244 CHAPTER 10. CHANGE OF VARIABLES

Now suppose U ⊆ Rp is open, h : U → Rp is continuous, and

mp (h(U \H)) = 0

where H ⊆U and H is Borel measurable. Suppose also that h is one to one and differen-tiable on H. Define for Lebesgue measurable E ⊆U

λ (E)≡ mp (h(E ∩H)) ,

Then it is clear that λ is indeed a measure on the σ algebra of Lebesgue measurable subsetsof U . Note that

mp (h(E))−mp (h(E ∩H))

≤ mp (h(E ∩H))+mp (h(E \H))−mp (h(E ∩H))

≤ mp (h(U \H)) = 0

Thus one could just as well let λ (E)≡ mp (h(E)).Since h is one to one on H, this along with Proposition 10.1.2 implies that λ ≪ mp

since if mp (E) = 0, then h(E ∩H) also has measure zero. Also λ and mp are finite onclosed balls so both are σ finite.

Therefore, for measurable E ⊆U, it follows from the Radon Nikodym theorem Corol-lary 7.11.12 that there is a real valued, nonnegative, measurable function f in L1 (K) forany compact set K such that

λ (E) = mp (h(E ∩H)) =∫

UXE f (x)dmp =

∫XE f (x)dmp (10.2)

So what is f (x)? To begin with, assume Dh(x)−1 exists. By differentiability, and usingDh(x)−1 exists as needed,

h(B(x,r))−h(x) ⊆ Dh(x)B(0,r)+Dh(x)B(0,εr)

⊆ Dh(x)(B(0,r (1+ ε)))

for all r small enough. Therefore, by translation invariance of Lebesgue measure,

mp (h(B(x,r))) ≤ mp (Dh(x)(B(0,r (1+ ε))))

= |det(Dh(x))|mp (B(0,r (1+ ε)))

Also, for |v|< r small enough,

h(x+v)−h(x) = Dh(x)v+o(v)Dh(x)−1 (h(x+v)−h(x)) = v+o(v)

and so if g(v) ≡ Dh(x)−1 (h(x+v)−h(x)) , |g(v)−v| < εr provided r is small enough.Therefore, from Lemma 10.2.1, for small enough r,

Dh(x)−1 (h(x+B(0,r))−h(x))⊇ B(0,(1− ε)r)

Thush(B(x,r))⊇ h(x)+Dh(x)B(0,(1− ε)r)