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
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