Classification and Properties of Atomic Classes in Lattices

Abstract

In this manuscript, we introduce a set of null sets, along with definitions for lattice measures of atoms and lattice semi-finite measures. Our key finding establishes that the lattice measure of any two atoms is either disjoint or identical. Additionally, we provide a proof demonstrating that the class encompassing all atoms within a lattice sigma algebra is countable. Ultimately, we affirm certain fundamental characteristics pertaining to atoms in a lattice sigma algebra.