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, φδ ≥ φδ' 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.
Read more about this topic: Outer Measure
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