Countable additivity

Countable additivity is the property that a measure assigns to a countable disjoint union the sum of the measures of its members.

Measure: A function assigning sizes to sets while satisfying specified mathematical properties. Countable additivity is the defining union property of a measure.
Sigma-algebra: A collection of subsets closed under complements and countable unions, including the whole space. A measure is defined on a sigma-algebra, which contains the unions required by countable additivity.
Probability measure: A measure on a sigma-algebra whose value on the whole sample space is one. Its countable additivity is the axiom that makes probabilities of disjoint alternatives sum.
Uncountable additivity: A proposed additivity condition requiring unions over arbitrary, possibly uncountable, disjoint families to equal the sum of their measures. Standard measures require countable additivity, not this stronger condition.
Disjoint union: A union of sets whose members have no elements in common. The additivity rule applies directly when the sets in the union are disjoint.
Measurable space: A set equipped with a sigma-algebra of subsets designated as measurable. Countable additivity is one of the conditions defining a measure on this structure.
Probability space: A sample space equipped with a sigma-algebra and a probability measure. Countable additivity governs probabilities of countably many disjoint events within this structure.
Complete additivity: The property that a set function is additive over every disjoint family, including uncountable families. It strengthens countable additivity and is generally not required of measures.
Finite additivity: The property that a measure of a finite disjoint union equals the sum of the individual measures. It is the finite version of the countable rule, but does not ensure continuity across infinite unions.
Set: A collection of distinct objects treated as a single mathematical object. The rule concerns measurable sets and their unions.

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