sigma-additivity

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Definition

A set function $\mu: \mathcal{A} \to [0, \infty]$ is $\sigma$-additive (or countably additive) if for any sequence $(A_n)_{n \in \mathbb{N}}$ of pairwise disjoint sets in $\mathcal{A}$,

$$ \mu\left(\bigcup_{n=1}^{\infty} A_n\right) = \sum_{n=1}^{\infty} \mu(A_n) $$

This is one of the defining properties of a measure.

$\sigma$-additivity is stronger than finite additivity and is essential for the theory of measures and integration.