Localizing sets and the structure of Sigma-algebras
James T. Campbell, Alan Lambert, Barnet M. Weinstock · Indiana University Mathematics Journal · 1998
Given a sigma-finite measure space (X, Σ,µ), we study the structure of sub-σ-algebras A of Σ.Our analysis is based on the concept of localizing set for A which was introduced by Lambert in 1991.Our basic result is that, given A ⊂ Σ, X may be partitioned as a countable union {B i } i>1 of sets in Σ (a maximal localizing partition) such that B 1 contains no localizing subsets (an antilocalizing set) and, for i > 1, B i is a maximal localizing set in {B i : 1 < j < i}.When (X, Σ,µ) is a Lebesgue space and ζ is Rohlin's measurable decomposition corresponding to the sub-σ-algebra A, localizing sets for A are Rohlin's sets which are one-sheeted for ζ.In Maharam's measure-algebra analysis, localizing sets for A are the sets of order 0 with respect to (the measure algebra of) A. Our approach via functional analysis is significantly more elementary than theirs.Further results include: a description of the kernel of the conditional expectation operator from L 1 (Σ) to L 1 (A) in terms of the maximal localizing partition, the representation of A as T -1 (Σ) when X is a Lebesgue space with no antilocalizing sets, and sufficient conditions for X to have no localizing sets.Introduction.This paper is an attempt to understand the structure of sigma-algebras and their associated conditional expectations encountered by the authors in their work on composition operators.In particular we are interested in the following questions.Given a sigma-finite measure space (X, Σ,µ) with a µ-sigma-finite subalgebra A ⊂ Σ, is there a canonical way of describing how A-sets decompose into Σ-subsets?If E A : L 1 → L 1 denotes conditional expectation with respect to A, is there a useful way of describing the kernel and range of E A ?