Outer Measure

An outer measure is a map that provides a concept of size to sets.

Description

For a given set X, an outer measure is a map μ* : P(X) -> [0,∞] (i.e., a map from all subsets of the given set to non-negative real numbers). Note that [0,∞] is shorthand for the more formally correct [0,∞) ∪ {∞}.

The concept follows from covers, i.e. a countable union of open sets defined as covering some other reference set. If these covers can be ordered then there is an infimum cover; the size of this is the size of the set. The ordering is achieved with a monotonic, non-negative function p that maps a set to a real number, and always maps the empty set (Ø) to zero. For example, a function that maps the empty set (Ø) to zero and all other sets to 1.

More generically, an outer measure must satisfy three conditions:

• μ*(Ø) = 0

• if A ⊂ B ⊂ X then μ*(A) ≤ μ*(B)

• σ sub-additivity: sub-additivity for finite and countably infinite elements, but not infinite elements

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  • There is no requirement for disjoint sets because clearly the union of two intersecting sets is smaller than the sum of those sets considered separately.

CategoryRicottone