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Measures on clans and on MV-algebras. (English) Zbl 1019.28009
Pap, E. (ed.), Handbook of measure theory. Vol. I and II. Amsterdam: North-Holland. 911-945 (2002).
The main aim of this Handbook chapter is to present in a unified approach the results about certain type of fuzzy measures, namely about \(T_\infty\)-valuations on clans of fuzzy sets. A clan \({\mathcal T}\) of fuzzy sets is a family of \([0,1]\)-valued functions defined on a set \(\Omega\) such that \(1\in{\mathcal T}\) and \((f-g)\vee 0\in{\mathcal T}\) whenever \(f,g\in{\mathcal T}\). Identifying sets with their characteristic functions, any algebra of sets is a clan of fuzzy sets. A \(T_\infty\) valuation on a class \({\mathcal T}\) of fuzzy sets is a real-valued function such that \(\mu(f+ g)= \mu(f)+ \mu(g)\), if \(f,g\in{\mathcal T}\) and \(f+ g<1\).
A natural generalization of clan of fuzzy sets is an MV-algebra. The authors study measures on MV-algebras in five sections. The first one gives information on MV-algebras including also some latest results, e.g., the Dvurečenskij and Mundici modification of the Loomis-Sikorski theorem. The second section is devoted to submeasures on MV-algebras. Some connections are explained between order properties and metric properties of corresponding structures (e.g., very well-known metric \(d(a,b)-\mu(a\Delta b)\)).
The rest of the chapter is devoted to measures on MV-algebras: section 3 to the real valued case (Jordan decomposition, Hahn decomposition, Lebesgue decomposition), section 4 to the system of uniformities on the given MV-algebra (decomposition theorem, the uniformity generated by a measure), section 5 to measures with values in a locally convex space (Hewitt-Yosida decomposition, Vitali-Hahn-Saks-Nikodým theorem, extension theorem).
For the entire collection see [Zbl 0998.28001].

28E10 Fuzzy measure theory
28C99 Set functions and measures on spaces with additional structure
06D35 MV-algebras