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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