33.6. CHANGE OF VARIABLES FOR NONLINEAR MAPS 663

Theorem 33.5.4 Let L be a linear transformation which is invertible. Then for anyBorel F, L(F) is Borel and

mp (L(F)) = |det(L)|mp (F)

Proof: From linear algebra, there are Li each elementary such that L = L1 ◦L2 ◦· · ·◦Ls.By Corollary 33.5.2, each Li maps Borel sets to Borel sets. Hence, using Lemma 33.5.3

mp (L(F)) = |det(L1)|mp (L2 ◦ · · · ◦Ls (F))

= |det(L1)| |det(L2)|mp (L3 ◦ · · · ◦Ls (F))

= · · ·=s

∏i=1

|det(Li)|mp (F) = |det(L)|mp (F)

the last claim from properties of the determinant.

33.6 Change of Variables for Nonlinear MapsAssume the following:

1. V = h(U) ,U,V open and bounded, h one to one.

2. h,h−1 are C1(Û),C1(V̂)

respectively where Û ⊇U ,V̂ ⊇V .

Let the balls be defined in terms of the norm

∥x∥ ≡ max{|xk| : k = 1, · · · , p}

Note that |x| ≥ ∥x∥ ≥ 1√p |x| so it doesn’t matter which norm you use in the definition of

differentiability. ∥·∥ happens to be a little more convenient here.Then define

φ (x,v)≡ ∥h(x+v)− (h(x)+Dh(x)v)∥∥v∥

(33.6)

Then φ is continuous on U × B(0,1) with the convention that φ (x,0) ≡ 0. Thus it isuniformly continuous on this compact set and so there exists δ > 0 such that if ∥v∥ < δ ,then

|φ (x,v)−φ (x,0)|= |φ (x,v)|< ε, (33.7)

this for all x ∈U .

h(x+v)−h(x) = Dh(x)v+o(v)

= Dh(x)(v+Dh−1 (h(x))o(v)

)Let f : V → R be a bounded, uniformly continuous function.

Let Bm be a collection of disjoint half open rectangles as in Lemma 32.1.6 such thateach has diameter no more than 2−m and each rectangle of Bm+1 is either a subset of arectangle of Bm or is equal to a rectangle of Bm such that ∪Bm = U . Let m be largeenough that the diameters of all these half open rectangles are less than δ . Denote therectangles of Bm as {Rm

i }∞

i=1 and let the center of these be denoted by x(mi). Also let mbe large enough that

| f (h(x(mi))) |det(Dh(x(mi)))|− f (h(x)) |det(Dh(x))||< ε for all x ∈ Rmi

A basic version of the theorems to be presented is the following.