On fuzzy integral inequalities and fuzzy expectation (Q1108397)

The authors consider some inequalities and some equalities for Sugeno's fuzzy integrals taking values in the positive extended reals. They may be regarded as extensions of the following theorem: \(\int h d\mu =k\quad \Leftrightarrow \quad \mu (\{h\geq k\})\geq k\geq \mu (\{h>k\}).\) Furthermore, the authors give a kind of fuzzy equation and its solution. Finally, an energy of a fuzzy set is defined as a nonnegative functional.