994 CHAPTER 28. HAUSDORFF MEASURE

which shows H sδ

is an outer measure. Now notice that H sδ(E) is increasing as δ → 0.

Picking a sequence δ k decreasing to 0, the monotone convergence theorem implies

H s(E)≤∞

∑i=1

H s(Ei).

The outer measure H s is called s dimensional Hausdorff measure when restricted tothe σ algebra of H s measurable sets.

Next I will show the σ algebra of H s measurable sets includes the Borel sets. This isdone by the following very interesting condition known as Caratheodory’s criterion.

28.1.1 PropertiesDefinition 28.1.3 For two sets, A,B in a metric space, we define

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

Theorem 28.1.4 Let µ be an outer measure on the subsets of (X ,d), a metric space. If

µ(A∪B) = µ(A)+µ(B)

whenever dist(A,B)> 0, then the σ algebra of measurable sets 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. LetK be closed and let S be a subset of Ω. Is µ(S)≥ µ(S∩K)+µ(S\K)? It suffices to assumeµ(S)< ∞. Let

Kn ≡ {x : dist(x,K)≤ 1n}

By Lemma 7.1.7 on Page 136, x→ dist(x,K) is continuous and so Kn is closed. By theassumption of the theorem,

µ(S)≥ µ((S∩K)∪ (S\Kn)) = µ(S∩K)+µ(S\Kn) (28.1.1)

since S∩K and S\Kn are a positive distance apart. Now

µ(S\Kn)≤ µ(S\K)≤ µ(S\Kn)+µ((Kn \K)∩S). (28.1.2)

If limn→∞ µ((Kn \K)∩ S) = 0 then the theorem will be proved because this limit alongwith 28.1.2 implies limn→∞ µ (S\Kn) = µ (S\K) and then taking a limit in 28.1.1, µ(S)≥µ(S∩K)+µ(S\K) as desired. Therefore, it suffices to establish this limit.

Since K is closed, a point, x /∈ K must be at a positive distance from K and so

Kn \K = ∪∞k=nKk \Kk+1.

Therefore

µ(S∩ (Kn \K))≤∞

∑k=n

µ(S∩ (Kk \Kk+1)). (28.1.3)