972 CHAPTER 26. INTEGRALS AND DERIVATIVES

V , {(xi,xi +hi)}∞

i=1 such that

G(xi +hi)−G(xi)

hi< p.

Next show there is a disjoint sequence of intervals{(

x′i,x′j +h′j

)}∞

j=1such that each

of these is contained in one of the former intervals and

G(

x′j +h′j)−G

(x′j)

h′j> q, ∑

jh′j ≥ m(Npq) .

Then

qm(Npq) ≤ q∑j

h′j ≤∑j

G(x′j +h′j

)−G

(x′j)≤∑

iG(xi +hi)−G(xi)

≤ p∑i

hi ≤ pm(V )≤ p(m(Npq)+ ε) .

Since ε was arbitrary, this shows m(Npq) = 0. Taking a union of all Npq for p,qrational, shows the derivative from the right exists a.e. Do a similar argument toshow the derivative from the left exists a.e. and then show the derivative from the leftequals the derivative from the right a.e. using a simlar argument. Thus G′ (x) existson (a,b) a.e. and so it exists a.e. on R because (a,b) was arbitrary.