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Pseudo-Additive Measures and Triangular-Norm-Based Conditioning

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Abstract

Conditioning in the framework of fuzzy measures (monotone normalized set functions vanishing in the empty set) is introduced. For every set B with non-null measure m(B) a conditional measure m B , based on a triangular norm T, is introduced. Universal conditioning preserving the lower semi-continuity is shown to be necessarily based on some strict triangular norm. Then also each conditional measure m B related to a pseudo-additive measure m is pseudo-additive. However, the pseudo-addition ⊕ B operating on the measures m B is in general different from the pseudo-addition ⊕ operating on the measure m. Specific cases of universal conditioning preserving the pseudo-addition ⊕ are characterized. Classical probabilistic conditioning is shown to be a special case.

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Benvenuti, P., Mesiar, R. Pseudo-Additive Measures and Triangular-Norm-Based Conditioning. Annals of Mathematics and Artificial Intelligence 35, 63–69 (2002). https://doi.org/10.1023/A:1014550212320

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