1549901369-Elements_of_Real_Analysis__Denlinger_

386 Chapter 7 111 The Riemann Integral

7 .4 Basic Existence and Additivity Theorems

Learning the results of this section may be more impor-

tant than learning their proofs. Let your instructor detemine

which proofs to study.

The first result of this section is a technical lemma that paves the way for an
important property of the integral known as the "additive property."

Lemma 7.4.1 (Additivity of Upper and Lower Integrals) Suppose f :
[a, b] --+JR is bounded, and a < c < b. Then

and (b) 1: f = 1: f +I: f.

Proof. Suppose f : [a, b] --+ JR is bounded, and a < c < b. Suppose P
is a partition of [a, b]. Let P' = P U { c}. Then P' is a partition of [a, b] that
refines P ; moreover, P' = P 1 U P2, where P 1 is a partition of [a, c] and P2 is
a partition of [c, d]. Then

S(f, P) ~ S(f, P') = S(f, P1) + S(f, P2)

~ 1:1+1:r

Hence, inf{S(f, P) : P is a partition of [a, b]} > J: f + J: f.

i.e., J: f ~ J: f + J: f · (11)
To prove the reverse inequality, let t: > 0. By the €-criterion for infimum
(Theorem 1.6.7) :J partitions P3 of [a, c], and P 4 of [c, b], such that

• ----c € - b €
  S(f, P3) < faf + 2 and S(f, P4) < f c f + 2·
  Then P3 U P4 is a partition of [a, b], and

But J: f :S S(f, P3 U P4). Therefore, J: f :SJ: f + J: f + t:.

Since this happens Ve > 0, we conclude by the forcing principle that

1: f ::; 1: f + 1: f. (12)

By (11) and (12) together, we conclude that

1: f = 1: f + 1: f.

(b) Exercise l. •