29.10. CHANGE OF VARIABLES 1045
= Cnm
∫B(0,1)
∫Rm
H n−m (f−1 (y)∩A)
dydw
= Cnmαn
∫Rm
H n−m (f−1 (y)∩A)
dy
Since ε is arbitrary in 29.9.60, this shows that∫Rm
H n−m (f−1 (y)∩A)
dy = 0 =∫
AJ∗ (x)dx.
Since this holds for arbitrary compact sets in S, it follows from Lemma 29.9.6 and innerregularity of Lebesgue measure that the equation holds for all measurable subsets of S. Thiscompletes the proof of the coarea formula.
There is a simple corollary to this theorem in the case of locally Lipschitz maps.
Corollary 29.9.9 Let f : Rn→ Rm where m≤ n and f is locally Lipschitz. This means thatfor each r > 0, f is Lipschitz on B(0,r) . Then the coarea formula, 29.9.48, holds for f.
Proof: Let A ⊆ B(0,r) and let fr be Lipschitz with f(x) = fr (x) for x ∈ B(0,r+1) .Then ∫
AJ∗f(x)dx =
∫A
J(Dfr (x))dx =∫Rm
H n−m (A∩ f−1r (y)
)dy
=∫
fr(A)H n−m (A∩ f−1
r (y))
dy =∫
f(A)H n−m (A∩ f−1 (y)
)dy
=∫Rm
H n−m (A∩ f−1 (y))
dy
Now for arbitrary measurable A the above shows for k = 1,2, · · ·∫A∩B(0,k)
J∗f(x)dx =∫Rm
H n−m (A∩B(0,k)∩ f−1 (y))
dy.
Use the monotone convergence theorem to obtain 29.9.48.From the definition of Hausdorff measure, it is easy to verify that H 0 (E) equals the
number of elements in E. Thus, if n = m, the Coarea formula implies∫A
J∗f(x)dx =∫
f(A)H 0 (A∩ f−1 (y)
)dy =
∫f(A)
#(y)dy
Note also that this gives a version of Sard’s theorem by letting S = A.
29.10 Change of VariablesWe say that the coarea formula holds for f : Rn→ Rm,n≥ m if whenever A is a Lebesguemeasurable subset of Rn, 29.9.48 holds. Note this is the same as∫
AJ∗ (x)dx =
∫f(A)
H n−m (A∩ f−1 (y))
dy, J∗ (x)≡ det(Df(x)Df(x)∗
)1/2