13.1. DEFINITIONS AND BASIC PROPERTIES 303

is decreasing in n. (If not, replace it with the minimum of itself and earlier gauges.) LetRεn denote the closure of all the sums, S (P, f ) where P is δ εn fine. From the condition,it follows diam(Rεn) ≤ εn and that these closed sets are nested in the sense that Rεn ⊇Rεn+1 because δ εn is decreasing in n. Therefore, there exists a unique, R ∈ ∩∞

n=1Rεn . Tosee this, let rn ∈ Rεn . Then since the diameters of the Rεn are converging to 0, {rn} isa Cauchy sequence which must converge to some R ∈ X . Since Rεn is closed, it followsR ∈ Rεn for each n. Letting ε > 0 be given, there exists εn < ε and for P a δ εn fine division,|S (P, f )−R| ≤ εn < ε.Therefore, R =

∫I f .

Are there examples of functions which are in R∗ [a,b]? Are there examples of functionswhich are not? It turns out the second question is harder than the first although it is veryeasy to answer this question in the case of the obsolete Riemann integral. The generalizedRiemann integral is a vastly superior integral which can integrate a very impressive collec-tion of functions. Consider the first question. It turns out that R [a,b]⊆ R∗ [a,b]. Recall thedefinition of the Riemann integral given above which is listed here for convenience.

Definition 13.1.6 A bounded function f defined on [a,b] is said to be RiemannStieltjes integrable if there exists a number I with the property that for every ε > 0, thereexists δ > 0 such that if

P≡ {x0,x1, · · · ,xn}

is any partition having ∥P∥< δ , and zi ∈ [xi−1,xi] ,∣∣∣∣∣I− n

∑i=1

f (zi)(F (xi)−F (xi−1))

∣∣∣∣∣< ε.

The number∫ b

a f (x) dFis defined as I.

First note that if δ > 0 is a number and if every interval in a division has length less thanδ then the division is δ fine. In fact, you could pick the tags as any point in the intervals.Then the following theorem follows immediately.

Theorem 13.1.7 Suppose that f is Riemann Stieltjes integrable according to Defin-ition 13.1.6. Then f is generalized Riemann integrable and the integrals are the same.

Proof: Just let the gauge functions be constant functions.In particular, the following important theorem follows from Theorem 9.3.7.

Theorem 13.1.8 Let f be continuous on [a,b] and let F be any increasing integra-tor. Then f ∈ R∗ [a,b] .

This integral can integrate almost anything you can imagine, including the functionwhich equals 1 on the rationals and 0 on the irrationals which is not Riemann integrable.This will be shown later.

The integral is linear. This will be shown next.

Theorem 13.1.9 Suppose α and β are constants and that f and g are in R∗ [a,b].Then α f +βg ∈ R∗ [a,b] and∫

I(α f +βg)dF = α

∫I

f dF +β

∫IgdF.