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Extension of fuzzy P-measure. (English) Zbl 0568.60003
The main goal of this paper is a generalization of the Carathéodory theorem for fuzzy P-measures introduced by the author [Probability of fuzzy events defined as denumerable additivity measure, to appear]. Let S(\({\hat \sigma}\)) be the smallest soft fuzzy \(\sigma\)-algebra containing the soft fuzzy algebra \({\hat \sigma}\) and \(C(\mu)=\{\{\mu_ n\}|\) \(\mu \leq \sup_{n}\{\mu_ n\}\), \(\forall n\in N\); \(\mu_ n\in {\hat \sigma}\}\) for all \(\mu\in S({\hat \sigma})\). Then the mapping \(p^*: S({\hat \sigma})\to [0,1]\), defined by \(p^*(\mu)=\inf \{\sum_{n}p(\mu_ n):\) \(\{\mu_ n\}\in C(\mu)\}\), is the unique fuzzy P-measure on S(\({\hat \sigma}\)) which is an extension of the given fuzzy P-measure \(p: {\hat \sigma}\to [0,1]\).

60A10 Probabilistic measure theory
03E72 Theory of fuzzy sets, etc.