Kenneth Kuttler

9.3. FUNDAMENTAL DEFINITIONS AND PROPERTIES 193

Without loss of generality Py contains x because from what was just shown you could addin the point x and the approximation of V (Py,g) to V[a,y] (g) would only be better. Thenfrom the definition,

V[a,y] (g)−V[a,x] (g)≥V (Py,g)−(V(P[a,x],g

)+ ε)

≥ |g(y)−g(x)|− ε ≥ g(y)−g(x)− ε

and since ε is arbitrary, this establishes 9.1. This proves the first part of the proposition.Now suppose g(x) = g1 (x)−g2 (x) where each gi is an increasing function. Why is g

of bounded variation? Letting x < y

|g(y)−g(x)| = |g1 (y)−g2 (y)− (g1 (x)−g2 (x))|≤ (g1 (y)−g1 (x))+(g2 (y)−g2 (x))

Therefore, if P = {x0, · · · ,xn} is any partition of [a,b]

∑i=1|g(xi)−g(xi−1)| ≤

∑i=1

(g1 (xi)−g1 (xi−1))+(g2 (xi)−g2 (xi−1))

= (g1 (b)−g1 (a))+(g2 (b)−g2 (a))

and this shows V[a,b] (g)≤ (g1 (b)−g1 (a))+(g2 (b)−g2 (a)) so g is of bounded variation.

The following is the definition of the Riemann Stieltjes integral.

Definition 9.3.3 A bounded function f defined on [a,b] is said to be Riemann Stielt-jes integrable if there exists a number I with the property that for every ε > 0, there existsδ > 0 such that if

P≡ {x0,x1, · · · ,xn} , a = x0 < · · ·< xn = b

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

∑i=1

f (zi)(g(xi)−g(xi−1))

∣∣∣∣∣< ε.

The number∫ b

a f (x) dg(x)is defined as I. I will denote this Riemann Stieltjes sum approx-imating I as ∑P f (zi)(g(xi)−g(xi−1)). When f is Riemann Stieltjes integrable on [a,b]with respect to g as just described, this is denoted as f ∈ R([a,b] ,g) or simply as R [a,b] ifthe definition is clear for g.

A special case is the following definition.

Definition 9.3.4 The Riemann integral is a special case of the above in which theintegrator function is g(x) = x. We write

∫ ba f (x)dg(x) in the form

∫ ba f (x)dx to signify the

Riemann integral.

There is only one possible number I satisfying the above definition.

Lemma 9.3.5 The integral∫ b

a f (x) dg(x) is well defined in the sense that if there is sucha number I, then there is only one.

9.3. FUNDAMENTAL DEFINITIONS AND PROPERTIES 193Without loss of generality P, contains x because from what was just shown you could addin the point x and the approximation of V (P,,g) to Viz.y) (g) would only be better. Thenfrom the definition,Viay] (g) ~~ Via.x| (g) 2V (P,,8) ~~ (V (Pasj+8) + €)> |g(y)—g(x)|-—€>8(y)—-g(x)—-e€and since € is arbitrary, this establishes 9.1. This proves the first part of the proposition.Now suppose g(x) = gi (x) — g2 (x) where each g; is an increasing function. Why is gof bounded variation? Letting x < ylg(y)—g(x)| = |gi(y)—82(y) — (81 &) — 82 (x))|< (gi (y)— 281 (4) +(g2(y) — 82 (4)Therefore, if P = {xo,--- ,x,} is any partition of [a,b]Ms:Ig (41) -g(i-1)| < (g1 (xi) — 81 (i-1)) + (82 i) — 82 (i-1))= (g1(b)—g1(@)) +(82(b) — 82 (a))and this shows Vig.) (g) < (g1 (b) — g1 (@)) + (82 (b) — g2 (a)) 80 g is of bounded variation.|The following is the definition of the Riemann Stieltjes integral.Ms:1llmnUDefinition 9.3.3 A bounded function f defined on |a,b| is said to be Riemann Stielt-Jes integrable if there exists a number I with the property that for every € > 0, there exists6 > 0 such that ifP= {x0,X1,°°° Xn}, A=X) <0 <x =Dis any partition having ||P|| < 6, and z; € [xj-1, xi],The number f{ ° f (x) dg (x)is defined as I. I will denote this Riemann Stieltjes sum approx-imating I as Vp f (zi) (g (ai) — g (xi-1)). When f is Riemann Stieltjes integrable on [a,b]with respect to g as just described, this is denoted as f € R(|a,b],g) or simply as R|a,b] ifthe definition is clear for g.A special case is the following definition.Definition 9.3.4 The Riemann integral is a special case of the above in which theintegrator function is g(x) =x. We write SF (x) dg (x) in the form L? f(x) dx to signify theRiemann integral.There is only one possible number / satisfying the above definition.Lemma 9.3.5 The integral [. P f (x) dg (x) is well defined in the sense that if there is sucha number I, then there is only one.