Constructive Set Function and Extraction of a k-dimensional Element

Abstract

We define a constructive non-additive set function as a generalization of a constructively k-additive set function ($k\in \mathbb {N}$). First, we prove that a distortion measure is a constructive set function if the distortion function is analytic.

A signed measure on the extraction space represents a constructive set function. This space is the family of all finite subsets of the original space. In the case where the support of the measure is included in the subfamily whose element’s cardinality is not more than k, the corresponding set function is constructively k-additive ($k\in \mathbb {N}$). For a general constructive set function $\mu$, we define the k-dimensional element of $\mu$, which is a set function, by restricting the corresponding measure on the extraction space to the above subfamily. We extract this k-dimensional element by using the generalized Möbius transform under the condition that $\sigma$-algebra is countably generated,