Page 283 - DMTH401_REAL ANALYSIS

Unit 22: Introduction to Riemann-Stieltjes Integration, using Riemann Sums

 A partition  can be thought of as “dividing” the interval [a, b] into subintervals. We may Notes
write |[a, b] and read this as “ divides [a, b],” or “partitions [a, b].” We will denote the
intervals of  by I : = [x , x ]. When we wish to work with 2 partitions at the same time we
i i – 1 i
will have to distinguish between them somehow, for example we can use y to denote the
j
other’s points and J to denote its intervals, etc.
j
 A real-valued function f(x) defined on the bounded and closed interval [a, b] is Riemann-
Stieltjes integrable on [a, b] with respect to  if there exists a number RSI such that for all
 > 0 there exists  > 0 such that for every partition  of [a, b],
mesh() <   |RS(f, ) – RSI| < .

22.8 Keywords

Cauchy Criterion for Riemann-Stieltjes Integrability: A function defined on [a, b] is Riemann-
Stieltjes integrable over [a, b] with respect to , defined on [a, b], if and only if for all  > 0 there
exists  > 0 such that for all partitions  and ¢ of [a, b], and for all selection vectors  and ¢
associated with  and ¢, respectively,
mesh() <  and mesh(¢) <   |RS(f, , , ) – RS(f, , ¢, ¢)| < .
Jordan: A function f Î BV[a, b] if and only if there exist functions g and h, both increasing on [a,
b], such that f(x) = g(x) – h(x) for a  x  b. If this is the case, then P(x)  g(x) – g(a), N(x)  h(x) – h(a)
and f(x) = f(a) + P(x) – N(x) for a  x  b.
22.9 Review Questions

1. Identify the properties of the integral.
2. Use them to find the Riemann stieltjes integral of functions.

Answers: Self Assessment

1. real-valued function 2. bounded on [a, b]
3. |(x ) –(x )| > 0 4. Riemann-Stieltjes integrable
1 2
5. discontinuity

22.10 Further Readings

Books Walter Rudin: Principles of Mathematical Analysis (3rd edition), Ch. 2, Ch. 3.
(3.1-3.12), Ch. 6 (6.1 - 6.22), Ch.7(7.1 - 7.27), Ch. 8 (8.1- 8.5, 8.17 - 8.22).

G.F. Simmons: Introduction to Topology and Modern Analysis, Ch. 2(9-13),
Appendix 1, p. 337-338.
Shanti Narayan: A Course of Mathematical Analysis, 4.81-4.86, 9.1-9.9, Ch.10,Ch.14,
Ch.15(15.2, 15.3, 15.4)
T.M. Apostol: Mathematical Analysis, (2nd Edition) 7.30 and 7.31.
S.C. Malik: Mathematical Analysis.

H.L. Royden: Real Analysis, Ch. 3, 4.

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