Inner variation and the SLi-functions

Abstract

In this paper, we investigate the Lusin Condition's by utilizing the concept of δ-fine. Using concepts of covering relations and inner variation, we gave an easier way of studying Lusin's condition (N ). Important results regarding full and fine covering relations are presented to help build the concept of inner variation zero. Inner variation zero gives an alternative approach to measure zero independent to the theory of Lebesgue measure. We use these concepts to define SLi-functions and give some results related to absolutely continuous functions, functions having negligible variations and functions satisfying the strong Lusin conditions. Mathematics Subject Classification: 26A46, 26A99, 28A05, 28A12