An axiomatic approach to the roughness measure of rough sets
  In Pawlak's rough set theory, a set is approximated by a pair of lower and
upper approximations. To measure numerically the roughness of an approximation,
Pawlak introduced a quantitative measure of roughness by using the ratio of the
cardinalities of the lower and upper approximations. Although the roughness
measure is effective, it has the drawback of not being strictly monotonic with
respect to the standard ordering on partitions. Recently, some improvements
have been made by taking into account the granularity of partitions. In this
paper, we approach the roughness measure in an axiomatic way. After
axiomatically defining roughness measure and partition measure, we provide a
unified construction of roughness measure, called strong Pawlak roughness
measure, and then explore the properties of this measure. We show that the
improved roughness measures in the literature are special instances of our
strong Pawlak roughness measure and introduce three more strong Pawlak
roughness measures as well. The advantage of our axiomatic approach is that
some properties of a roughness measure follow immediately as soon as the
measure satisfies the relevant axiomatic definition.
