A.6. THE CHANGE OF VARIABLES FORMULA 867

·∣∣DF

(Gn ◦ · · · ◦G2

(x′))∣∣ ∣∣DGn

(Gn−1 ◦ · · · ◦G2

(x′))∣∣ ·∣∣DGn−1

(Gn−2 ◦ · · · ◦G2

(x′))∣∣ · · · ∣∣DG2

(x′)∣∣dVn.

Now use Fubini’s theorem again to make the inside integral taken with respect to x2. Notethat the term |DG1 (x)| disappeared. Exactly the same process yields∫

Rn−1

∫R

XQ−p j

(G−1

1 ◦G−12(x′′))(

ψ j f)

(h(pi)+F 1 ◦ · · · ◦F n−1 ◦Gn ◦ · · · ◦G3

(x′′))

·∣∣DF

(Gn ◦ · · · ◦G3

(x′′))∣∣ ∣∣DGn

(Gn−1 ◦ · · · ◦G3

(x′′))∣∣ ·∣∣DGn−1

(Gn−2 ◦ · · · ◦G3

(x′′))∣∣ · · ·dy2dVn−1.

Now F is just a composition of flips, so |DF (Gn ◦ · · · ◦G3 (x′′))|= 1, and so this term can

be replaced with 1. Continuing this process, eventually yields an expression of the form∫Rn

XQ−p j

(G−1

1 ◦ · · · ◦G−1n−2 ◦G

−1n−1 ◦G

−1n ◦F−1 (y)

)(ψ j f

)(h(pi)+y)dVn. (1.31)

Denoting by G−1 the expression, G−11 ◦ · · · ◦G

−1n−2 ◦G

−1n−1 ◦G

−1n ,

XQ−p j

(G−1

1 ◦ · · · ◦G−1n−2 ◦G

−1n−1 ◦G

−1n ◦F−1 (y)

)= 1

exactly when G−1 ◦F−1 (y) ∈ Q−p j. Now recall that

h(p j +x

)−h

(p j)= F ◦G(x)

and so the above holds exactly when

y = h(p j +G−1 ◦F−1 (y)

)−h

(p j)∈ h

(p j +Q−p j

)−h

(p j)

= h(Q)−h(p j).

Thus (1.31) reduces to ∫Rn

Xh(Q)−h(p j)(y)(

ψ j f)(h(pi)+y)dVn

=∫Rn

Xh(Q) (z)(

ψ j f)(z)dVn.

It follows from (1.28),

UG (g)− ε ≤ LG (g)≤∫

XU (x) f (h(x)) |detDh(x)|dVn

≤ ∑Q∈G ′

∫XQ (x) f (h(x)) |detDh(x)|dx

= ∑Q∈G ′

q

∑j=1

∫XQ (x)

(ψ j f

)(h(x)) |detDh(x)|dx

= ∑Q∈G ′

q

∑j=1

∫Rn

Xh(Q) (z)(

ψ j f)(z)dVn

= ∑Q∈G ′

∫Rn

Xh(Q) (z) f (z)dVn =∫

Xh(U) (z) f (z)dVn