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.