Abstract: The concepts of the lower approximation integral, the upper approximation integral and rough integrals are given on the basis of function rough sets. Based on these concepts, the relation of the lower approximation integrals, the relation of the upper approximation integrals, the relation of rough integrals, and the double median theorem of rough integrals are discussed. Rough integrals have finite contraction characteristic and finite extension characteristic.

Key words: function rough sets, the lower approximation integral, the upper approximation
integral, rough integrals