5.5 Measures And Outer Measures
There is also something called an outer measure which is defined on the set of all subsets.
Definition 5.5.1 Let Ω be a nonempty set and let λ : P
) satisfy the following:
- λ = 0
- If A ⊆ B, then λ
Then λ is called an outer measure.
Every measure determines an outer measure. For example, suppose that μ is a measure on ℱ a σ
algebra of subsets of Ω. Then define
This is easily seen to be an outer measure. Also, we have the following Proposition.
Proposition 5.5.2 Let μ be a measure as just described. Then
as defined above, is an outer
measure and also, if E ∈ℱ, then
Proof: The first two properties of an outer measure are obvious. What of the third? If any
then there is nothing to show so suppose each of these is finite. Let Fi ⊇ Ei
such that Fi ∈ℱ
is arbitrary, this establishes the third condition. Finally, if E ∈ℱ
then by definition,
E ⊇ E
. Also, μ
such that F ⊇ E
. If follows that μ
lower bound of all such