Kenneth Kuttler

29.12. THE CASE OF W 1,p 1055

Since a.e. point is a Lebesgue point, it follows that in the case where d (y,Ω,h) =−1,

−det(Dh(x)) = |det(Dh(x))| a.e. x ∈Ω

The case where the degree equals 1 is similar. Thus det(Dh(x)) has the same sign onh(Ω).

Now let O be an open set. Then by invariance of domain, h(O) is also an open set. LetVk denote a decreasing sequence of open sets, Vk ⊇Vk+1 whose intersection is the compactset h(∂Ω) such that m

(Vk)< 1/k. Then if f ≺ h(O)\Vk, it follows since h(O)\Vk is an

open set which is at a positive distance from h(∂Ω) , Lemma 29.11.2 implies∫h(Ω)

f (y)dy =∫

Ω

|det(Dh(x))| f (h(x))dx

Taking a sequence of such f increasing to Xh(O)\Vk, it follows from monotone convergence

theorem in the above that∫h(Ω)

Xh(O)\Vk(y)dy =

∫Ω

|det(Dh(x))|Xh(O)\Vk(h(x))dx

=∫

Ω

|det(Dh(x))|XO\h−1(Vk) (x)dx

Now letting k→ ∞, it follows from the monotone convergence theorem that∫h(Ω)

Xh(O)\h(∂Ω) (y)dy =∫

Ω

|det(Dh(x))|XO\∂Ω (x)dx

Since both ∂Ω and h(∂Ω) have measure zero, this implies∫h(Ω)

Xh(O) (y)dy =∫

Ω

|det(Dh(x))|XO (x)dx

Now let G denote the Borel sets E with the property that∫h(Ω)

Xh(E) (y)dy =∫

Ω

|det(Dh(x))|XE (x)dx

It follows easily that if E ∈ G then so does EC. This is because h(Ω) has finite measureand |det(Dh(x))| is in L1 (Ω). If Ei is a sequence of disjoint sets of G then the monotoneconvergence theorem implies ∪Ei is also in G . It was shown above that the π system ofopen sets is contained in G . Therefore, it follows from the lemma on π systems, Lemma12.12.3 on Page 329, G equals the Borel sets. Now the desired result follows from ap-proximating f ≥ 0 and Borel measurable with a sequence of Borel simple functions whichconverge pointwise to f . This proves the theorem.

The following corollary follows right away by splitting f into positive and negativeparts of real and imaginary parts.

29.12. THE CASE OFW!? 1055Since a.e. point is a Lebesgue point, it follows that in the case where d(y,Q,h) = —1,— det (Dh (x)) = |det (Dh (x))| ae. x EQThe case where the degree equals | is similar. Thus det(Dh(x)) has the same sign onh(Q).Now let O be an open set. Then by invariance of domain, h (QO) is also an open set. LetV, denote a decreasing sequence of open sets, Vk > Vi, whose intersection is the compactset h (AQ) such that m (Vi) < 1/k. Then if f < h(O) \ Vj, it follows since h(O) \ Vi is anopen set which is at a positive distance from h (QQ) , Lemma 29.11.2 implies[fare ff eth 9)1 Fh (9) aTaking a sequence of such f increasing to Bo)\Ve> it follows from monotone convergencetheorem in the above thatwo, (y) dI, |det (Dh (x))| 2p(oy yy (h(x) dx| i ldet(Dh(x))| Zon (yp) (*) axNow letting k — 9, it follows from the monotone convergence theorem that[ 2n(0)\n(9Q) (Y) dy = [ |det (Dh (x))| 2o\aa (x) dxJh(Q) JQSince both dQ and h(dQ) have measure zero, this implies|. Zino (wy = ff [dee (Dh (x))| % (w) axh(Q) QNow let Y denote the Borel sets E with the property thatJ, Zine) Way = ff [dee (Dh (x))| Mie (x) axh(Q) QIt follows easily that if E <¢ Y then so does E©. This is because h(Q) has finite measureand |det (Dh(x))| is in L! (Q). If E; is a sequence of disjoint sets of Y then the monotoneconvergence theorem implies UE; is also in Y. It was shown above that the 7 system ofopen sets is contained in Y. Therefore, it follows from the lemma on 7 systems, Lemma12.12.3 on Page 329, Y equals the Borel sets. Now the desired result follows from ap-proximating f > 0 and Borel measurable with a sequence of Borel simple functions whichconverge pointwise to f. This proves the theorem.The following corollary follows right away by splitting f into positive and negativeparts of real and imaginary parts.