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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