1. IDEALS OVER UNCOUNTABLE SITS
1.1. Let K be a regular uncountable cardinal number. A set I of subsets
of K is an ideal over K if
(1.1) (i) 0 I and
K
t I
(ii) if X I and Y £ X then Y I
(iii) if X I and Y U then X U Y 6 I,
In this paper we deal only with nontrivial K-complete ideals over K:
(iv) {a} I for all a
K
(v) if y K and if X I for all a y, then U X 6 I.
a ay
Throughout the paper, ideal means a nontrivial K-complete ideal over K.
The least ideal over K is the ideal
I = {X c
K
: |x|
K}
.
If K is a measurable cardinal, and if y is K-additive two-valued measure
on K, then
(1.2) I * {X c
K
: y(X) = 0}
is an ideal. This ideal is prime, i.e. for every X £ K either X I or
K-X I. More generally, if K is a real-valued measurable cardinal and y
is a K-additive measure on K, then (1.2) is an ideal.
The example (1.2) motivates the following terminology: let I be a
given ideal, and let X £ K. If X £ I then we say that X has measure 0,
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