466 CHAPTER 17. THE AREA FORMULA

Since this holds for any bounded E, this implies∫Rm H n−m

(f−1 (y1)∩ (A\A+)

)= 0.

Thus the desired formula holds in this case also.Borel measurability of A can be replaced with Lebesgue measurability. Let A be

Lebesgue measurable and let F ⊆ A ⊆ G where mn (G\F) = 0 and F is Fσ and G isGδ . From the above,

∫Rm H n−m

(G∩f−1 (y)

)dy =

∫Rm H n−m

(F ∩f−1 (y)

)dy and so

from the above arguments, H n−m(F ∩f−1 (y)

)= H n−m

(G∩f−1 (y)

)a.e. Thus y→

H n−m(A∩f−1 (y)

)is measurable by completeness of the measure and

∫Rm

H n−m (A∩f−1 (y))

dy

∈[∫Rm

H n−m (F ∩f−1 (y))

dy,∫Rm

H n−m (G∩f−1 (y))

dy]

=

[∫F

J∗ (x)dx,∫

GJ∗ (x)dx

]=

[∫A

J∗ (x)dx,∫

AJ∗ (x)dx

]

We don’t need to restrict A to be contained in NC where N is the set of Lebesguemeasure 0 where Df does not exist. Consider NC ∩B(0,k) .(

NC ∩B(0,k)∩f−1 (y))∪(N∩B(0,k)∩f−1 (y)

)= B(0,k)∩f−1 (y)

and the ends are H n−m measurable so it follows that so is N ∩B(0,k)∩ f−1 (y). Alsoy→H n−m

(N∩B(0,k)∩f−1 (y)

)is measurable by similar reasoning and

∫Rm

H n−m (N∩B(0,k)∩f−1 (y))

dy

=∫Rm

H n−m (B(0,k)∩f−1 (y))

dy−∫Rm

H n−m (NC ∩B(0,k)∩f−1 (y))

dy = 0

So, passing to a limit and the monotone convergence theorem, we get∫Rm

H n−m (N∩f−1 (y))

dy = 0.

Therefore, the set of points where f fails to be differentiable is irrelevant and can be ig-nored. ■

Also note that by definition,∫Rm

H n−m (A∩f−1 (y))

dy =∫f(A)

H n−m (A∩f−1 (y))

dy.

Recall that H 0 (E) equals the number of elements in E. Thus, if n = m, the coareaformula implies∫

AJ∗f (x)dx =

∫f(A)

H 0 (A∩f−1 (y))

dy =∫f(A)

#(y)dy≥∫f(A)

1dy

Thus, this gives a version of Sard’s theorem by letting the singular set S be A in the above.