29.9. THE COAREA FORMULA 1041
Proof: First note that det(Df(x)Df(x)∗
)=∑i∈Λ(n,m) det
(Dfi (x)
)2 by the Binet Cauchytheorem. Let S ≡ {x : J∗ (x) = 0}. For each i, fi ({x : det
(Dfi (x)
)= 0})
has measurezero due to Sard’s theorem and so it will follow from the argument presented below thatSi ≡ fi ({x : det
(Dfi (x)
)= 0})
has measure zero. Thus Si \S can be neglected. For N ≡{x : Df(x) does not exist} ,mn (N) = 0 by Rademacher’s theorem.Thus in what follows, wecan always assume that either Dfi (x) does not exist or det
(Dfi (x)
)exists and is not 0. This
will be clear from the argument. Let A be a closed subset of Rn \ {S∪N}. By Lemma
29.4.1, there exist disjoint Borel measurable sets{
F ij
}∞
j=1such that fi is one to one on F i
j ,(fi)−1 is Lipschitz on fi
(F i
j
), and
∪∞j=1F i
j ={
x : Dfi (x) exists anddetDfi (x) ̸= 0}.
If x ∈ Rn \{S∪N}, it follows x ∈ F ij for some i and j. Hence ∪i, jF i
j ⊇ A.
Now let{
E ij
}be measurable sets such that E i
j ⊆ F ik for some k, the sets E i
j are disjoint,
and their union coincides with ∪i, jF ij . Let g :Rn→Rn be a Lipschitz function which equals(
fi)−1 on fi(
E ij
). I am supressing the dependence on i. Then for any x ∈ E i
j,g(fi (x)
)= x.
In particular, gic(fi (x)
)= xic where
gi (y)≡(
gi1 (y) · · · gim (y))T
for i≡ (i1, · · · , im) with gic (y) defined similarly and x ∈ E ij, with
y≡(
y1y2
)≡(
f(x)xic
)≡ fi (x) ∈ fi
(E i
j
),
xi = gi
(fi (x)
), y2 ≡ xic = gic
(fi (x)
)(29.9.49)
Then, by definition, ∫A
J∗ (x)dx≡∫
Adet(Df(x)Df(x)∗
)1/2 dx (29.9.50)
First, using Theorem 29.8.2, and the fact that Lipschitz mappings take sets of measure zeroto sets of measure zero, replace E i
j with Ẽ ij ⊆ E i
j such that E ij \ Ẽ i
j has measure zero and
Dfi (g(y))Dg(y) = I, |det(Dg(y))|=∣∣∣detDfi (g(y))
∣∣∣−1(29.9.51)
on fi(
Ẽ ij
). Changing the variables using the area formula and 29.9.51, the expression in
29.9.50 equals∫A
J∗ (x)dx =∞
∑j=1
∑i ∈Λ(n,m)
∫Ẽ i
j∩A
(det(Df(x)Df(x)∗
))1/2 dx
=∞
∑j=1
∑i ∈Λ(n,m)
∫E i
j∩A
(det(Df(x)Df(x)∗
))1/2 dx