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 ( 1 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

0

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