240 CHAPTER 10. CHANGE OF VARIABLES

There is an open set V ⊇ Nk such that mp (V ) < ε . For each x ∈ Nk, there is a ball Bxcentered at x with radius 5rx < 1 such that B̂x ⊆ V, where Bx = B(x,rx) , B̂x = B(x,5rx)and for y ∈ B̂x,

h(y) ∈ h(x)+Dh(x)B(0,5rx)+B(0,ε5rx)

⊆ h(x)+B(0,∥Dh(x)∥5rx)+B(0,ε5rx)

≤ B(h(x) ,(k+ ε)5rx)

So h(B(x,5rx))≤ B(h(x) ,(k+ ε)5rx) and so

mp(h(B̂x))≤ (k+ ε)p mp (B(x,5rx)) .

Then, the balls B(x,rx) for x ∈ Nk,cover Nk and so by the Vitali covering theorem, thereare disjoint balls Bi = B(xi,rxi) such that for B̂i the ball with same center and 5 times theradius as Bi,Nk ⊆ ∪kB̂k. Thus

mp (h(Nk)) ⊆ mp(∪k(h(B̂k)))≤∑

kmp(h(B̂k))

≤ ∑k(k+ ε)p mp

(B̂k)= ∑

k(k+ ε)p 5pmp (Bk)

≤ 5p (k+ ε)p mp (V )< ε5p (k+ ε)p

Since ε > 0 is arbitrary, it follows that mp (h(Nk)) = 0 and so h(Nk) is measurable and hasmeasure zero. Now let k→ ∞ to conclude that mp (h(N)) = 0.

Now the other claim is shown as follows. By Proposition 8.3.2, if E is Lebesgue mea-surable, E ⊆ H, there is an Fσ set F ⊆ E such that mp (E \F) = 0. Then h(F) is clearlymeasurable because h is continuous and F is a countable union of compact sets. Thush(E) = h(F)∪h(E \F) and the second was just shown measurable while the first is an Fσ

set so it is actually a Borel set. ■From Linear Algebra,(My Elementary Linear Algebra book has the necessary theorems

carefully proved.) if A is an invertible linear transformation, it is the composition of finitelymany invertible linear transformations which are of the following form.(

x1 · · · xr · · · xs · · · xp)T →

(x1 · · · xr · · · xs · · · xp

)T

(x1 · · · xr · · · xp

)T →(

x1 · · · cxr · · · xp)T

,c ̸= 0

(x1 · · · xr · · · xs · · · xp

)T

→(

x1 · · · xr · · · xs + xr · · · xp)T

where these are the actions obtained by multiplication by elementary matrices. Denotethese special linear transformations by E (r↔ s) ,E (cr) ,E (s→ s+ r) .

Let R = ∏pi=1 (ai,bi) . Then it is easily seen that

mp (E (r↔ s)(R)) = mp (R) = |det(E (r↔ s))|mp (R)

since this transformation just switches two sides of R.

mp (E (cr)(R)) = |c|mp (R) = |det(E (cr))|mp (R)