Localizing Sets and the Structure of Sigma-Algebras


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 {Bi}i>1 of sets in Σ (a maximal localizing partition) such that B1 contains no localizing subsets (an antilocalizing set) and, for 1 > 1, Bi is a maximal localizing set in ∪{Bi : 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 Roblin'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 L1 (Σ) to L1 (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.

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Indiana University Mathematics Journal