Editing Measure (mathematics) (section)

===Additivity===
Measures are required to be countably additive. However, the condition can be strengthened as follows.
For any set <math>I</math> and any set of nonnegative <math>r_i,i\in I</math> define:
<math display=block>\sum_{i\in I} r_i=\sup\left\lbrace\sum_{i\in J} r_i : |J|<\infty, J\subseteq I\right\rbrace.</math>
That is, we define the sum of the <math>r_i</math> to be the supremum of all the sums of finitely many of them.

A measure <math>\mu</math> on <math>\Sigma</math> is <math>\kappa</math>-additive if for any <math>\lambda<\kappa</math> and any family of disjoint sets <math>X_\alpha,\alpha<\lambda</math> the following hold:
<math display=block>\bigcup_{\alpha\in\lambda} X_\alpha \in \Sigma</math>
<math display=block>\mu\left(\bigcup_{\alpha\in\lambda} X_\alpha\right) = \sum_{\alpha\in\lambda}\mu\left(X_\alpha\right).</math>
The second condition is equivalent to the statement that the [[Ideal (set theory)|ideal]] of null sets is <math>\kappa</math>-complete.