On Dense Subsets of the Measure Algebra

We show that the minimal cardinality of a dense subset of the measure algebra is the same as the minimal cardinality of a base of the ideal of Lebesgue measure zero subsets of the real line. 0. Introduction. Let (F, <) be a given partial ordering. A subset D ç P is called dense if for any p g F there exists d g D such that d < p. A subset D is called upward dense if D is dense in (F, > ). Let A(F, < ) denote the minimal cardinality of a dense subset of ( F, < ). Recall that with any boolean algebra fé'we can associate a natural partial ordering < v. Let 0 and 1 denote the minimal and the maximal elements in this ordering. A dense or upward dense subset of ^ is a subset of <$\ (0,1} which is dense or upward dense with respect to < v. Note that A(»\{0,1},<,)-A(*\{0,1},,>). We call this cardinal number AC^). Let J be an ideal of subsets of a set X. A base of ./is an upward dense subset of (•/, ÇZ). Let A(./) = A(./, 2). ./is a a-ideal if for any countable sic J we have U J^g J. Let Jc denote the dual filter {X \ A: A <aJ). In this paper 3$ denotes the a-field of Borel subsets of the Cantor set "2 (w denotes the set of natural numbers and 2 denotes the set {0,1}). Let JTdenote the ideal of subsets of "2 of first Baire category. The canonical product measure on "2 is called the Lebesgue measure on "2. Let i? denote the ideal of subsets of "2 of Lebesgue measure zero. Note that both ideals JTand »S?have Borel bases (i.e. bases contained in SS). If ./is an ideal on "2 then JO Sä is an ideal in the a-field 38. We denote the quotient boolean algebra 38/(JO 38) by 31/J. A boolean algebra is a-saturated if there is no uncountable family of pairwise disjoint elements of this algebra. Recall that the algebras 88/X and 88/¡P are a-saturated. Suppose that #is a boolean algebra and aei'. Then by <€a we denote the boolean algebra with the universe {b g <$; b <^a} endowed with the operations canonically defined from the operations in <€. 1. The main result. We show a method of constructing a dense subset of the boolean algebra á?/./from a base of ./for some class of ideals on "2. Received by the editors July 29, 1983. 1980 Mathematics Subject Classification. Primary 04A15; Secondary 28A05. ©1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page