Chapter 28

Hausdorff Measure28.1 The Definition

This chapter is on Hausdorff measures. First I will discuss some outer measures. In all thatis done here, α (n) will be the volume of the ball in Rn which has radius 1.

Definition 28.1.1 For a set E, denote by r (E) the number which is half the diameter of E.Thus

r (E)≡ 12

sup{|x−y| : x,y ∈ E} ≡ 12

diam(E)

Let E ⊆ Rn.

H sδ(E)≡ inf{

∞

∑j=1

β (s)(r (C j))s : E ⊆ ∪∞

j=1C j,diam(C j)≤ δ}

H s(E)≡ limδ→0+

H sδ(E).

Note that H sδ(E) is increasing as δ → 0+ so the limit clearly exists.

In the above definition, β (s) is an appropriate positive constant depending on s. It willturn out that for n an integer, β (n) = α (n) where α (n) is the Lebesgue measure of the unitball, B(0,1) where the usual norm is used to determine this ball.

Lemma 28.1.2 H s and H sδ

are outer measures.

Proof: It is clear that H s( /0) = 0 and if A ⊆ B, then H s(A) ≤H s(B) with similarassertions valid for H s

δ. Suppose E = ∪∞

i=1Ei and H sδ(Ei)< ∞ for each i. Let {Ci

j}∞j=1 be

a covering of Ei with∞

∑j=1

β (s)(r(Cij))

s− ε/2i < H sδ(Ei)

and diam(Cij)≤ δ . Then

H sδ(E) ≤

∞

∑i=1

∞

∑j=1

β (s)(r(Cij))

s

≤∞

∑i=1

H sδ(Ei)+ ε/2i

≤ ε +∞

∑i=1

H sδ(Ei).

It follows that since ε > 0 is arbitrary,

H sδ(E)≤

∞

∑i=1

H sδ(Ei)

993