Optimal approximations with Rough Sets and similarities in measure spaces

• A concept of Optimal Approximation has been introduced and analyzed within the framework of Rough Sets.
• Five similarity measure axioms have been provided and several similarity indexes have been analyzed.
• An efficient algorithm for finding optimal simulation with Marczewski–Steinhaus similarity has been proposed and analyzed.

When arbitrary sets are approximated by more structured sets, it may not be possible to obtain an exact approximation that is equivalent to a given set. Presented here, is a new proposal for a ‘metric’ approach to Rough Sets. We assume some finite measure space is defined on a given universe, and then use it to define various similarity indexes. A set of axioms and the concept of consistency for similarity indexes are also proposed. The core of the paper is a definition of the ‘optimal’ or ‘best’ approximation with respect to any particular similarity index, and an algorithm to find this optimal approximation by using the Marczewski–Steinhaus Index. This algorithm is also shown to hold for a class of similarity indexes that are consistent with the Marczewski–Steinhaus Index.