Kenneth Kuttler

11.9. CHANGE OF VARIABLES, NONLINEAR MAPS 333

Regard Dh(x) as an n× n matrix, the matrix of the linear transformation Dh(x) withrespect to the usual coordinates. Since x∈ Z, it follows that there exists an invertible matrixM such that MDh(x) is in row reduced echelon form with a row of zeros on the bottom.Therefore, using Theorem 11.7.4 about taking out the determinant of a transformation,

mp (h(B(x,r))) =∣∣det

(M−1)∣∣mp (M (h(B(x,r))))

≤∣∣det

(M−1)∣∣mp (M (Dh(x))(B(0,r))+MB(0,rη))

≤∣∣det

(M−1)∣∣α p−1 ∥M (Dh(x))∥p−1 (2r+2ηr)p−1 ∥M∥2rη

≤ C(∥M∥ ,

∣∣det(M−1)∣∣ ,∥Dh(x)∥)4p−1rp2η

Here αn is the volume of the unit ball in Rn. This is because M (Dh(x))(B(0,r)) +MB(0,rη) in the third line up is contained in a cylinder, the base in Rp−1 which hasradius ∥M (Dh(x))∥(2r+2ηr) and height ∥M∥2rη . Thus its measure is no more than∫Rp−1

∫ ∥Mrη∥−∥Mrη∥ dxpdmp−1.Then letting δx be still smaller if necessary, corresponding to suf-

ficiently small η ,mp (h(B(x,r)))≤ εmp (B(x,r)) .

The balls of this form constitute a Vitali cover of Z. Hence, by the covering theorem Corol-lary 9.12.5, there exists {Bi}∞

i=1 ,Bi =Bi (xi,ri) , a collection of disjoint balls, each of whichis contained in V, such that mp (h(Bi)) ≤ εmp (Bi) and mp (Z \∪iBi) = 0. Hence fromLemma 11.8.1,

mp (h(Z)\∪ih(Bi))≤ mp (h(Z \∪iBi)) = 0

Therefore,

mp (h(Z))≤∑i

mp (h(Bi))≤ ε ∑i

mp (Bi)≤ ε (mp (V ))≤ ε (mp (Z)+ ε) .

Since ε is arbitrary, this shows mp (h(Z)) = 0. What if A is not bounded? Then considerZn = Z∩B(0,n)⊆ A∩B(0,n) . From what was just shown, h(Zn) has measure 0 and so itfollows that h(Z) also does, being the countable union of sets of measure zero. ■

11.9 Change of Variables, Nonlinear MapsThis preparation leads to a good change of variables formula. First is a lemma which islikely familiar by now.

Lemma 11.9.1 Let h : Ω→ Rp where (Ω,F ) is a measurable space and suppose h iscontinuous. Then h−1 (B) ∈F whenenver B is a Borel set.

Proof: Measurability applied to components of h shows that h−1 (U) ∈F wheneverU is an open set. If G is consists of the subsets G of Rp for which h−1 (G) ∈F , then G isa σ algebra and G contains the open sets. ■

Definition 11.9.2 Let h : U → h(U) be continuous, U open, and let H ⊆ U bemeasurable and h is one to one and differentiable on H. Define λ (F)≡ mp (h(F ∩H)) .

Lemma 11.9.3 λ is a well defined measure on measurable subsets of U and λ ≪ mp.

11.9. CHANGE OF VARIABLES, NONLINEAR MAPS 333Regard Dh (a) as an n x n matrix, the matrix of the linear transformation Dh (a) withrespect to the usual coordinates. Since x € Z, it follows that there exists an invertible matrixM such that MDh (2) is in row reduced echelon form with a row of zeros on the bottom.Therefore, using Theorem 11.7.4 about taking out the determinant of a transformation,|det (M~') | my (M (h (B(a,r))))|det (M~') | mp (M (Dh (a)) (B(O,r)) + MB (0,rn))Mp (h (B(a,r)))lAIAJdet (M~')| 1M (Dh (@)) |?! (2r + 2mr)? | |[M||2rnC (|||), [det (M-!)|, |b (a)|)) 4”-1r?2nlAHere a, is the volume of the unit ball in R”. This is because M (Dh (a)) (B(0,r)) +MB(0,rn) in the third line up is contained in a cylinder, the base in R?~! which hasradius ||M(Dh(a))||(2r+2nr) and height ||M||2rn. Thus its measure is no more thanJip Tut dxpdmp—,.Then letting 6, be still smaller if necessary, corresponding to suf-ficiently small 7 ,mp (hu(B (ayr))) < emp (B(a,r)).The balls of this form constitute a Vitali cover of Z. Hence, by the covering theorem Corol-lary 9.12.5, there exists {B;};°, ,B; = B; (a;,1r;) , a collection of disjoint balls, each of whichis contained in V, such that m, (h(B;)) < Em, (B;) and m, (Z\U;B;) = 0. Hence fromLemma 11.8.1,My (h(Z) \ Ujh (Bi)) < mp (h(Z\U;Bi)) = 0Therefore,mp (R(Z)) < Yip (h(Bi)) < € Y mp (Bi) < € (mp (V)) < € (mp (Z) +8).iSince € is arbitrary, this shows m, (h(Z)) = 0. What if A is not bounded? Then considerZn = ZB (0,n) CAMB(O0,n). From what was just shown, h (Z,,) has measure 0 and so itfollows that h (Z) also does, being the countable union of sets of measure zero.11.9 Change of Variables, Nonlinear MapsThis preparation leads to a good change of variables formula. First is a lemma which islikely familiar by now.Lemma 11.9.1 Leth: Q— R? where (Q,F) is a measurable space and suppose h iscontinuous. Then h' (B) € ¥ whenenver B is a Borel set.Proof: Measurability applied to components of h shows that h! (U) € ¥ wheneverU is an open set. If Y is consists of the subsets G of R? for which h~' (G) € F, then F isa o algebra and Y contains the open sets. HlDefinition 11.9.2 Leth: u > h(U) be continuous, U open, and let H CU bemeasurable and h is one to one and differentiable on H. Define 1 (F) =m, (h(FNA#)).Lemma 11.9.3 A is a well defined measure on measurable subsets of U and A < Mp.