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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].

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