On natural density, orthomodular lattices, measure algebras and non-distributive $L^p$ spaces

TL;DR

This paper explores the extension of natural density to sigma-algebras within Boolean algebras, employing quantum-inspired submeasures on orthomodular lattices to construct non-distributive $L^p$ spaces.

Contribution

It introduces a method to extend natural density to sigma-algebras using submeasures on orthomodular lattices, connecting measure theory with quantum structures.

Findings

Natural density extends to sigma-algebras within Boolean algebras.

Submeasures on orthomodular lattices can be used to construct non-distributive $L^p$ spaces.

The approach bridges measure theory, quantum logic, and non-distributive functional spaces.

Abstract

In this note we show, roughly speaking, that if $\mathcal{B}$ is a Boolean algebra included in the natural way in the collection $\mathcal{D}/_\sim$ of all equivalence classes of natural density sets of the natural numbers, modulo null density, then $\mathcal{B}$ extends to a $\sigma$-algebra $\Sigma \subset \mathcal{D}/_\sim$ and the natural density is $\sigma$-additive on $\Sigma$. We prove the main tool employed in the argument in a more general setting, involving a kind of quantum state function, more precisely, a group-valued submeasure on an orthomodular lattice. At the end we discuss the construction of `non-distributive $L^p$ spaces' by means of submeasures on lattices.