308 CHAPTER 13. THE GENERALIZED RIEMANN INTEGRAL

Corollary 13.1.16 Suppose f ∈ R∗ [a,b] has values in R and that∣∣∣∣S (P, f )−∫

If dF

∣∣∣∣≤ ε

for all P which is δ fine. Then if P = {(Ii, ti)}ni=1 is δ fine,

n

∑i=1

∣∣∣∣ f (ti)∆Fi−∫

Iif dF

∣∣∣∣≤ 2ε. (13.1)

Proof: Let I ≡{

i : f (ti)∆Fi ≥∫

Ii f dF}

and let I C ≡ {1, · · · ,n}\I . Then by Hen-stock’s lemma ∣∣∣∣∣∑i∈I f (ti)∆Fi− ∑

i∈I

∫Ii

f dF

∣∣∣∣∣= ∑i∈I

∣∣∣∣ f (ti)∆Fi−∫

Iif dF

∣∣∣∣≤ ε

and ∣∣∣∣∣ ∑i∈I C

f (ti)∆Fi− ∑i∈I C

∫Ii

f dF

∣∣∣∣∣= ∑i∈I C

∣∣∣∣ f (ti)∆Fi−∫

Iif dF

∣∣∣∣≤ ε

so adding these together yields 13.1.This generalizes immediately to the following.

Corollary 13.1.17 Suppose f ∈ R∗ [a,b] has values in C and that∣∣∣∣S (P, f )−∫

If dF

∣∣∣∣≤ ε (13.2)

for all P which is δ fine. Then if P = {(Ii, ti)}ni=1 is δ fine,

n

∑i=1

∣∣∣∣ f (ti)∆Fi−∫

Iif dF

∣∣∣∣≤ 4ε. (13.3)

Proof: It is clear that if 13.2 holds, then |S (P,Re f )−Re∫

I f dF | ≤ ε, which showsthat Re

∫I f dF =

∫I Re f dF . Similarly Im

∫I f dF =

∫I Im f dF . Therefore, using Corollary

13.1.16, ∑ni=1

∣∣∣Re f (ti)∆Fi−∫

Ii Re f dF∣∣∣ ≤ 2ε and ∑

ni=1

∣∣∣i Im f (ti)∆Fi−∫

Ii i Im f dF∣∣∣ ≤ 2ε.

Adding and using the triangle inequality, yields 13.3.

13.2 Monotone Convergence TheoremThere is nothing like the following theorem in the context of Riemann integration.

Example 13.2.1 Let {rn}∞

m=1 be the rational numbers in [0,1]. Let fn (x) be 1 if x ∈{r1, · · · ,rn} and 0 elsewhere and F (x) = x. Then fn is Riemann integrable and convergespointwise to the function f (x) which is 1 on all rationals in [0,1] and zero elsewhere. How-ever, f is not Riemann integrable. Indeed, there is a gap between the upper and lowersums. See Theorem 9.3.10 on Page 198.

In contrast to this example, here is the monotone convergence theorem.