Countable Additivity

A set function mu possesses countable additivity if, given any countable disjoint collection of sets {E_k}_(k=1)^n on which mu is defined,

 mu( union _(k=1)^inftyE_k)=sum_(k=1)^inftymu(E_k).

A function having countable additivity is said to be countably additive.

Countably additive functions are countably subadditive by definition. Moreover, provided that mu(emptyset)=0 where emptyset is the empty set, every countably additive function mu is necessarily finitely additive.

See also

Countable Additivity Probability Axiom, Countable Subadditivity, Disjoint Union, Finite Additivity, Set Function

This entry contributed by Christopher Stover

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Royden, H. L. and Fitzpatrick, P. M. Real Analysis. Pearson, 2010.

Cite this as:

Stover, Christopher. "Countable Additivity." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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