0959v2

Mathematics

Probability

Scientific paper

In the paper we define the convergence of compact fuzzy sets as a convergence of alpha-cuts in the topology of compact subsets of a metric space. Furthermore we define typical convergences of fuzzy variables and show relations with convergence of their fuzzy distributions. In this context we prove a general formulation of the Strong Law of Large Numbers for fuzzy sets and fuzzy variables with Archimedean t-norms. Next we dispute a structure of fuzzy logics and postulate a new definition of necessity measures. Finally, we prove fuzzy version of the Glivenko-Cantelli theorem and use it for a construction of a complete fuzzy measurement theory.

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