Measure (mathematics)
Additivity

Measures are required to be countably additive. However, the condition can be strengthened as follows.For any set I {\displaystyle I} and any set of nonnegative r i, i ∈ I {\displaystyle r_{i},i\in I} define: ∑ i ∈ I r i = sup { ∑ i ∈ J r i: | J | < ∞, J ⊆ I }. {\displaystyle \sum _{i\in I}r_{i}=\sup \left\lbrace \sum _{i\in J}r_{i}:|J|<\infty,J\subseteq I\right\rbrace.} That is, we define the sum of the r i {\displaystyle r_{i}} to be the supremum of all the sums of finitely many of them.

A measure μ {\displaystyle \mu } on Σ {\displaystyle \Sigma } is κ {\displaystyle \kappa } -additive if for any λ < κ {\displaystyle \lambda <\kappa } and any family of disjoint sets X α, α < λ {\displaystyle X_{\alpha },\alpha <\lambda } the following hold: ⋃ α ∈ λ X α ∈ Σ {\displaystyle \bigcup _{\alpha \in \lambda }X_{\alpha }\in \Sigma } μ (⋃ α ∈ λ X α) = ∑ α ∈ λ μ (X α). {\displaystyle \mu \left(\bigcup _{\alpha \in \lambda }X_{\alpha }\right)=\sum _{\alpha \in \lambda }\mu \left(X_{\alpha }\right).} The second condition is equivalent to the statement that the ideal of null sets is κ {\displaystyle \kappa } -complete.
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