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

Abstract

We study $\sigma$-spanned lattices, that is, complete lattices where the complete join or meet of any set is the join or meet of a countable subset. We derive that for a $\sigma$-finite measure space, the set of equi\-valence classes of measurable functions (with values in a closed subset of $\mathbb{R}\cup\{-\infty,+\infty\}$) modulo equality almost everywhere (a.e.) is an infinitely distributive $\sigma$-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.