18.4 Continuous state space
For continuous state space the fuzzy probability distribution P* on Θ is given by a fuzzy probability density f* on Θ (cf. Definition 8.1 in Section 8.1). In general for fuzzy utility U*(., d) with δ-level functions (·, d) and δ(·, d), the fuzzy expected utility
is calculated in the following way: Denoting the δ-cuts of P* U*(θ, d) by
the endpoints are obtained by
By the construction lemma the characterizing function of P* U*(, d) is obtained.
In general the resulting generalized expected utility is a fuzzy interval. Contrary to the classical situation, where expected utilities are real numbers, it is not always simple to come to an optimal decision. ...