29.9. THE COAREA FORMULA 1043
det[(
Dy2gi (y)∗ Dy2gic (y)∗ )( Dy2gi (y)
Dy2gic (y)
)]1/2
= det(Dy2g(y)∗Dy2g(y)
)1/2
Therefore, 29.9.53 reduces to∫
Kij∩A det
(Df(x)Df(x)∗
)1/2 dx =
∫Rm
∫f−1(y1)∩Ki
j∩Adet(Dy2g(y)∗Dy2g(y)
)1/2 dy2dy1. (29.9.57)
Then z ∈ g(
y1, f−1 (y1)∩Kij ∩A
)if and only if
fi (z) =(
f(z)zic
)∈(
y1f−1 (y1)∩Ki
j ∩A
)if and only if z ∈ f−1 (y1) and zic ∈ f−1 (y1)∩Ki
j ∩ A. Letting ĝ be the function y2 →g(y1,y2) , this shows that z ∈ ĝ
(f−1 (y1)∩Ki
j ∩A)
if and only if y2 = zic ∈ f−1 (y1)∩Kij∩A
and so ĝ(
f−1 (y1)∩Kij ∩A
)= f−1 (y1)∩Ki
j ∩A. Of course ĝ actually depends on y1 butthis is suppressed here. Therefore,
g(
y1, f−1 (y1)∩Kij ∩A
)= f−1 (y1)∩Ki
j ∩A
By this observation and the area formula, the equations 29.9.53, 29.9.57 imply∫Ki
j∩Adet(Df(x)Df(x)∗
)1/2 dx =∫Rm
H n−m(
f−1 (y1)∩Kij ∩A
)dy1.
Using Lemmas 29.9.6 and 29.9.5, along with the inner regularity of Lebesgue measure, Kij
can be replaced with E ij. Therefore, summing the terms over all i and j,∫
Adet(Df(x)Df(x)∗
)1/2 dx =∫Rm
H n−m (f−1 (y)∩A)
dy
which verifies the coarea formula whenever A is a closed subset of Rn \{S∪N} .By Lemma 29.9.6 again, this formula is true for all A a closed subset of Rn \S. Using
the same two lemmas again, we see this coarea formula holds for all A a measurable subsetof Rn \S.
It remains to verify the formula for all measurable sets A, regardless of whether theyintersect S. Recall
S≡
{x : ∑
idet(
Dfi (x))2
= 0
}= {x : detU (x)≡ J∗ (x) = 0} .
Consider the case where A ⊆ S. Let A be compact so that by Lemma 29.9.5, y →H n−m
(A∩ f−1 (y)
)is Borel measurable. For ε > 0, define kε , p : Rn×Rm → Rm by
kε (x,z)≡ f(x)+ εz, p(x,z)≡ z.