996 CHAPTER 28. HAUSDORFF MEASURE

Letting δ → 0, and noting ε > 0 was arbitrary, yields

H s(A∪B)≥H s(A)+H s(B).

Equality holds because H s is an outer measure. By Caratheodory’s criterion, H s is aBorel measure.

To verify the second assertion, note there is no loss of generality in letting H s (E)< ∞.Let

E ⊆ ∪∞j=1C j, r(C j)< δ ,

and

H sδ(E)+δ >

∞

∑j=1

β (s)(r (C j))s.

LetFδ = ∪∞

j=1C j.

Thus Fδ ⊇ E and

H sδ(E) ≤ H s

δ(Fδ )≤

∞

∑j=1

β (s)(r(C j))s

=∞

∑j=1

β (s)(r (C j))s < δ +H s

δ(E).

Let δ k→ 0 and let F = ∩∞k=1Fδ k

. Then F ⊇ E and

H sδ k(E)≤H s

δ k(F)≤H s

δ k(Fδ )≤ δ k +H s

δ k(E).

Letting k→ ∞,H s(E)≤H s(F)≤H s(E)

A measure satisfying the conclusion of Theorem 28.1.5 is called a Borel regular mea-sure.

28.2 H p and mpNext I will compare H p and mp. To do this, recall the following covering theorem whichis a summary of Corollary 13.4.5 found on Page 350.

Theorem 28.2.1 Let E ⊆Rp and let F be a collection of balls of bounded radii such thatF covers E in the sense of Vitali. Then there exists a countable collection of disjoint ballsfrom F , {B j}∞

j=1, such that mp(E \∪∞j=1B j) = 0.

In the next lemma, the balls are the usual balls taken with respect to the usual distancein Rp coming from the Euclidean norm.

Lemma 28.2.2 If S⊆Rp and mp (S) = 0, then H p (S) =H pδ(S) = 0. Also, there exists a

constant k such that H p (E)≤ kmp (E) for all E Borel k ≡ β (p)α(p) . Also, if Q0 ≡ [0,1)p, the

unit cube, then H p ([0,1)p)> 0.