1020 CHAPTER 29. THE AREA FORMULA
is Lebesgue measurable and∫h(A)
g(y)dH n =∫
Ag(h(x))J∗ (x)dmn
where J∗ (x) = det(U (x)) = det(Dh(x)∗Dh(x)
)1/2.
Since H n = mn on Rn, this is just a generalization of the usual change of variablesformula. This is much better because it is not limited to h having values in Rn. Alsonote that you could replace A with G since they differ by a set of measure zero thanksto Rademacher’s theorem. Note that if you assume that h is Lipschitz on G then it has aLipschitz extension to Rn. The conclusion has to do with integrals over G. It is not reallynecessary to have h be Lipschitz continuous on Rn, but you might as well assume thisbecause of the existence of the Lipschitz extension. Here is another interesting change ofvariables theorem.
Theorem 29.5.4 Let h : G⊆Rn→Rm be continuous where G is an open set and let A⊆Gwhere A is the Borel measurable set consisting of x where Dh(x) exists. Suppose h isdifferentiable and one to one on A. Also let g : h(G)→ [0,∞] be H n measurable. Then
x→ (g◦h)(x)J∗ (x)
is Lebesgue measurable and∫h(A)
g(y)dH n =∫
Ag(h(x))J∗ (x)dmn (29.5.24)
where J∗ (x) = det(U (x)) = det(Dh(x)∗Dh(x)
)1/2.
Proof: By Lemma 29.4.4, ν (E)≡H n (h(E ∩A)) is a measure defined on the Lebes-gue measurable sets contained in G and ν ≪ mn. The reason it is a measure is
ν (∪iEi) ≡ H n (h(∪iEi∩A)) = H n (h(∪iEi∩A))
= H n (∪ih(Ei∩A)) = ∑i
H n (Ei∩A) = ∑i
ν (Ei)
This measure is finite on compact sets. Therefore, by Corollary 11.6.8, it is a regularmeasure. By the Radon Nikodym theorem for Radon measures, Theorem 31.3.5,
ν (E) =∫
EDmnνdmn
where Dmnν is the symmetric derivative given by
limr→0
H n (B(x,r)∩A)mn (B(x,r))
However, from Theorem 29.4.3 this limit equals J∗ (x) described above as
det(Dh(x)∗Dh(x)
)1/2
Now the rest of the argument is identical to that presented above leading to Theorem29.5.3.
Note that from 29.5.24, H n (h(A\A+)) = 0 so this also gives a generalization ofSard’s theorem used earlier in the case that h is one to one.