Kenneth Kuttler

13.4. INTEGRALS OF DERIVATIVES 317

Thenmn

∑k=1

h(tnk )g(tn

k )(F (xn

k)−F(xn

k−1))

=mn

∑k=1

(Hn

k −Hnk−1)

g(tnk )

=mn

∑k=1

Hnk g(tn

k )−mn−1

∑k=0

Hnk g(tnk+1)=

mn−1

∑k=1

Hnk(g(tn

k )−g(tnk+1))

+Hnmng(tnmn

)(13.17)

Applying 13.16, this equals

mn−1

∑k=1

(∫ xnk

ahdF

)(g(tn

k )−g(tnk+1))

+Hnmng(tnmn

)(13.18)

+mn−1

∑k=1

(enk

)(g(tn

k )−g(tnk+1))

(13.19)

I want to estimate the last term. To do this, suppose g is either increasing or decreasing.Then this last term is dominated by

εn |g(b)−g(a)|

and so it converges to 0 as n→ ∞. The function x→∫ x

a hdF is continuous. See Problem4 on Page 318 and the following problem. Therefore, since δ n ≤ εn the last term in 13.18converges to (∫ b

ahdF

)g(b−)

The remaining term in 13.18 is just a Riemann sum for a continuous function having anintegrator function given by an increasing function. Therefore, since δ n→ 0 the norm ofthe partitions consisting of the division points converges to 0 and so this Riemann sum,added to the other terms in 13.18 and 13.19 converges to∫ b

∫ x

ahdFdg+

(∫ b

ahdF

)g(b−)

thanks to Theorem 9.3.7 about the existence of the Riemann Stieltjes integral.Of course you can also use Proposition 9.3.2 about functions of bounded variation being

the difference of two increasing functions to conclude g could be a real valued boundedvariation function. This proves the following theorem.

Theorem 13.4.6 Let f ∈ R∗ [a,b] where the increasing integrator function F is con-tinuous and suppose g is of bounded variation. Then f g ∈ R∗ [a,b] also.

The proof of this theorem, patterned after the proof of the Dirichlet test for convergenceof series, shows why you have to assume something more on g. This requirement, alongwith the fact that f ∈ R∗ [a,b] does not imply | f | ∈ R∗ [a,b] is really OBNOXIOUS. Thereason for this is that the generalized Riemann integral can be like conditional convergence.Recall how strange things could take place. In the next chapter I will present a generalabstract framework for Lebesgue integration. This is like absolutely convergent series andso many of the strange things will disappear and the resulting integral is much easier to usein applications. It also is the integral for the study of probability.

13.4. INTEGRALS OF DERIVATIVES 317ThenVre)e(n) (FO) —F (1) = Y (Ae - He) 8 (0)k=1 k=1Mn my—1 my—1= Last) — Y Hes (ha) = YL AE (g (eh) 9 (eh) +HM,8(M,) 13.17)k=l k=0 k=lApplying 13.16, this equalsmn—1 nxtL (/ ‘ndF ) (8 (4h) — 8 s1)) + Ang 8 (tn) (13.18)k=1 \Vamy—1|+ ¥ (en) (¢ (2) —8 (thy1)) (13.19)k=1I want to estimate the last term. To do this, suppose g is either increasing or decreasing.Then this last term is dominated byEn|g(b) — 8 (a)|and so it converges to 0 as n — ce. The function x > |*hdF is continuous. See Problem4 on Page 318 and the following problem. Therefore, since 6, < €, the last term in 13.18converges to-b( | ha ) 2(b-)aThe remaining term in 13.18 is just a Riemann sum for a continuous function having anintegrator function given by an increasing function. Therefore, since 6, — 0 the norm ofthe partitions consisting of the division points converges to 0 and so this Riemann sum,added to the other terms in 13.18 and 13.19 converges to[ [rarag+ ( [nar @(b-)thanks to Theorem 9.3.7 about the existence of the Riemann Stieltjes integral.Of course you can also use Proposition 9.3.2 about functions of bounded variation beingthe difference of two increasing functions to conclude g could be a real valued boundedvariation function. This proves the following theorem.Theorem 13.4.6 Lez f © R* [a,b] where the increasing integrator function F is con-tinuous and suppose g is of bounded variation. Then fg € R* [a,b] also.The proof of this theorem, patterned after the proof of the Dirichlet test for convergenceof series, shows why you have to assume something more on g. This requirement, alongwith the fact that f € R* [a,b] does not imply |f| € R* [a,b] is really OBNOXIOUS. Thereason for this is that the generalized Riemann integral can be like conditional convergence.Recall how strange things could take place. In the next chapter I will present a generalabstract framework for Lebesgue integration. This is like absolutely convergent series andso many of the strange things will disappear and the resulting integral is much easier to usein applications. It also is the integral for the study of probability.