In mathematics, in particular in measure theory, an outer measure is a construction used to define measurable sets. Measures are generalizations of length, area and volume, but are useful for much more abstract and complicated sets than mere intervals or open balls in . An outer measure is defined as a function from all the subsets of the underlying space to the extended reals. The initial idea is to define a generalized measuring function that fulfils three requirements:
- Any interval of reals has measure .
- All sets have a definite measure in .
- Countable additivity, i.e.
where is the measure function.
It turns out the second and third requirements together for all sets are incompatible conditions; so the purpose of constructing an outer measure is to define which sets are measurable, and fulfil the countably additivity axiom.
Formally, an outer measure is defined as a function
such that
- (empty set has zero measure)
- (every subset has an outer measure)
- (monotonity)
- (countable sub-additivity)
This allows us to define the concept of measurability which is used to extend the countable sub-additivity given by 4. into a strict countable additivity as long as the sets being manipulated fulfill certain measurability axioms.
This method is known as the Caratheodory construction and is one way of arriving at the concept of Lebesgue measure that is so important for measure theory and the theory of integrals.
