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).