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