19.5 When Is A Measure A Borel Measure?
You have an outer measure defined on the power set of some metric space. How can you tell that the σ
algebra of measurable sets includes the Borel sets? This is what is discussed here. This is a very important
idea because, from the above, you can then assert regularity of the measure if, for example it is finite on
Definition 19.5.1 For two sets, A,B in a metric space, we define
Theorem 19.5.2 Let μ be an outer measure on the subsets of (X,d), a metric space. If
whenever dist(A,B) > 0, then the σ algebra of measurable sets S contains the Borel sets.
Proof: It suffices to show that closed sets are in S, the σ-algebra of measurable sets, because
then the open sets are also in S and consequently S contains the Borel sets. Let K be closed
and let S be a subset of Ω. Is μ(S) ≥ μ(S ∩ K) + μ(S ∖ K)? It suffices to assume μ(S) < ∞.
By Lemma 11.1.14 on Page 685, x → dist
is continuous and so
is closed. By the assumption of
since S ∩ K and S ∖ Kn are a positive distance apart. Now
If limn→∞ μ((Kn ∖K) ∩S) = 0 then the theorem will be proved because this limit along with 19.7 implies
and then taking a limit in
) ≥ μ
(S ∩ K
) + μ
(S ∖ K
) as desired.
Therefore, it suffices to establish this limit.
Since K is closed, a point, x
must be at a positive distance from K
then μ(S ∩ (Kn ∖ K)) → 0 because it is dominated by the tail of a convergent series so it suffices to show
By the construction, the distance between any pair of sets, S ∩ (Kk ∖Kk+1) for different even values of k is
positive and the distance between any pair of sets, S ∩ (Kk ∖Kk+1) for different odd values of k is positive.
and so for all M, ∑
k=1Mμ(S ∩ (Kk ∖ Kk+1)) ≤ 2μ