On dense subsets of the measure algebra

Abstract

Abstract.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.

On the additivity and cofinality of Radon measures

“…Proof. These results also were given to me by J. Cichori; part (b) appears in [2], with a different proof. For each a e 2I L \{0}> choose a compact set F a £ [0, 1] such that 0 ^= F' a £ a in 3I L .…”

“…Consequently y4 + Q is simultaneously co 2 -Sierpiriski and coj-Sierpinski. Evidently cf(,yT L ) = c = to 2 . Now suppose that B £ A is a set of cardinal (o 1 , and set X = B + Q, with /x the subspace measure on X.…”

“…(ii) The next step is to adapt the functions above to obtain operations /h-> F r : N N -• ,/K L and Et-> R E : .JV L^ £f 0 such that / £ * R E whenever V f £ £. To do this, set L(n) = {i:ieN, (n + 1) 2 < i < (n + 2) 2 Remark. The fact that this characterizes add(Al) and cf(./4 L ) will be evident from the results below, but we shall not need it in a formal sense.…”

“…For /e N N set V f = V' g where g(n) = 0 n (/ f L(n)) for each neN. If V f £ £ then g ^* R' E , i.e.there is an meN such that (n, 6 n (/ f L{n))) e R' E for every n ^ m; now (i, /(i)) e R E for every i ^ (m+ I)2 , so (iii) Suppose that F <= N N and that #(F) < add (,/fl). Then £ = {J JeF V f e ./V L so / £* /?…”

The left side of Cichoń’s diagram

“…Cichoń's diagram (see [CKP85], [Fre84], [BJ95]) is the following table of 12 cardinals: The arrows show provable inequalities between these cardinals, such as ℵ 1 = non(countable) ≤ add(N ) ≤ cov(N ) ≤ 2 ℵ 0 = cov(countable).…”