Based on the semantic analysis, a general axiomatic definition of uncertainty measure for rough set is proposed. By extending the Shannon entropy function to strictly concave function, a class of uncertainty measures based on strictly concave function are put forward. They are weighted average of strictly concave function, whose variable is a conditional probability. It follows that a series of measuring methods are developed. The measuring methods based on fuzzy entropy are discussed under the view of strictly concave function. It is proved that they are the special cases of the method proposed in this paper. The difference and relationship among roughness measure, modified rough measure and the uncertainty measure based on strictly concave function are discussed. Finally, some examples are designed to compare the methods discussed in this paper. It is found that the proposed uncertainty measures based on strictly concave function are consistent with the semantics of uncertainty of rough set.