countable additivity

Countable additivity is the defining promise of a measure: if you break a set into countably many non-overlapping pieces, the sizes of the pieces add up exactly to the size of the whole — and this works for an infinite list of pieces, not merely a finite one. It is the property that lets measure theory commute with limits.

Formally, a set function mu defined on a sigma-algebra is countably additive (or sigma-additive) if for every sequence A_1, A_2, A_3, ... of pairwise disjoint measurable sets, mu(union of all A_n) = sum over n of mu(A_n). The sum may be a finite number or +infinity; either way the equality must hold exactly. Finite additivity is the special case where all but finitely many A_n are empty.

The step from finite to countable additivity is the whole difference between elementary geometry and analysis. It is genuinely a stronger condition: there exist finitely additive set functions that are not countably additive, and they fail to support a good integration theory. Countable additivity is equivalent, for a finitely additive measure, to continuity from below — that mu(union of an increasing sequence) is the limit of the mu values — and this continuity is exactly what the great convergence theorems exploit.