12 THOMAS JECH AND KAREL PRIKRY For each n, let S = U Z . Since Z isanI-partition of S, we n n n oo °° have S-S I, andtherefore P S is nonempty. Let a ( S be n n n n n n=0 n=0 arbitrary. For each n there exists i $ such that a Y. . We let J n n i n x = xn . n l n Since a X and X W for each n, itisenough toshow that n n n X„ 3 X, 3 . . . 3 X 3 ... . Since each Z isa disjoint family, itfollows 0 1 n n that each in isunique, and that if m n then Y. 2. Y. . Hence by (ii), m n X fl X has positive measure, and since W W , i t follows that X 3 X . o m n m n m n Let I beanideal over K andlet S bea set ofpositive measure. A functional on S isa collection F offunctions such that W = {dom(f) : f ( F} isanI-partition of S and dom(f) ^ dom(g) whenever f and g are distinct elements of F. A functional F on S isordinal-valued if the values ofeach f F are ordinal numbers. Let F and G betwo ordinal-valued functionals on S. Wesay that F G if (1.9) (i) Wp WG, and (ii) if f F and g G are such that dom(f) c_ dom(g) then f(ot) g(a) for all a dom(f) . Lemma 1.4.2. Anideal I isprecipitous ifand only iffor noset S of positive measure there exists a sequence offunctionals on S such that Fn F, ... F .... 0 1 n Proof, a) Let S bea set ofpositive measure and let F F ... F ... bea sequence offunctionals on S. Weclaim that I is not n precipitous. Let usconsider the I-partitions W ,W ,...,W ,... of S. 0 1 n Let Xrt D X- ^ . . . 3 X 3 . . . be such that X W ^ for each n we 0 n n oo n claim that fl X is empty. n=0
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