\(P\)-measures on soft fuzzy \(\sigma\)-algebras (Q1313012)

The authors establish a certain isomorphism between the space of all \(P\)- measures of soft fuzzy \(\sigma\)-algebras [\textit{K. Piasecki}, Fuzzy Sets Syst. 17, 271-284 (1985; Zbl 0604.60005)] and the space of all probability measures on a Boolean \(\sigma\)-algebra. The proof utilizes the well-known Loomis-Sikorski theorem. As a by-product, the authors provide a negative answer to a question of \textit{A. Dvurečenskij} [Fuzzy Sets Syst. 43, No. 2, 173-181 (1991; Zbl 0742.28009)] by showing that there are soft fuzzy \(\sigma\)-algebras that do not admit \(P\)- measures.