IDEALS OVER UNCOUNTABLE SETS 7 if X f. I then we say that X has positive measure, and if K-X € I, then we say that X has measure 1. The phrase almost all a means that the set of all contrary a*s has measure 0. Note that the set F = {X £ K : K-X € 1} is a (nontrivial K-complete) filter over K, the dual of I, and that if I is prime then F is an ultrafilter. An ideal I is normal if it is closed under diagonal unions: (1.3) if X € I for all a K, then {a £ K : a € Xg for some 6 a} € I. A function f on S £ « is regressive if f(a) a for all a € S, a ^ 0. An ideal I is normal if and only if for every set S of positive measure, if f is a regressive function on S, then f is constant on some set T £ S of positive measure. The least normal ideal is the ideal of thin sets a set X £ K is thin if the complement of X contains a closed unbounded subset of K. In this case, the sets of positive measure are the stationary subsets of K, the sets which meet every closed unbounded set. 1.2. Let X be a cardinal number. A ideal I over K is X-saturated if the Boolean algebra B = P(X)/I is A-saturated if every pairwise disjoint family of elements of B has size less than X. Thus I is X-saturated just in case there exists no collection W of size X of sets of positive measure such that X 0 Y 6 I whenever X and Y are distinct members of W. Let us denote by (1.4) sat(I) the least cardinal number X such that I is K-saturated.

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