Triangular norm-based measures and their Markov kernel representation

Abstract We approach the problem whether left-continuous triangular norm-based valuations (called T -measures or T -probability measures) defined on triangular normbased 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).

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