, Volume 20, Issue 2, pp 399–412 | Cite as

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

  • Jarno TalponenEmail author


In this note we first 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.


Orthomodular lattice Boolean algebra Measure algebra Group-valued submeasure Natural density Density set Content extension Banach space Function space \(L^p\) space 

Mathematics Subject Classification

06C15 28A60 11B05 46E99 



This research was financially supported by the Academy of Finland Project #268009, the Finnish Cultural Foundation and Väisälä Foundation.


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Authors and Affiliations

  1. 1.Department of Physics and MathematicsUniversity of Eastern FinlandJoensuuFinland

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