Interval 2 1 n n2 0 we conclude that the pair of

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Unformatted text preview: ion for integrability. Suppose that  0. Choose a finite set S such that fx  var , a, b whenever x a, b S and define P to be the partition of a, b whose points are the numbers a and b and the members of S arranged in increasing order. Since l P, f is nonnegative and u P, f never exceeds the value / b a , we have b Þ a w P, f d  Þ a u P, f b b l P, f d Þ a u P, f d b . Since f satisfies the first criterion for integrability, f is integrable on a, b and the same b b from which we deduce that Þ fd  0. argument shows that whenever  0 we have Þ fd a a b. Prove that if f is the ruler function that was introduced in an earlier example then f is a Riemann integrable function on the interval 0, 1 , even though f is discontinuous at every rational number in the interval. The ruler function obviously has the property described in part a. 6. Given that  is an increasing function and that f is a bounded nonnegative function defined on an interval 309 a, b , prove that the following conditions are equivalent: a. The function f is integrable with respect to  on the interval a, b and Þ fd  0. b a  0 there exists an elementary set E such that var , E  x a, b fx E. b. For every number and such that To show that condition a implies condition b we assume that f is integrable with respect to  on a, b . Suppose that  0. To obtain the desired set E we shall use the same sort of technique as was used in the proof of Theorem 11.9.4. Choose a step function S f such that Þ a Sd  b 2 and define E x a, b . Sx We observe that x a, b fx E. Now since S is a step function, the set E is elementary and we have 2  Þ a Sd ÞE Sd ÞE b d  var , E from which we deduce that var , E  . To show that condition b implies condition a we assume that condition b holds. Once again we borrow from the proof of proof of Theorem 11.9.4. Using the fact that f is bounded we choose a number k such that f x  k for every x a, b . For each positive integer n we choose an elementary set E n such that var , E n  1 and such that n 1 x a, b fx En. n For each n we have Þa u Pn, f b d  ÞE ÞE n n u P n , f d  Þ kd  Þ a ,b En a ,b En 1 d n u P n , f d b k var , E n  Þ 1 d an var , a, b k n n Since the latter expression approaches 0 as n Þa S b inf Ý we deduce that S is a step function and f S 0 and this shows that f is integrable with respect to  and that Þ fd  0. b a This chapter provides a special group of exercises that are designed to be done as a special project and which depend upon the special group of exercises on elementary sets that appeared earlier. The main purpose of these exercises is to invite you to prove the following interesting fact about integrals: If f is a nonnegative function on an interval a, b and f is Riemann-Stieltjes integrable with respect to an b increasing function  and if var , a, b  0 and if Þ f d  0 then there must be at least one number a x a, b for which f x  0. To reach this special group of exercises, cli...
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