IDEALS OVER UNCOUNTABLE SETS 11 0 1 n such that X W for each n, and (IXn is nonempty, n n A n=0 Clearly, if I is prime then I is precipitous (we recall our assumption that I is K-complete). More generally, we have: Theorem 1.4.1. If I is K -saturated then I is precipitous. Proof. Let S be a set of positive measure and let W W ... be a sequence of I-partitions of S. We shall find X £X £..... such that 00 X W for each n and that f X is nonempty, n n n n=n0 For each n, let d - |W i since I is K -saturated, we have d S n ' n1 n for all n. Let W = {X. : i # }. We shall construct I-partitionsv n n I - {Yn : i £ } of S such that for all n,i,Jj n i n (1.8) (i) Y j f l Y ^ if i * j, (ii) Y* c X* and X*-Y*€I, (iii) Z n^ Z n + r Suppose that we have constructed Z , then we construct Z = {Y. : i d } n-1 n i n as follows: For each i d , we let n Y n - (X? - U Xn) f Y11"1 ji 3 where K is the unique K d . such that X, c X n-1 i ~ It is clear that the Z f s satisfy (i), and (ii) follows easily by in- duction on n. That each Z is an I-partition of S is a consequence of each W being an I-partition of S, and of (ii). It is easy to verify that for each n, Z ,- is a refinement of Z . n+1 n
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