Let X be a set and S a collection of subsets of X. A set function μ:S->[0, ∞] is said to possess countable monotonicity provided that, whenever a set E element S is covered by a countable collection {E_k}_(k = 1)^∞ of sets in S, μ(E)<= sum_(k = 1)^∞ μ(E_k). A function which possesses countable monotonicity is said to be countably monotone. One can easily verify that any set function μ which is both monotone (in the sense of mapping subsets of the domain to subsets of the range) and countably additive is necessarily countably monotone. The converse is not true in general.
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