  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.
