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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 $$p: {\hat \sigma}\to [0,1]$$.

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