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 I S is nonempty. Let a € ( 1 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 X0

rt

D X- ^ . . . 3 X 3 . . . be such that X € W ^ for each n; we

— 1 — —

n

— n F

oo n

claim that fl X is empty.

n=0