10 THOMAS JECH AND KAREL PRIKRY

X and an ordinal y such that yf = y for all f € G . Clearly, the

family

{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

measure.)

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

n

fcL W- . . . W . . .

0 1 n

then there exists a sequence of sets