A.6. THE CHANGE OF VARIABLES FORMULA 861

≤∫Rn

∫Rm

φ (x,y) dVy dVx =∫Rn+m

φ dV ≤UG ( f ) .

Therefore, ∣∣∣∣∫Rn

∫Rm

f (x,y) dVy dVx−∫Rn+m

f (z) dV∣∣∣∣≤ ε

and since ε > 0 is arbitrary, this proves Fubini’s theorem2. ■

Corollary A.5.5 Suppose E is a bounded contented set in Rn and let φ ,ψ be continuousfunctions defined on E such that φ (x) ≥ ψ (x). Also suppose f is a continuous boundedfunction defined on the set

P≡ {(x,y) : ψ (x)≤ y≤ φ (x)} ,

It follows f XP ∈R(Rn+1

)and∫P

f dV =∫

E

∫φ(x)

ψ(x)f (x,y) dydVx.

Proof: Since f is continuous, there is no problem in writing f (x, ·)X[ψ(x),φ(x)] (·) ∈R(R1). Also, f XP ∈R

(Rn+1

)because P is contented thanks to Corollary A.4.9. There-

fore, by Fubini’s theorem ∫P

f dV =∫Rn

∫R

f XP dydVx

=∫

E

∫φ(x)

ψ(x)f (x,y) dydVx

proving the corollary. ■Other versions of this corollary are immediate and should be obvious whenever en-

countered.

A.6 The Change Of Variables FormulaFirst recall Theorem 26.3.2 on Page 492 which is listed here for convenience.

Theorem A.6.1 Let h : U → Rn be a C1 function with h(0) = 0,Dh(0)−1 exists. Thenthere exists an open set V ⊆U containing 0 flips, F 1, · · · ,F n−1, and primitive functionsGn,Gn−1, · · · ,G1 such that for x ∈V,

h(x) = F 1 ◦ · · · ◦F n−1 ◦Gn ◦Gn−1 ◦ · · · ◦G1 (x) .

Also recall Theorem 14.6.5 on Page 272.

Theorem A.6.2 Let φ : [a,b]→ [c,d] be one to one and suppose φ′ exists and is continuous

on [a,b]. Then if f is a continuous function defined on [a,b] ,∫ d

cf (s) ds =

∫ b

af (φ (t))

∣∣φ ′ (t)∣∣ dt

2Actually, Fubini’s theorem usually refers to a much more profound result in the theory of Lebesgue integra-tion.