Fuzzy signed measure

TL;DR

This paper introduces a fuzzy signed measure on sigma-algebras, extending classical measure theory, and proves a fuzzy Hahn Decomposition Theorem that decomposes spaces into positive and negative parts.

Contribution

It defines fuzzy signed measures and positive/negative sets, and generalizes the Hahn Decomposition Theorem to fuzzy measure spaces.

Findings

Defined fuzzy signed measures on sigma-algebras.

Proved the fuzzy Hahn Decomposition Theorem.

Decomposed any space into positive and negative sets with measure zero intersection.

Abstract

we will define a fuzzy signed measure on $\sigma$-algebras, as well as positive and negative sets. Herein, we will show that the Fuzzy Hahn Decomposition Theorem, which is a generalization of the classical Hahn Decomposition Theorem, decompose any space X into a positive set A and a negative set B such that A+B=X and the signed measure of $A \wedge B $ is 0.