A decomposition theorem for fuzzy set-valued random variables

In this paper, a decomposition theorem for a (square integrable) fuzzy random variable FRV is proposed. The paper is mainly divided in two parts. In the first part, for any FRV X, we define the Hukuhara set as the family of (deterministic) fuzzy sets C for which the Hukuhara difference X ⊖H C exists almost surely; in particular, we prove that such a family is a closed (with respect to different well known metrics) convex subset of the family of all fuzzy sets. In the second part, we prove that any square integrable FRV can be decomposed, up to a random translation, as the sum of a FRV Y and an element C chosen uniquely (thanks to a minimization argument) in the Hukuhara set. This decomposition allows us to characterize fuzzy random translations; in particular, a FRV is a fuzzy random translation if and only if its Aumann expectation equals C (given by the above decomposition) up to a deterministic translation. Examples and open problems are also presented.