1250 CHAPTER 35. WEAK DERIVATIVES
on the Lebesgue measurable subsets of Ω. By Lemma 35.6.8 µ≪mn and so by the RadonNikodym theorem, there exists a nonnegative function, J (x) in L1
loc (Rn) such that wheneverE is Lebesgue measurable,
µ (E) = mn (h(E ∩Ω)) =∫
E∩Ω
J (x)dmn. (35.6.22)
Extend J to equal zero off Ω.
Lemma 35.6.11 The function, J (x) equals |detDh(x)| a.e.
Proof: Define
Q≡ {x ∈Ω : x is not a point of density of Ω}∪N∪
{x ∈Ω : x is not a Lebesgue point of J}.
Then Q is a set of measure zero and if x /∈ Q, then by 35.6.17, and 35.6.16,
|detDh(x)|
= limr→0
mn (h(B(x,r)))mn (B(x,r))
= limr→0
mn (h(B(x,r)))mn (h(B(x,r)∩Ω))
mn (h(B(x,r)∩Ω))
mn (B(x,r))
= limr→0
1mn (B(x,r))
∫B(x,r)∩Ω
J (y)dmn
= limr→0
1mn (B(x,r))
∫B(x,r)
J (y)dmn = J (x) .
the last equality because J was extended to be zero off Ω. This proves the lemma.Here is the change of variables formula for Lipschitz mappings. It is a special case of
the area formula.
Theorem 35.6.12 Let Ω be a Lebesgue measurable set, let f ≥ 0 be Lebesgue measurable.Then for h a Lipschitz mapping defined on Rn which is one to one on Ω,∫
h(Ω)f (y)dmn =
∫Ω
f (h(x)) |detDh(x)|dmn. (35.6.23)
Proof: Let F be a Borel set. It follows that h−1 (F) is a Lebesgue measurable set.Therefore, by 35.6.22,
mn(h(h−1 (F)∩Ω
))(35.6.24)
=∫
h(Ω)XF (y)dmn =
∫Ω
Xh−1(F) (x)J (x)dmn
=∫
Ω
XF (h(x))J (x)dmn.