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It turns out the second and third requirements together for all sets are incompatible conditions; see non-measurable set. 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 defined on all subsets of a set X
such that
This allows us to define the concept of measurability as follows: A subset E of x is φ-measurable iff for every subset A of X
Theorem. The φ-measurable sets form a σ-algebra and φ restricted to the measurable sets is a countably additive measure.
For a proof of this theorem see the Halmos reference, section 11.
This method is known as the Carathéodory 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.
Suppose (X, d) is a metric space and φ an outer measure on X. If φ has the property that
whenever
then φ is called a metric outer measure. The Borel sets of X are the elements of the smallest σ-algebra generated by the open sets.
Theorem. If φ is a metric outer measure on X, then every Borel subset of X is φ-measurable.
There are several procedures for constructing outer measures on a set. The classic Munroe reference below describes two particularly useful ones which are referred to as Method I and Method II.
Let X be a set, C a subset of 2X which contains the emptyset and p an extended real valued function on C which vanishes on the emptyset.
Theorem. Suppose the class C and the function p are as above and define
where the infimum extends over all sequences {Ai}i of elements of C which cover E (with the convention that if no such sequence exists, then the infimum is infinite). Then φ is an outer measure on X.
The second technique is more suitable for constructing outer measures on metric spaces, since it yields metric outer measures.
Suppose (X,d) is a metric space. As above C is a subset of 2X which contains the emptyset and p an extended real valued function on C which vanishes on the emptyset. For each δ > 0, let
and
where the infimum extends over all sequences {Ai}i of elements of Cδ which cover E. Obviously, φδ ≥ φδ' when δ ≤ δ' since the infimum is taken over a smaller class as δ decreases. Thus
exists.
Theorem. φ0 is a metric outer measure on X.
This is the construction used in the definition of Hausdorff measures for a metric space.
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