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

On natural density, orthomodular lattices, measure algebras and non-distributive $$L^p$$ L p... In this note we first show, roughly speaking, that if $$\mathcal {B}$$ B is a Boolean algebra included in the natural way in the collection $$\mathcal {D}/_\sim$$ D / ∼ of all equivalence classes of natural density sets of the natural numbers, modulo null density, then $$\mathcal {B}$$ B extends to a $$\sigma$$ σ -algebra $$\Sigma \subset \mathcal {D}/_\sim$$ Σ ⊂ D / ∼ and the natural density is $$\sigma$$ σ -additive on $$\Sigma$$ Σ . 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 $$L^p$$ L p spaces’ by means of submeasures on lattices. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

, Volume 20 (2) – Sep 3, 2015
14 pages

/lp/springer_journal/on-natural-density-orthomodular-lattices-measure-algebras-and-non-OUVdK0bSeW
Publisher
Springer International Publishing
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-015-0363-3
Publisher site
See Article on Publisher Site

### Abstract

In this note we first show, roughly speaking, that if $$\mathcal {B}$$ B is a Boolean algebra included in the natural way in the collection $$\mathcal {D}/_\sim$$ D / ∼ of all equivalence classes of natural density sets of the natural numbers, modulo null density, then $$\mathcal {B}$$ B extends to a $$\sigma$$ σ -algebra $$\Sigma \subset \mathcal {D}/_\sim$$ Σ ⊂ D / ∼ and the natural density is $$\sigma$$ σ -additive on $$\Sigma$$ Σ . 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 $$L^p$$ L p spaces’ by means of submeasures on lattices.

### Journal

PositivitySpringer Journals

Published: Sep 3, 2015

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