An extension on Sugeno integral and Pettis-Sugeno integral

We extend the concept of Sugeno fuzzy integral from nonnegative fuzzy measurable functions to extended real-valued fuzzy measurable functions; several necessary and sufficient conditions of absolute (S)-integrability for extended real-valued fuzzy measurable functions are given. Meanwhile, the space (S(/spl mu/), /spl rho/(.,.)) of all fuzzy measurable functions is proved to be a metric space under a necessary and sufficient condition. Finally, as an application the Pettis-Sugeno integral is introduced.