You have an outer measure defined on the set of all subsets of ℝ^{p}. 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
ball.

Definition 5.9.1For two sets, A,B we define

dist(A,B ) ≡ inf{d(x,y) : x ∈ A,y ∈ B }.

Theorem 5.9.2Let μ be an outer measure on the subsets of (X,d), a metric space. If

μ(A∪ B ) = μ(A)+ μ (B )

wheneverdist(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) < ∞.
Let

By the construction, the distance between any pair of sets, S ∩ (K_{k}∖K_{k+1}) for different even values of k is
positive and the distance between any pair of sets, S ∩ (K_{k}∖K_{k+1}) for different odd values of k is positive.
Therefore,