Question

5. Let f : [a, b] → R be bounded, and a : [a, b] → R monotonically increasing, (a) For a partion P of (a, b), define the upper and lower Riemann-Stieltjes sums with respect to a. (b) (i) Define what it means for f to be Riemann-Stieltjes integrable with respect to a. (ii) State Riemann's Integrability Criterion. (C) Suppose f is both bounded and monotonic, and that a is both monotonically increasing and continuous. Prove that then f is Riemann-Stieltjes integrable with respect to a. (Hint: Use Riemann's Integrability Criterion.]

Answer 1

(a) Given a function f that is bounded and defined on the closed interval I = [a, b], a function a that is defined and monotonically increasing on I, and a partition P = {xo, X1, ......Xn-1, xn} of I. Let supxel;f(x); m; = infxer; f(x), for 1; = [xj-1, x;]. Then, upper and lower Riemann Stieltjes sum of f over a with respect to the partition P is defined by Mj n U(P, f, a) = M;Aaj, L(P, f,a) = m;Aaj j=1 j=1 Scanned with CS where Aa= (a(xi) - (xj-1)).

(b) Suppose that f is a real valued bounded function defined on 1 = [a, b], P = P[a, b] be the set of all partitions of [a, b] and a a monotonically increasing function defined on 1. Then the upper and lower Riemann Sieltjes integrals are defined by Lºr(x)da(x) = infpU(P, f,a); °r<)da(x) = suppL(P, f, a), respectivelyitilf so f(x)da(x) = sa f(x)da(x) then f is said to be CS Riemann Stieltjes integrable.

(b) (ii Let f be a function defined on a bounded, closed interval (a, b). We want to consider the Riemann integral of f on [a, b]. We will see that this is not always possible; those for which it is possible are called (Riemann) integrable functions on [a, b]. A partition of [a, b], P, is a finite collection of points, a = lo < X1 <...< In = b, which divides [a, b] into n many subintervals I; = [Xj-1,X;], j = 1, ..., n. The length of a partition is given by ||P|| = max(x; – ;-1). A tagged partition is the pair (P, 21, ..., Zn) where zi E I;. We shall use to denote a tagged partition. Given any tagged partition P, we define the Riemann sum of f with respect to P by S(5, 1) = f(z;)AX;; where Av; = x; – 2j-1. n j=1 Geometrically, S(S,P) is an approximate area of the region bounded by x = a, x = b, y = 0 and y = f(x) (assuming f is non-negative). We call f Riemann integrable on (a, b) if there exists L ER so that for every e > 0, there exists some 8 >0 st. |S(f, P) – L[ <E, VP, ||P|| <d, for any tag P on P. It is easy to show that such L is uniquely determined whenever it exists. It is called the Riemann integral of f over [a, b] and is denoted by Saf. We use R[a, b] to denote the set of all Riemann integrable functions on [a, b]. It can be shown that any Riemann integrable functions on a closed and bounded interval [a, b] are bounded functions; see textbook for a proof. Hence- forth we will work only with bounded functions.

(c) Proof. Proof of part (b) For any positive integer n choose a partition P = {X0,X1,...,Xn} of (a,b) such that Ad= a(xi) - Q(x;-1) = la(b) – a(a) Since a is continuous, such a choice is possible by the intermediate value theorem. Assume f is monotone increasing on (a,b). Then M = f(xi) and m; = S(x-1). Therefore U(P.f.a) - L(P.J.a) \(xi) - S(x-1)] Adi (alb) - a(a)" (S(xi) - 16: [a(b) – a(a)) [(b) - f(a)] Given e > 0, choose ne N such that (a(b) – a (a)) tr(b) – f(a)] < €. 11 For this n and corresponding partition P, UP,.a) - L(Pl.a) <€. 18 Properties of the Riemann-Stieltjes Integral are given below. For a given monotone increasing function a on (a,b), R(a) denotes the set of bounded real-valued functions f on (a, b) that are Riemann-Stieltjes integrable with respect to a.