302 CHAPTER 13. THE GENERALIZED RIEMANN INTEGRAL

With this proposition and definition, it is time to define the generalized Riemann inte-gral. The functions being integrated typically have values inR orC but there is no reason torestrict to this situation and so in the following definition, X will denote the space in whichf has its values. For example, X could be Rp which becomes important in multivariablecalculus. For now, just think C. It will be assumed Cauchy sequences converge and thereis a norm although it is likely possible to generalize even further.

Definition 13.1.3 For a = xi−1 < · · · < xn = b, and F an increasing function, ∆Fiwill be defined as F (xi)−F (xi−1). Let X be a complete normed vector space. (For exam-ple, X = R or X = C or X = Rp.) Then f : [a,b]→ X is generalized Riemann integrable,written as f ∈ R∗ [a,b] if there exists R ∈ X such that for all ε > 0, there exists a gauge δ ,such that if P≡ {(Ii, ti)}n

i=1 is δ fine, then defining S (P, f ) by

S (P, f )≡n

∑i=1

f (ti)∆Fi,

it follows |S (P, f )−R|< ε. If such an R exists, then the integral is defined as follows.∫I

f dF ≡∫ b

af dF ≡ R.

Here |·| refers to the norm on X . For R, this is just the absolute value.

Note that if P is δ 1 fine and δ 1 ≤ δ then it follows P is also δ fine. Because of this, itfollows that the generalized integral is unique if it exists.

Proposition 13.1.4 If R, R̂ both work in the above definition of the generalized Rie-mann integral, then R̂ = R.

Proof: Let ε > 0 and let δ correspond to R and δ̂ correspond to R̂ in the above defini-tion. Then let δ 0 = min

(δ , δ̂

). Let P be δ 0 fine. Then P is both δ and δ̂ fine. Hence,∣∣R− R̂∣∣≤ |R−S (P, f )|+

∣∣S (P, f )− R̂∣∣< 2ε

Since ε is arbitrary, it follows that R = R̂.Is there a simple way to tell whether a given function is in R∗ [a,b]? The following

Cauchy criterion is useful to make this determination. It looks just like a similar conditionfor Riemann Stieltjes integration.

Proposition 13.1.5 A function f : [a,b]→ X is in R∗ [a,b] if and only if for every ε > 0,there exists a gauge function δ ε such that if P and Q are any two divisions which are δ ε

fine, then |S (P, f )−S (Q, f )|< ε.

Proof: Suppose first that f ∈ R∗ [a,b]. Then there exists a gauge, δ ε , and an element ofX , R, such that if P is δ ε fine, then |R−S (P, f )| < ε/2. Now let P,Q be two such δ ε finedivisions. Then

|S (P, f )−S (Q, f )| ≤ |S (P, f )−R|+ |R−S (Q, f )|< ε

2+

ε

2= ε.

Conversely, suppose the condition of the proposition holds. Let εn→ 0+ as n→∞ andlet δ εn denote the gauge which goes with εn. Without loss of generality, assume that δ εn