336 CHAPTER 11. REGULAR MEASURES

Therefore, for Qr ≡mp(h(B(x,r)∩H))

mp(h(B(x,r)))≥ 1|detDh(x)|mp(B(x,r))(1+ε)p

∫B(x,r) gdmp so

1mp (B(0,r))(1+ ε)p

∫B(x,r)

g|detDh(x)|

dmp ≤ Qr

≤ 1mp (B(0,r))(1− ε)p

∫B(x,r)

g|detDh(x)|

dmp

and so for Lebesgue points of g, a.e. x with |detDh(x)| ̸= 0,

1(1+ ε)p ≤

g(x)|detDh(x)|

≤ 1(1− ε)p

Then for such x, 1(1+ε)p

g|detDh(x)| ≤ liminfr→0 Qr ≤ limsupr→0 Qr,≤ 1

(1−ε)pg

|detDh(x)| so,

since ε is arbitrary, limr→0 Qr =g(x)

|detDh(x)| . ■

Lemma 11.9.7 For a.e. x ∈ H,g(x) = |detDh(x)|.

Proof: First considerx such that |det(Dh(x))| ̸= 0. Then by Lemmas 11.9.5 and 11.9.6

limr→0

mp (h(B(x,r)∩H))

mp (B(x,r))= lim

r→0

mp (h(B(x,r)∩H))

mp (h(B(x,r)))mp (h(B(x,r)))

mp (B(x,r))

=g(x)

|detDh(x)||detDh(x)|= g(x)

for a.e. x where |det(Dh(x))| ̸= 0.If |detDh(x)|= 0 then for r small enough,

1mp (B(x,r))

∫B(x,r)

gdmp =mp (h(B(x,r)∩H))

mp (B(x,r))

≤mp (h(x)+Dh(x)B(0,r)+B(0,εr))

mp (B(x,r))=

mp (Dh(x)B(0,r)+B(0,εr))mp (B(x,r))

Now Dh(x)B(0,r) + B(0,εr) has finite diameter and lies in a p− 1 dimensional sub-set. Therefore, from Theorem 11.7.4 on linear mappings, there is an orthogonal matrix Qpreserving all distances such that

|detQ|mp (Dh(x)B(0,r)+B(0,εr)) = mp (QDh(x)B(0,r)+B(0,εr))

where QDh(x)B(0,r) lies in a ball in Rp−1 of some radius r̂ = ∥Dh(x)∥r,. Thus the seton the right side is contained in a cylinder of radius r̂+ εr and height 2rε so its measure isno more than α p−1 (r̂+ rε)p−1 2εr for α p−1 = mp−1 (B(0,1)) . Thus,

1mp (B(x,r))

∫B(x,r)

gdmp ≤(∥Dh(x)∥+1)p

α p−1 (r+ rε)p−1 2εrα prp

= 2(∥Dh(x)∥+1)p α p−1

α p(1+ ε)p−1

ε

Since ε is arbitrary, for every Lebesgue point where |detDh(x)| = 0, it follows g = 0 =|detDh(x)| . ■

Here is the change of variables formula which follows from Lemma 11.9.4 now that ghas been identified.