848 APPENDIX A. THE THEORY OF THE RIEMANNN INTEGRAL∗

Proof: Each of the combinations of functions described above is Riemannn integrableby Theorem A.3.1. For example, to see a f +bg ∈R (Rn) consider φ (y,z)≡ ay+bz. Thisis clearly a continuous function of (y,z) such that φ (0,0) = 0. To obtain | f | ∈R (Rn), letφ (y,z)≡ |y|. It remains to verify the formulas. To do so, let G be a grid with the propertythat for k = f ,g, | f | and a f +bg,

UG (k)−LG (k)< ε. (1.9)

Consider (1.7). For each Q ∈ G pick a point in Q, xQ. Then

∑Q∈G

k (xQ)v(Q) ∈ [LG (k) ,UG (k)]

and so ∣∣∣∣∣∫

k dx− ∑Q∈G

k (xQ)v(Q)

∣∣∣∣∣< ε.

Consequently, since∑

Q∈G(a f +bg)(xQ)v(Q)

= a ∑Q∈G

f (xQ)v(Q)+b ∑Q∈G

g(xQ)v(Q) ,

it follows ∣∣∣∣∫ (a f +bg) dx−a∫

f dx−b∫

gdx∣∣∣∣≤∣∣∣∣∣

∫(a f +bg) dx− ∑

Q∈G(a f +bg)(xQ)v(Q)

∣∣∣∣∣+∣∣∣∣∣ a ∑Q∈G

f (xQ)v(Q)−a∫

f dx

∣∣∣∣∣+∣∣∣∣∣b ∑

Q∈Gg(xQ)v(Q)−b

∫gdx

∣∣∣∣∣≤ ε + |a|ε + |b|ε.

Since ε is arbitrary, this establishes (1.7) and shows the integral is linear.It remains to establish the inequality (1.8). By (1.9), and the triangle inequality for

sums, ∫| f | dx+ ε ≥ ∑

Q∈G| f (xQ)|v(Q)

≥

∣∣∣∣∣ ∑Q∈G f (xQ)v(Q)

∣∣∣∣∣≥∣∣∣∣∫ f dx

∣∣∣∣− ε.

Then since ε is arbitrary, this establishes the desired inequality. ■