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Let be a finite, non-negative, finitely additive set function on a measurable space . Show that is countably additive if and only if it satisfies the Axiom of Continuity: For .
(=>) Assume is countably additive. Let , . Then,
Suppose . Then implies .
(<=) Assume satisfies Axiom of Continuity. Let be mutually disjoint sets. Define .
Then . , . .