5.9 When Is A Measure A Borel Measure?
You have an outer measure defined on the set of all subsets of ℝp. 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 any
Definition 5.9.1 For two sets, A,B we define
Theorem 5.9.2 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 2.2.46 on Page 165, 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 5.12 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
class=”left” align=”middle”(ℝ)5.10. ONE DIMENSIONAL LEBESGUE MEASURE