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 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

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