1026 CHAPTER 29. THE AREA FORMULA

Lemma 29.7.8 Suppose U is a bounded open set as described above. Then there exists aunique function in L∞

(∂U,H p−1

)p, n(y) for y ∈ ∂U such that |n(y)| = 1,n is H p−1

measurable, (meaning each component of n is H p−1 measurable) and for every w ∈ Rp

satisfying |w|= 1, and for every f ∈C1c (Rp) ,

limt→0

∫U

f (x+ tw)− f (x)t

dx =∫

∂Uf (n ·w)dH p−1

Proof: Let U ⊆V ⊆V ⊆ ∪Ni=0Qi and let {ψ i}

Ni=0 be a C∞ partition of unity on V such

that spt(ψ i)⊆ Qi. Then for all t small enough and x ∈U ,

f (x+ tw)− f (x)t

=1t

N

∑i=0

ψ i f (x+ tw)−ψ i f (x) .

Thus using the dominated convergence theorem and Rademacher’s theorem,

limt→0

∫U

f (x+ tw)− f (x)t

dx

= limt→0

∫U

(1t

N

∑i=0

ψ i f (x+ tw)−ψ i f (x)

)dx

=∫

U

N

∑i=0

p

∑j=1

D j (ψ i f )(x)w jdx

=∫

U

p

∑j=1

D j (ψ0 f )(x)w jdx+N

∑i=1

∫U

p

∑j=1

D j (ψ i f )(x)w jdx (29.7.31)

Since spt(ψ0)⊆ Q0, it follows the first term in the above equals zero. In the second term,fix i. Without loss of generality, suppose the k in the above definition equals p and 29.7.29holds. This just makes things a little easier to write. Thus gi is a function of

(x1, · · · ,xp−1) ∈p−1

∏j=1

(ai

j,bij)≡ Bi

Then ∫U

p

∑j=1

D j (ψ i f )(x)w jdx

=∫

Bi

∫ gi(x1,··· ,xp−1)

aip

p

∑j=1

D j (ψ i f )(x)w jdxpdx1 · · ·dxp−1

=∫

Bi

∫ gi(x1,··· ,xp−1)

−∞

p

∑j=1

D j (ψ i f )(x)w jdxpdx1 · · ·dxp−1