IDEALS OVER UNCOUNTABLE SETS 13 For each n, there exists a function f F such that X = dom(f ). n n n n If a 0 X , then because F F . .. , we have f_(a) f0(a) ..., A n rr. U 1 0 1 n=0 °° a contradiction. Hence ( X = 0. n=0 b) Let us assume that I is not precipitous. Let S be a set of positive measure and let W_ W_ ... W ... be I-partitions of S r 0 1 n CO such that 0 X is empty whenever L £ X1 3 ... £ X 2.• is a sequence n=0 of sets such that X ( W for each n. We shall construct functionals on S n n such that F„ F, ... F . . . . 0 1 n Without loss of generality let us assume that if X W , Y £ W and X £ Y, then X 4 Y. Let T = U W^ note that the partially ordered set (T,c) is an upside down tree. n n=n For each z S, let us consider the set T = {X T : z X}. Since z every descending sequence Xn D X- D ... 3 X 3 ... in T has empty inter- section, it follows that for every z 6 S, T has no infinite descending sequence Xrt o X- D ... 3 X 3 ... thus the relation c on T is well n 0 1 n z founded. For each z S, let p be the rank function for the well founded z relation c on T if X,Y T and X c Y, then p (X) p (Y). Hence for each n, if X W .. . , Y W and z 6 X c Y, then p (X) p (Y). n+l n z z Thus we define, for each X T, a function f on X as follows: f(z) « p (X) (all z X). X z For each n, we let F ^{f^rX^W}. It follows that each F is an n X n n ordinal valued functional on S, and that F^ F. , .. . F ... . a 0 1 n Corollary. If I is precipitous then I is weakly normal. Proof. Assume that I is not weakly normal. Then there exists a set S of positive measure such that for every unbounded I-function f with dom(f) £ S
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