Hahn Decomposition Theorem of Signed Lattice Measure

TL;DR

This paper extends the Hahn Decomposition Theorem to signed lattice measures on sigma-algebras, demonstrating a decomposition of space into positive and negative lattices with specific measure properties.

Contribution

It introduces the concept of signed lattice measures and proves a Hahn Decomposition Theorem within this new framework.

Findings

Decomposition of space into positive and negative lattices

Definition of signed lattice measures on sigma-algebras

Proof of Hahn Decomposition Theorem for signed lattice measures

Abstract

In this paper, we will define a signed Lattice measure on $\sigma$-algebras, as well as give the definition of positive and negative Lattice. Herein, we will show that the Hahn Decomposition Theorem decomposes any space X into a positive lattice A and a negative Lattice B such that $A \vee B$ =X and the signed Lattice measure of $A \wedge B $ is 0.