Uncertainty measures---Problems concerning additivity

Authors:
Siegfried Weber
Affiliations:
Institut für Mathematik, FB 08, Universität Mainz, Germany
Venue:
Fuzzy Sets and Systems
Year:
2009

Citing 2
Cited 4

Conditioning on MV-algebras and additive measures—I

Fuzzy Sets and Systems - Special issue on fuzzy measures and integrals
A complete characterization of all weakly additive measures and of all valuations on the canonical extension of any finite MV-chain

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Representation theorem for probabilities on IFS-events

Information Sciences: an International Journal
A complete characterization of all weakly additive measures and of all valuations on the canonical extension of any finite MV-chain

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Measure-free conditioning and extensions of additive measures on finite MV-algebras

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Extensions of states on MV-quantales

Fuzzy Sets and Systems

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Abstract

Additivity of an uncertainty measure on an MV-algebra has a clear meaning. If the divisibility is dropped, we come up to a so-called Girard algebra. There we discuss strong resp. weak additivity based on so-called divisible disjoint unions resp. on additivity for all sub-MV-algebras. We obtain a description of those extensions from additive measures on an MV-algebra to the canonical Girard algebra extension of pairs which are strongly additive and valuation measures. Finally, we prove the non-existence of strongly additive measure extensions, if the underlying MV-algebra is a finite chain with more than two non-trivial elements.