An Axiomatic Definition of Non-discrete Möbius transform

Some global analysis for Möbius transforms must be useful in providing certain analysis tools. It may provide some approximation methods, for example, in the analysis of discrete set functions including numerical analysis. A constructive set function was defined as a set function defined on nondiscrete space, the corresponding generalized Möbius transform can be represented as a signed measure. We defined the covering measure for this set function, which dominates the set function value. This set function is a non-negative subadditive set function, and the corresponding function spaces have rich topological properties. On the other hand, there are several approaches to generalize the Möbius transform. We attempt to summarize several of them to provide an axiomatic definition. Using this formulation, we defined the notion of bounded variation for the generalized Möbius transform and provide a covering measure using generalized Jordan decomposition of the Möbius transform.