X and an ordinal y such that yf = y for all f G . Clearly, the
{Sf : f G}
is a family of size A of almost disjoint sets of positive measure. n
An I-function f : S » K is unbounded if for every y K, the set
{a S : f(a) 5 y} has measure 0. An unbounded I-function f : S -. K is
a minimal unbounded function if there exists no unbounded I-function g with
dom(g) £ S such that g(a) f(.a) for all a dom(g). An ideal I is
weakly normal if for every set S of positive measure there exists a minimal
unbounded I-function h with dom(h) S. If I is normal then I is weakly
normal (the diagonal function d(a) = a is minimal on every set of positive
1.4. Let I be an ideal over K, and let S be a set of positive measure.
An I-partition of S is a maximal collection W of subsets of S of positive
measure such that X f l Y ( I for any distinct X,Y W. An I-partition W
of S is a refinement of an I-partition W of S,
(1.7) W]L 5 W2,
if every X £ W is a subset of some Y W .
The ideal I is precipitous if whenever S is a set of positive measure
and {W : n co} are I-partitions of S such that
fcL W- . . . W . . .
0 1 n
then there exists a sequence of sets
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