Outer Measure - Construction of Outer Measures

Construction of Outer Measures

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 family of subsets of X which contains the empty set and p a non-negative extended real valued function on C which vanishes on the empty set.

Theorem. Suppose the family C and the function p are as above and define

That is, the infimum extends over all sequences of elements of C which cover E, with the convention that the infimum is infinite if no such sequence exists. 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 family of subsets of X which contains the empty set and p a non-negative extended real valued function on C which vanishes on the empty set. For each δ > 0, let

and

Obviously, φ_δ ≥ φ_δ' when δ ≤ δ' since the infimum is taken over a smaller class as δ decreases. Thus

exists (possibly infinite).

Theorem. φ₀ is a metric outer measure on X.

This is the construction used in the definition of Hausdorff measures for a metric space.