A.3. BASIC PROPERTIES 847

∑Pk

(MQ (k)−mQ (k))v(Q)≥ δ ∑Pk

v(Q)

and so for k = f ,g,ε > δ > ∑

Pk

v(Q) . (1.6)

Suppose for k = f ,g,MQ (k)−mQ (k)≤ δ .

Then if x1,x2 ∈ Q,

| f (x1)− f (x2)|< δ , and |g(x1)−g(x2)|< δ .

Therefore,

|h(x1)−h(x2)| ≡ |φ ( f (x1) ,g(x1))−φ ( f (x2) ,g(x2))|< ε

and it follows that|MQ (h)−mQ (h)| ≤ ε.

Now letS ≡ {Q ∈ G : 0 < MQ (k)−mQ (k)≤ δ , k = f ,g} .

Thus the union of the boxes in S is contained in some large box, R, which depends onlyon f and g and also, from the assumption that φ (0,0) = 0, MQ (h)−mQ (h) = 0, unlessQ⊆ R. Then

UG (h)−LG (h)≤ ∑Q∈P f

(MQ (h)−mQ (h))v(Q)+

∑Q∈Pg

(MQ (h)−mQ (h))v(Q)+ ∑Q∈S

δv(Q) .

Now since K is compact, it follows φ (K) is bounded and so there exists a constant C,depending only on h and φ such that MQ (h)−mQ (h)<C. Therefore, the above inequalityimplies

UG (h)−LG (h)≤C ∑Q∈P f

v(Q)+C ∑Q∈Pg

v(Q)+ ∑Q∈S

δv(Q) ,

which by (1.6) implies

UG (h)−LG (h)≤ 2Cε +δv(R)≤ 2Cε + εv(R) .

Since ε is arbitrary, the Riemannn criterion is satisfied and so h ∈R (Rn). ■

Corollary A.3.2 Let f ,g ∈R (Rn) and let a,b ∈ R. Then a f + bg, f g, and | f | are all inR (Rn). Also, ∫

Rn(a f +bg) dx = a

∫Rn

f dx+b∫Rn

gdx, (1.7)

and ∫| f | dx≥

∣∣∣∣∫ f dx∣∣∣∣ . (1.8)