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
x =
n l
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
precipitous. Let usconsider the I-partitions W ,W ,...,W ,... of S.
0 1 n
Let X0
D X- ^ . . . 3 X 3 . . . be such that X W ^ for each n; we
n F
oo n
claim that fl X is empty.
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