Then,we define the vector-valued ap-C-Stieltjes integral and prove the convergence theorems for the vector-valued ap-C-Stieltjes integral.

On the basis of interior operators, closure operators and approximate operators, the compound and overlapping compound of interior operators, closure operators and approximate operators are studied in reflexives and transitive rough sets.

First the basic ideas and framework of the rough set theory and the different views of knowledge representation in rough set theory were introduced,and then the upper and lower approximate operators definitions were discussed respectively in view of element based,granular based,subsystem based,and probability.

On covering-based rough approximation operators;

Constructive and Algebraic Methods of Approximation Operators over Two Universes;

On the relationship between rough approximation operators and possibility measures;

Topological properties of approximation operators based on reflexive and symmetric relations;

A general formula is given for the approximation operators of fuzzy sets using the triangular norm and its conorm.

But the base algebra systems, on which approximation operators are defined, are confined to Boolean algebra or special Boolean algebras, including set algebra, Nelson algebra(quasi-pseudo Boolean algebra), and atomic Bool.