Kenneth Kuttler

246 CHAPTER 10. REGULAR MEASURES

Now let F be a Borel set so that h−1 (F)∩H is measurable and plays the role of F inthe above. Then

λ(h−1 (F)

)≡ mp

(h(h−1 (F)∩H

))=∫

UXh−1(F)∩H (x)g(x)dmp (x) =

∫H

XF (h(x))g(x)dmp (x)

Thus also for s a Borel measurable nonnegative simple function,∫h(H)

s(y)dmp (y) =∫

Hs(h(x))(x)g(x)dmp (x)

Using a sequence of nonnegative simple functions to approximate a nonnegative Borelmeasurable f , we obtain from the monotone convergence theorem that∫

h(H)f (y)dmp (y) =

∫H

f (h(x))(x)g(x)dmp (x)

If f is only Lebesgue measurable, then there are nonnegative Borel measurable functionsk, l such that k (y)≤ f (y)≤ l (y) with equality holding off a set of mp measure zero. Thenk (h(x))g(x) ≤ f (h(x))g(x) ≤ l (h(x))g(x) and the two on the ends are Lebesguemeasurable which forces the function in the center to also be Lebesgue measurable bycompleteness of Lebesgue measure because∫

Hl (h(x))g(x)− k (h(x))g(x)dmp =

∫h(H)

l (y)dmp−∫h(H)

k (y)dmp

=∫h(H)

f (y)dmp−∫h(H)

f (y)dmp = 0

Thus l (h(x))g(x)− k (h(x))g(x) = 0 a.e. Then for f nonnegative and Lebesgue mea-surable, ∫

Hf (h(x))g(x)dmp =

∫h(H)

f (y)dmp.

This shows the following lemma.

Lemma 10.5.4 Let h : U→h(U) be continuous, U open, and let H ⊆U be measurableand h is one to one and differentiable on H. Then there exists nonnegative measurableg ∈ L1

loc such that whenever f is nonnegative and Lebesgue measurable,∫h(H)

f (y)dmp =∫

Hf (h(x))g(x)dmp

where all necessary measurability is obtained.

It remains to identify g.

Lemma 10.5.5 For a.e. x, satisfying |detDh(x)|> 0, and r small enough,

Dh(x)B(0,(1− ε)r) ⊆ h(B(x,r))⊆ h(

B(x,r))⊆ Dh(x)B(0,(1+ ε)r),

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

∈ [|detDh(x)|(1− ε)p , |detDh(x)|(1+ ε)p]

limr→0

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

= |detDh(x)|

246 CHAPTER 10. REGULAR MEASURESNow let F be a Borel set so that h~' (F) OH is measurable and plays the role of F inthe above. ThenA(h!(F))=m “ (h! (F) mu=f Zereew (@ x) g(a) dmyp (x =| Arh (a) dmy (x)Thus also for s a Borel measurable nonnegative simple function,[8 amo) = [, 8) @)8(@) Amp)Using a sequence of nonnegative simple functions to approximate a nonnegative Borelmeasurable f, we obtain from the monotone convergence theorem that[yf Damo) = [Fa (@)) (@) (a) Amp)If f is only Lebesgue measurable, then there are nonnegative Borel measurable functionsk,l such that k(y) < f(y) </(y) with equality holding off a set of m, measure zero. Thenk(h(a))g(a) < f(h(x)) g(a) <1 (h(x))g (a) and the two on the ends are Lebesguemeasurable which forces the function in the center to also be Lebesgue measurable bycompleteness of Lebesgue measure because[10 (@)) g(a) Km (@))g(@)dm =f Hy)dimy— J. Cu) drmhon f(y)dmy — haw f(y)dm, =0Thus /(h(ax)) g(a) —k(h(a)) g(a) =O ae. Then for f nonnegative and Lebesgue mea-surable,[fo @aeeamy =f. foamThis shows the following lemma.Lemma 10.5.4 Leth :U — h(U) be continuous, U open, and let H CU be measurableand h, is one to one and differentiable on H. Then there exists nonnegative measurablegE Li, . such that whenever f is nonnegative and Lebesgue measurable,haw fy Jdmp= | f(r(w ax) dmpwhere all necessary measurability is obtained.It remains to identify g.Lemma 10.5.5 For ae. x, satisfying |det Dh (x)| > 0, and r small enough,Dh(«)B(0,(1—)r) © h(B(w,r)) Ch (B(w,7)) C Dh (x) BO,1+8)7),)mp (h (B(a,1)))me Oe r))w,1))a[|\det Dh (a)| (1 — €)? , |detDh (x)| (1 +e)?Lg ea |\det Dh (a)|