17.6. CHANGE OF VARIABLES 467

17.6 Change of VariablesNow let s(x) = ∑

pi=1 ciXEi (x) where Ei is Lebesgue measurable and ci ≥ 0. Then

∫Rn

s(x)J∗ (f)(x)dx =p

∑i=1

ci

∫Ei

J∗ (f)(x)dx =p

∑i=1

ci

∫Rm

H n−m (Ei∩f−1(y))

dy

=∫Rm

p

∑i=1

ciHn−m (Ei∩f−1(y)

)dy =

∫Rm

[∫f−1(y)

s dH n−m]

dy

=∫Rm

[∫f−1(y)

s dH n−m]

dy =∫f(Rn)

[∫f−1(y)

s dH n−m]

dy. (17.26)

Theorem 17.6.1 Let g≥ 0 be Lebesgue measurable and let

f : Rn→ Rm, n≥ m, f being Lipschitz

Then ∫Rn

g(x)J∗ (f)(x)dx =∫f(Rn)

[∫f−1(y)

g(u)dH n−m (u)

]dy. (17.27)

Proof: Let si ↑ g where si is a simple function satisfying 17.26. Then let i→ ∞ and usethe monotone convergence theorem to replace si with g. This proves the change of variablesformula. ■

Note how if m = n this will end up reducing to the conclusion of Theorem 11.10.2.The following is an easy example of the use of the coarea formula to give a familiar

relation.

Example 17.6.2 Let f : Rn → R be given by f (x) ≡ |x| . Then J∗ (x) ends up being 1.Then by the coarea formula,∫

B(0,r)dmn =

∫ r

0H n−1 (B(0,r)∩ f−1 (y)

)dy =

∫ r

0H n−1 (∂B(0,y))dy

Then mn (B(0,r))≡ αnrn =∫ r

0 H n−1 (∂B(0,y))dy. Then differentiate both sides to obtainnαnrn−1 =H n−1 (∂B(0,r)) . In particular H 2 (∂B(0,r)) = 3 4

3 πr2 = 4πr2. Of course αnwas computed earlier. Recall from Theorem 14.4.1 on Page 405

αn = πn/2(Γ(n/2+1))−1

Therefore, the n−1 dimensional Hausdorf measure of the boundary of the ball of radius rin Rn is nπn/2(Γ(n/2+1))−1rn−1.

I think it is clear that you could generalize this to other more complicated situations.The above is nice because J∗ (x) = 1. This won’t be so in general when considering otherlevel surfaces.