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 l 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 l 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 l

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

fs

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