Pseudo-Additive Measures and Triangular-Norm-Based Conditioning

Abstract

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