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Finite Monotonicity


Let X be a set and S a collection of subsets of X. A set function mu:S->[0,infty] is said to possess finite monotonicity provided that, whenever a set E in S is covered by a finite collection {E_k}_(k=1)^n of sets in S,

 mu(E)<=sum_(k=1)^nmu(E_k).

A set function possessing finite monotonicity is said to be finitely monotone. Note that a set function mu which is countably monotone is necessarily finitely monotone provided that emptyset in S and mu(emptyset)=0, where emptyset is the empty set.


See also

Countable Monotonicity, Cover, Monotone, Set Function

This entry contributed by Christopher Stover

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References

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

Cite this as:

Stover, Christopher. "Finite Monotonicity." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/FiniteMonotonicity.html

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