29.4. ESTIMATES AND A LIMIT 1017

where D is a diagonal matrix having the nonnegative eigenvalues of Dh(x)∗ Dh(x) downthe main diagonal, Q an orthogonal matrix. A is where h is differentiable. However, it isA+ when referring to k. Then Lemma 29.4.2 implies

limr→0

H n (k(B(x,r)∩A))mn (B(x,r))

= J∗k(x)

This is true for each choice of ε > 0. Pick such an ε small enough that J∗k(x)< J∗h(x)+δ .Let

T ≡{(h(w) ,0)T : w ∈ B(x,r)∩A

},

Tε ≡{(h(w) ,εw)T : w ∈ B(x,r)∩A

}≡ k(B(x,r)∩A) ,

then T =(

PTε 0)T where P is the projection map defined by P

(xy

)≡ x. Since P

decreases distances, it follows from Lemma 29.1.1

H n (h(B(x,r)∩A)) = H n (PTε)≤H n (Tε) = H n (k(B(x,r))∩A) .

Thus for a.e. x ∈ A+,

J∗h(x)+δ ≥ J∗k(x) = limr→0

H n (k(B(x,r)∩A))mn (B(x,r))

≥ limsupr→0

H n (h(B(x,r)∩A))mn (B(x,r))

≥ lim infr→0

H n (h(B(x,r)∩A))mn (B(x,r))

≥ limr→0

H n (h(B(x,r)∩A+))

mn (B(x,r))= J∗h(x) (29.4.21)

Thus, since δ is arbitrary, limr→0H n(h(B(x,r)∩A))

mn(B(x,r)) = det(Dh(x)∗Dh(x)

)1/2 when x ∈ A+.If x /∈ A+, the above 29.4.21 shows that

J∗h(x) = 0≥ limsupr→0

H n (h(B(x,r)∩A))mn (B(x,r))

≥ 0

and so this has shown that for a.e.x ∈ A,

limr→0

H n (h(B(x,r)∩A))mn (B(x,r))

= J∗ (x)

Another good idea is in the following lemma.

Lemma 29.4.4 Let k be as defined above. Let A be the set of points where Dh exists soA = A+ relative to k. Then if F is Lebesgue measurable, h(F ∩A) is H n measurable. AlsoH n (h(N∩A)) = 0 if mn (N) = 0.

Proof: By Lemma 29.4.1, there are disjoint Borel sets Ek such that k is Lipschitz oneach Ek and ∪kEk = A = A+ where A+ refers to k. Thus

Pk(Ek ∩F ∩A) = h(Ek ∩F ∩A)

is Hn measurable by Lemma 29.1.2. Hence h(F ∩A) = ∪kh(Ek ∩F ∩A) is H n measur-able. The last claim follows from Lemma 29.1.2.

H n (h(Ek ∩N∩A))≤H n (k(Ek ∩N∩A)) = 0

and so the result follows.