The Sugeno integral and functional representation of the monad of lattice-valued capacities

The Sugeno integral of an upper semicontinuous function from a compactum to a compact Hausdorff–Lawson lattice with respect to a lattice-valued capacity is introduced, and its characterization and properties are presented. It is proved that in a family of fuzzy integrals, the Sugeno integral is unique that provides a functional representation of the monad of lattice-valued capacities.