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Triangular norm-based measures and their Markov kernel representation. (English) Zbl 0751.60003

Summary: We approach the problem whether left-continuous triangular norm-based valuations (called \(T\)-measures or \(T\)-probability measures) defined on triangular norm-based tribes of the unit cube can be disintegrated by Markov kernels. We prove that each \(T\)-measure based on a “fundamental” triangular norm (these triangular norms \(T\), together with their corresponding triangular conorms \(S\), satisfy the functional equation \(T(x,y)+S(x,y)=x+y\)) can be uniquely represented as a sum of a “disintegrable” \(T\)-measure and a “hard core” which is either identically zero or which is monotonically irreducible (i.e., cannot be disintegrated).

MSC:

60A99 Foundations of probability theory
28E10 Fuzzy measure theory
03E72 Theory of fuzzy sets, etc.
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