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

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Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer Basel
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

References

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