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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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##### References:
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