Two-valued sigma-maxitive measures and Mesiar's hypothesis

TL;DR

This paper investigates the conditions under which two-valued sigma-maxitive measures can be derived from sigma-additive measures, specifically focusing on the role of sigma-principality in the context of Mesiar's hypothesis.

Contribution

It reformulates Mesiar's hypothesis and proves that two-valued sigma-maxitive measures can be induced by sigma-additive measures if they are sigma-principal, clarifying the conditions for such measures.

Findings

Two-valued sigma-maxitive measures can be induced by sigma-additive measures.

Sigma-principality is a key condition for this induction.

The reformulation clarifies the validity of Mesiar's hypothesis.

Abstract

We reformulate Mesiar's hypothesis [Possibility measures, integration and fuzzy possibility measures, Fuzzy Sets and Systems 92 (1997) 191-196], which as such was shown to be untrue by Murofushi [Two-valued possibility measures induced by $\sigma$-finite $\sigma$-additive measures, Fuzzy Sets and Systems 126 (2002) 265-268]. We prove that a two-valued $\sigma$-maxitive measure can be induced by a $\sigma$-additive measure under the additional condition that it is $\sigma$-principal.