364 CHAPTER 13. LEBESGUE MEASURE

where Φ is a continuous function of φ⃗ and θ .1 Then if f is nonnegative and Lebesguemeasurable,∫

Rpf (y)dmp =

∫hp(A)

f (y)dmp =∫

Af(

hp

(⃗φ ,θ ,ρ

))ρ

p−1Φ

(⃗φ ,θ

)dmp (13.9.21)

Furthermore whenever f is Borel measurable and nonnegative, one can apply Fubini’stheorem and write∫

Rpf (y)dy =

∫∞

0ρ

p−1∫

Af(

h(⃗

φ ,θ ,ρ))

Φ

(⃗φ ,θ

)dφ⃗dθdρ (13.9.22)

where here dφ⃗ dθ denotes dmp−1 on A. The same formulas hold if f ∈ L1 (Rp) .

Proof: Formula 13.9.20 is obvious from the definition of the spherical coordinatesbecause in the matrix of the derivative, there will be a ρ in p−1 columns. The first claimis also clear from the definition and math induction or from the geometry of the abovedescription. It remains to verify 13.9.21 and 13.9.22. It is clear hp maps Ā× [0,∞) ontoRp. Since hp is differentiable, it maps sets of measure zero to sets of measure zero. Then

Rp = hp (N∪A× (0,∞)) = hp (N)∪hp (A× (0,∞)) ,

the union of a set of measure zero with hp (A× (0,∞)) . Therefore, from the change ofvariables formula,∫

Rpf (y)dmp =

∫hp(A×(0,∞))

f (y)dmp

=∫

A×(0,∞)f(

hp

(⃗φ ,θ ,ρ

))ρ

p−1Φ

(⃗φ ,θ

)dmp

which proves 13.9.21. This formula continues to hold if f is in L1 (Rp). Finally, if f ≥ 0or in L1 (Rn) and is Borel measurable, then it is F p measurable as well. Recall that F p

includes the smallest σ algebra which contains products of open intervals. Hence F p

includes the Borel sets B (Rp). Thus from the definition of mp∫A×(0,∞)

f(

hp

(⃗φ ,θ ,ρ

))ρ

p−1Φ

(⃗φ ,θ

)dmp

=∫(0,∞)

∫A

f(

hp

(⃗φ ,θ ,ρ

))ρ

p−1Φ

(⃗φ ,θ

)dmp−1dm

=∫(0,∞)

ρp−1

∫A

f(

hp

(⃗φ ,θ ,ρ

))Φ

(⃗φ ,θ

)dmp−1dm

Now the claim about f ∈ L1 follows routinely from considering the positive and negativeparts of the real and imaginary parts of f in the usual way.

1Actually it is only a function of the first but this is not important in what follows.