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Article Dans Une Revue Bulletin of the Belgian Mathematical Society - Simon Stevin Année : 2024

A class of lattices applicable to measurable sets and functions (modulo equality a.e.)

Résumé

We study σ-spanned lattices, that is, complete lattices where the complete join or meet of any set is the join or meet of a countable subset. From our framework, we derive that for a σ-finite measure space, the set of equivalence classes of measurable functions (with values in a closed subset of R ∪ {−∞, +∞}) modulo equality almost everywhere (a.e.) is an infinitely distributive σ-spanned lattice, and the same holds for equivalence classes of measurable sets; this slightly improves a result from functional analysis. Finally, we give an interpretation of the essential supremum and infimum in terms of adjunctions. Our results can provide a basis for a lattice-theoretical form of functional analysis and harmonic analysis, in particular in mathematical morphology.
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Dates et versions

hal-04467054 , version 1 (11-07-2024)

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Christian Ronse. A class of lattices applicable to measurable sets and functions (modulo equality a.e.). Bulletin of the Belgian Mathematical Society - Simon Stevin, 2024, 31 (2), ⟨10.36045/j.bbms.240105⟩. ⟨hal-04467054⟩
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