1244 CHAPTER 35. WEAK DERIVATIVES

35.6 Change Of Variables Formula Lipschitz MapsWith Rademacher’s theorem, one can give a general change of variables formula involvingLipschitz maps. First here is an elementary estimate.

Lemma 35.6.1 Suppose V is an n− 1 dimensional subspace of Rn and K is a compactsubset of V . Then letting

Kε ≡ ∪x∈KB(x,ε) = K +B(0,ε) ,

it follows thatmn (Kε)≤ 2n

ε (diam(K)+ ε)n−1 .

Proof: Let an orthonormal basis for V be {v1, · · · ,vn−1} and let

{v1, · · · ,vn−1,vn}

be an orthonormal basis for Rn. Now define a linear transformation, Q by Qvi = ei. ThusQQ∗ = Q∗Q = I and Q preserves all distances because∣∣∣∣∣Q∑

iaiei

∣∣∣∣∣2

=

∣∣∣∣∣∑iaivi

∣∣∣∣∣2

= ∑i|ai|2 =

∣∣∣∣∣∑iaiei

∣∣∣∣∣2

.

Letting k0 ∈ K, it follows K ⊆ B(k0,diam(K)) and so,

QK ⊆ Bn−1 (Qk0,diam(QK)) = Bn−1 (Qk0,diam(K))

where Bn−1 refers to the ball taken with respect to the usual norm in Rn−1. Every point ofKε is within ε of some point of K and so it follows that every point of QKε is within ε ofsome point of QK. Therefore,

QKε ⊆ Bn−1 (Qk0,diam(QK)+ ε)× (−ε,ε) ,

To see this, let x ∈ QKε . Then there exists k ∈ QK such that |k−x| < ε . Therefore,|(x1, · · · ,xn−1)− (k1, · · · ,kn−1)|< ε and |xn− kn|< ε and so x is contained in the set on theright in the above inclusion because kn = 0. However, the measure of the set on the right issmaller than

[2(diam(QK)+ ε)]n−1 (2ε) = 2n [(diam(K)+ ε)]n−1ε.

This proves the lemma.Next is the definition of a point of density. This is sort of like an interior point but not

as good.

Definition 35.6.2 Let E be a Lebesgue measurable set. x ∈ E is a point of density if

limr→0

m(E ∩B(x,r))m(B(x,r))

= 1.