On the essential union and intersection of families of measurable sets
Zsolt Páles · Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae Sectio computatorica · 2020
In this short note, we demonstrate that in a σ-finite measure space the union and the intersection of arbitrary (i.e., not necessary countable) families of measurable sets can be defined in a natural manner.By examples, we also show that the σ-finiteness of the underlying measure space is only sufficient for this property but not necessary.The notions that we discuss below turn out to be significant if one tries to extend the Radon-Nikodym Theorem for non-σ-finite measure spaces.For details, we refer to Section 1.1 and pages 65-71 of the recent monograph [1] by Fonseca and Leoni.More comments will be given at the end of this note.Definition 1.Given a measure space (X, A, μ), we say that a setIt is easy to see that = μ is an equivalence relation on the σ-algebra A, and ⊆ μ is a partial ordering on the equivalence classes.It can also be shown that the function d μ defined by(A, B ∈ A) Key words and phrases: Essential union and intersection of measurable sets.