Fuzzy probability space and extension theorem (Q1078885)

This paper deals with fundamental mathematical theory of fuzzy probability. As classical measure theory, fuzzy algebra, fuzzy \(\sigma\)- algebra and fuzzy probability, whose \(\sigma\)-additivity is characterized by the countable uncoincident fuzzy events and the operation \(\oplus\), are introduced. The main result of this paper is an extension theorem of fuzzy probability from a fuzzy algebra to the fuzzy \(\sigma\)-algebra generated by it. Finally, the relation between fuzzy probability and classical probability is expressed by an integral formula, which was offered by \textit{L. A. Zadeh} [J. Math. Anal. Appl. 23, 421-427 (1968; Zbl 0174.490)] as the definition of fuzzy probability.