Weakly dense subsets of the measure algebra

Abstract

Abstract.We introduce the notion of the weak density of a Boolean algebra and show that for homogeneous measure algebras it coincides with the density (=least size of a coinitial set). From this we obtain a partial liffîng of the measure algebra of [0, 1 ] of minimal size which does not extend to a lifting. It also follows that the ;r-character of each point and the ^-weight are the same for the Stone space of a homogeneous measure algebra 0. Notation For set-theoretic and topological notation not described below, see [3,8,9,10]. For measure theory see [7]. If B is a Boolean algebra and a e B , ac is the complement of a . Ba = B\a = {b € B: b < a} . B is homogeneous if Ba is isomorphic to B for all a e B\{0}. If / is a function which assigns an ordinal to each Boolean algebra, then B is homogeneous in f if f{Ba) = f(B) for all a 6 B\{0}. If (X,~L,n) is a probability space and A, B eZ then we write A ç* B, A =* B to mean ß(A\B) = 0, ß(AAB) -0, respectively. If / is any nonempty set, then (X ,//) is the product space with product measure denoted also by fi. For a e / let na: X1 -* XAM be defined by nJJ) = /|A{a} . Then for any set E c X , supp(£) == {a € /: E £ n~ n"aE}.For cardinals k > co, ß is the usual product measure on {0,1}*, ZK is the cr-algebra of measurable sets, and ¿VK = {E e ZK : ¡iE = 0} . srfK = E^/yT is the measure algebra of ({0,1}K ,2X ,/u). If E e ZK then [E] e sfK is the equivalence class of E. If B is a subalgebra of stfK, then a partial lifting p: B -> Z^ is a Boolean homomorphism satisfying />(a) e a for each a € B .

Combinatorics of Ultrafilters on Cohen and Random Algebras

“…The fact that is a consequence of the work of Burke [3]. More precisely, in Case 1 of [3, Theorem 1], he showed the following: if has the property that for all there exists such that either or , then there exists a cofinal subset such that . Therefore, the inequality follows from the observation that, whenever U is an ultrafilter on and is cofinal, clearly X satisfies the assumption of Burke’s result.…”

“…The fact that is a consequence of the work of Burke [3]. More precisely, in Case 1 of [3, Theorem 1], he showed the following: if has the property that for all there exists such that either or , then there exists a cofinal subset such that .…”

Combinatorics of ultrafilters on Cohen and random algebras

“…The fact that cof(N ) ≤ u(B ω ) is a consequence of the work of Burke [3]. More precisely, in Case 1 of [3, Theorem 1], he showed the following: if X ⊆ B ω \ {0} has the property that for all b ∈ B ω there exists x ∈ X such that either x ≤ b or x ∧ b = 0, then there exists a cofinal subset Y ⊆ N such that |X| = |Y |.…”

On the relationship between density and weak density in Boolean algebras

“…Recently, there has been interest in determining a bound for d(77) in terms of wd(77). M. Burke has shown wd(77) = d(77) for measure algebras [2]. Recall that X is an antichain means X c 77\{0} and consists of pairwise disjoint elements and the cellularity of 77, c(B) = sup{\A\: A is an antichain} [4].…”