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Commutative L-algebras and measure theory

  • Wolfgang Rump EMAIL logo
Veröffentlicht/Copyright: 25. September 2021
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 33 Heft 6

Abstract

Measure and integration theory for finitely additive measures, including vector-valued measures, is shown to be essentially covered by a class of commutative L-algebras, called measurable algebras. The domain and range of any measure is a commutative L-algebra. Each measurable algebra embeds into its structure group, an abelian group with a compatible lattice order, and each (general) measure extends uniquely to a monotone group homomorphism between the structure groups. On the other hand, any measurable algebra X is shown to be the range of an essentially unique measure on a measurable space, which plays the role of a universal covering. Accordingly, we exhibit a fundamental group of X, with stably closed subgroups corresponding to a special class of measures with X as target. All structure groups of measurable algebras arising in a classical context are archimedean. Therefore, they admit a natural embedding into a group of extended real-valued continuous functions on an extremally disconnected compact space, the Stone space of the measurable algebra. Extending Loomis’ integration theory for finitely additive measures, it is proved that, modulo null functions, each integrable function can be represented by a unique continuous function on the Stone space.

Keywords: structure group; vector measure
MSC 2010: 28B15; 06F20; 46B42; 46G10

Dedicated to B. V. M.

Communicated by Manfred Droste

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Received: 2020-11-05
Revised: 2021-08-13
Published Online: 2021-09-25
Published in Print: 2021-11-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. On the topological complexity of manifolds with abelian fundamental group
  3. Reconstructing étale groupoids from semigroups
  4. Fiberwise linear differential operators
  5. Foliations with isolated singularities on Hirzebruch surfaces
  6. On a stronger reconstruction notion for monoids and clones
  7. Cochain level May–Steenrod operations
  8. Commutative L-algebras and measure theory
  9. Rankin–Selberg integrals for principal series representations of GL(n)
  10. A simple proof of the generalized Leibniz rule on bounded Euclidean domains
  11. Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras
  12. Seshadri constants on some Quot schemes
  13. Hörmander Fourier multiplier theorems with optimal Besov regularity on multi-parameter Hardy spaces
  14. On spectral and non-spectral problem for the planar self-similar measures with four element digit sets
  15. C-minimal topological groups
  16. Cevian properties in ideal lattices of Abelian ℓ-groups
  17. Study of twisted Bargmann transform via Bargmann transform
https://doi.org/10.1515/forum-2020-0317
Schlagwörter für diesen Artikel
structure group; vector measure

Artikel in diesem Heft

  1. Frontmatter
  2. On the topological complexity of manifolds with abelian fundamental group
  3. Reconstructing étale groupoids from semigroups
  4. Fiberwise linear differential operators
  5. Foliations with isolated singularities on Hirzebruch surfaces
  6. On a stronger reconstruction notion for monoids and clones
  7. Cochain level May–Steenrod operations
  8. Commutative L-algebras and measure theory
  9. Rankin–Selberg integrals for principal series representations of GL(n)
  10. A simple proof of the generalized Leibniz rule on bounded Euclidean domains
  11. Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras
  12. Seshadri constants on some Quot schemes
  13. Hörmander Fourier multiplier theorems with optimal Besov regularity on multi-parameter Hardy spaces
  14. On spectral and non-spectral problem for the planar self-similar measures with four element digit sets
  15. C-minimal topological groups
  16. Cevian properties in ideal lattices of Abelian ℓ-groups
  17. Study of twisted Bargmann transform via Bargmann transform
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