a. The fuzzy valued function
has δ-level functions
By assumption the integral exists and is finite for all δ ∈ (0; 1].
Now the fuzzy marginal density f*(·) is defined by its δ-level functions δ(·) and δ(·) respectively. Furthermore by Therefore f*(·) is a fuzzy density.
b. Let f(·) be a classical probability density with existing expectation, i.e. The δ-level functions of this density are all equal to f(·). Using the definition of Section 22.1 for the expectation we obtain
The characterizing function ψ (·) of μ* is given by the family of single point sets