35.6. CHANGE OF VARIABLES FORMULA LIPSCHITZ MAPS 1249

which impliesmn (B(x,r)\Ω)< εα (n)rn. (35.6.21)

Then for such r,

1≥ mn (h(B(x,r)∩Ω))

mn (h(B(x,r)))

≥ mn (h(B(x,r)))−mn (h(B(x,r)\Ω))

mn (h(B(x,r))).

From Lemma 35.6.8, 35.6.21, and 35.6.13, this is no larger than

1− Lip(h)nεα (n)rn

mn (Dh(x)B(0,r (1− ε))).

By the theorem on the change of variables for a linear map, this expression equals

1− Lip(h)nεα (n)rn

|det(Dh(x))|rnα (n)(1− ε)n ≡ 1−g(ε)

where limε→0g(ε) = 0. Then for all r small enough,

1≥ mn (h(B(x,r)∩Ω))

mn (h(B(x,r)))≥ 1−g(ε)

which shows 35.6.16 since ε is arbitrary. It remains to verify 35.6.17.In case x ∈ S, for small |v| ,

h(x+v) = h(x)+Dh(x)v+o(|v|)

where |o(|v|)|< ε |v| . Therefore, for small enough r,

h(B(x,r))−h(x)⊆ K +B(0,rε)

where K is a compact subset of an n− 1 dimensional subspace contained in Dh(x)(Rn)which has diameter no more than 2 ||Dh(x)||r. By Lemma 35.6.1 on Page 1244,

mn (h(B(x,r))) = mn (h(B(x,r))−h(x))≤ 2n

εr (2 ||Dh(x)||r+ rε)n−1

and so, in this case, letting r be small enough,

mn (h(B(x,r)))mn (B(x,r))

≤ 2nεr (2 ||Dh(x)||r+ rε)n−1

α (n)rn ≤Cε.

Since ε is arbitrary, the limit as r→ 0 of this quotient equals 0.If x /∈ S, use 35.6.13 - 35.6.15 along with the change of variables formula for linear

maps. This proves the Lemma.Since h is one to one, there exists a measure, µ, defined by

µ (E)≡ mn (h(E))