Title: Localizing sets and the structure of sigma-algebras

Authors: James T. Campbell, Alan Lambert and Barnet M. Weinstock

Issue: Volume 47 (1998), Issue 3, 913-938

Abstract: Given a sigma-finite measure space $(X, \Sigma, \mu)$, we study the structure of sub-$\sigma$-algebras $\mathcal{A}$ of $\Sigma$. Our analysis is based on the concept of \emph{localizing set for} $\mathcal{A}$, which was introduced by Lambert in 1991. Our basic result is that, given $\mathcal{A} \in \Sigma$, $X$ may be partitioned as a countable union $\{B_i\}_{i > 1}$ of sets in $\Sigma$ (a \emph{maximal localizing partition}) such that $B_i$ contains no localizing subsets (an \emph{antilocalizing set}) and, for $i > 1$, $B_i$ is a maximal localizing set in $\bigcup \{B_i: 1 < j < i\}$. When $(X, \Sigma, \mu)$ is a Lebesgue space and $\zeta$ is Rohlin's measurable decomposition corresponding to the sub-$\sigma$-algebra $\mathcal{A}$, localizing sets for $\mathcal{A}$ are Rohlin's \emph{sets which are one-sheeted for} $\zeta$. In Maharam's measure-algebra analysis, localizing sets for $\mathcal{A}$ are the \emph{sets of order $0$ with respect to} (the measure algebra of ) $\mathcal{A}$. 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 $L^1(\Sigma)$ to $L^1(\mathcal{A})$ in terms of the maximal localizing partition, the representation of $\mathcal{A}$ as $T^{-1}(\Sigma)$ when $X$ is a Lebesgue space with no antilocalizing sets, and sufficient conditions for $X$ to have no localizing sets.