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Outer Measure


Given a set X, a set function mu^*:2^X->[0,infty] is said to be an outer measure provided that mu^*(emptyset)=0 and that mu^* is countably monotone, where emptyset is the empty set.

Given a collection S of subsets of X and an arbitrary set function mu:S->[0infty], one can define a new set function mu^* by setting mu^*(emptyset)=0 and defining, for each non-empty subset E subset X,

 mu^*(E)=infsum_(k=1)^inftymu(E_k)

where the infimum is taken over all countable collections {E_k}_(k=1)^infty of sets in S which cover E. The resulting function mu^*:2^X->[0infty] is an outer measure and is called the outer measure induced by mu.


See also

Carathéodory Extension, Carathéodory Measure, Countable Monotonicity, Measurable Set, Measurable Space, Measure, Measure Space, Premeasure

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. "Outer Measure." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/OuterMeasure.html

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