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
→ [0
,∞)
satisfy the following:
 λ = 0
 If A ⊆ B, then λ ≤ λ
 λ ≤∑
_{i=1}^{∞}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
ˆμ (S ) ≡ inf {μ(E) : E ⊇ S, E ∈ ℱ }


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 =
μ.
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
F_{i} ⊇ E_{i} such that
F_{i} ∈ℱ and
+
> μ. Then
∞ ∞ ∑∞
ˆμ(∪i=1Ei) ≤ μ(∪i=1Fi) ≤ μ (Fi)
∞ ( i=1) ∞
< ∑ ˆμ (E )+ ε = ∑ ˆμ (E ) + ε
i=1 i 2i i=1 i
Since
ε is arbitrary, this establishes the third condition. Finally, if
E ∈ℱ,
then by definition,
≤ μ because
E ⊇ E. Also,
μ ≤ μ for all
F ∈ℱ such that
F ⊇ E. If follows that
μ is a
lower bound of all such
μ and so
≥ μ.■