Chapter 17

The Area FormulaI am grateful to those who have found errors in this material, some of which were egregious.I would not have found these mistakes because I never teach this material and I don’t use itin my research. I do think it is wonderful mathematics however.

17.1 Estimates for Hausdorff MeasureThis section is on estimates which relate Hausdorff measure to Lebesgue measure. Thiswill allow a geometric motivation for measures on Lipschitz manifolds.

The main case will be for h a Lipschitz function, |h(x)−h(y)| ≤ K |x−y| definedon Rn. This is no loss of generality because of Theorem 16.3.6. However, the main presen-tation will include more general situations than this. One uses the differentiability of h offa set of measure zero to show the existence of disjoint Borel sets E on which h is Lipschitzwith its inverse also being Lipschitz on h(E).

The following lemma states that Lipschitz maps take sets of measure zero to sets ofmeasure zero. It also gives a convenient estimate. This involves the consideration of H n

as an outer measure. Thus it is not necessary to know the set B is measurable.In fact, one only needs to have h locally Lipschitz in much of what follows.

Definition 17.1.1 Let h : Rn→ Rm. This function is said to be locally Lipschitz iffor every x ∈ Rn, there exists a ball Bx containing x and a constant Kx such that for ally,z ∈ Bx,

|h(z)−h(y)| ≤ Kx |z−y|

Lemma 17.1.2 If h is Lipschitz with Lipschitz constant K then for B⊆ Rn,

H n (h(B))≤ KnH n (B)

Also, if T is a set in Rn, mn (T ) = 0, then H n (h(T )) = 0. It is not necessary that h beone to one.

Proof: If H n (B) = ∞, there is nothing to show. Assume H n (B) < ∞. Let {Ci}∞

i=1cover B with each having diameter less than δ and let this cover be such that

∑i

β (n)12

diam(Ci)n < H n

δ(B)+ ε

Then {h(Ci)} covers h(B) and each set has diameter no more than Kδ . Then

H nKδ

(h(B)) ≤ ∑i

β (n)(

12

diam(h(Ci))

)n

≤ Kn∑

iβ (n)

(12

diam(Ci)

)n

≤ Kn (H nδ(B)+ ε

)Since ε is arbitrary, this shows that H n

Kδ(h(B))≤ KnH n

δ(B). Now take a limit as δ → 0.

The second claim follows from mn = H n on Lebesgue measurable sets of Rn. ■

Lemma 17.1.3 If h is locally Lipschitz and mn (T ) = 0, then H n (h(T )) = 0. It is notnecessary that h be one to one.

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