Algebraic Characterizations of Measure Algebras

We present necessary and sufficient conditions for the existence of a countably additive measure on a Boolean σ-algebra. For instance, a Boolean σ-algebra B is a measure algebra if and only if B−{0} is the union of a chain of sets C1 ⊂ C2 ⊂ ... such that for every n, (i) every antichain in Cn has at mostK(n) elements (for some integerK(n)), (ii) if {an}n is a sequence with an / ∈ Cn for each n, then limn an = 0, and (iii) for every k, if {an}n is a sequence with limn an = 0, then for eventually all n, an / ∈ Ck. The chain {Cn} is essentially unique. 1. Statement of results A Boolean algebra is an algebra B of subsets of a given nonempty set X, with Boolean operations a∪ b, a∩ b, −a = X − a, and the zero and unit elements 0 = ∅ and 1 = X. A Boolean σ-algebra is a Boolean algebra B such that every countable set A ⊂ B has a supremum supA = ∨ A (and an infimum inf A = ∧ A) in the partial ordering of B by inclusion. Definition 1.1. A measure (more precisely, a strictly positive σ-additive probability measure) on a Boolean σ-algebra B is a real valued function m on B such that (i) m(0) = 0, m(a) > 0 for a = 0, and m(1) = 1, (ii) m(a) ≤ m(b) if a ⊂ b, (iii) m(a ∪ b) = m(a) +m(b) if a ∩ b = 0, (iv) m( ∨∞ n=1 an) = ∑∞ n=1 m(an) whenever the an are pairwise disjoint. A measure algebra is a Boolean σ-algebra that carries a measure. Let B be a Boolean algebra and let B = B−{0}. A set A ⊂ B is an antichain if a ∩ b = 0 whenever a and b are distinct elements of A. A partition W (of 1) is a maximal antichain, i.e. an antichain with ∨ W = 1. B satisfies the countable chain condition (ccc) if it has no uncountable antichains. B is weakly distributive if for every sequence {Wn}n of partitions there exists a partition W with the property that each a ∈ W meets only finitely many elements of each Wn. If B is a measure algebra, then B satisfies the ccc and is weakly distributive. Below we present additional, purely algebraic, conditions that characterize measure algebras. Received by the editors December 11, 2006. 2000 Mathematics Subject Classification. Primary 28A60, 06E10. This work was supported in part by GAAV Grant IAA100190509. c ©2007 American Mathematical Society