1873_solutions

Positive integer n we define the function s n as

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Unformatted text preview: r every x a, b . For each positive integer n we choose an elementary set E n such that m E n  1 and such that n 1 x a, b fx En. n For each n we have Þa u Pn, f b  ÞE ÞE n n u Pn, f  Þ kÞ a ,b En a ,b 1 n En u Pn, f b km E n  Þ 1 an k  n  bna 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 and that Þ f  0. b a If you are reading the on-screen version of this book you can find a special group of exercises that are designed to be done as a special project. These exercises require you to have read some of the chapter on infinite series and they 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: b If f is a nonnegative integrable function on an interval a, b , where a  b, and if Þ f  0 then there must be a at least one number x a, b for which f x  0. To reach this special group of exercises, click on the link . Positive Integrable Functions Have Positive Integrals Suppose that f is a nonnegative integrable function on an interval a, b where a  b and that Þ a f  0. b 1. Prove that for every number  0 there exists an elementary set E such that m E  x a, b fx E. and such that This exercise is a duplicate of the last exercise in the exercises on exercises on integrability. 274 2. Prove that if, for every positive integer n, we choose an elementary set E n such that m En  b n a 2 and such that ba x a, b fx En 2n then for every elementary E satisfying Ý  En E n1 we have m E  b a. To obtain this proof you will need to make use of the special group of exercises on . elementary sets that can be reached by clicking on the icon The existence of the sets E n follows from Exercise 1. Now if Ý  En E n1 then the special exercises on elementary sets guarantee that Ý Ý m En  mE n1 b n1 2n a b a. 3. Prove that if the sets E n are defined as in Exercise 2, the set Ý  En a, b n1 must be nonempty and deduce that there must exist a number x a, b such that f x  0. Ý The elementary set a, b can’t be a subset of  n1 E n because we do not have m a, b  b a. Thus there must be numbers x in a, b that do not belong to any of the sets E n and since any such number x must satisfy the inequality f x  b na 2 for every positive integer n we have f x  0 for such numbers x. 4. Improve on the preceding exercises by proving that for every number elementary sets such that the following two conditions hold:  0 there exists a sequence E n of a. For every number satisfying Ý x  En a, b n1 we have f x  0. b. For every elementary set E satisfying Ý E  En n1 we have m E  . Suppose that  0. We choose a sequence E n of elementary sets such that the conditions m En  n 2 and x a, b 275 fx 2n En hold for each n. Continue as above. Of course, we don’t have a guarantee that a, b nonempty unless  b a. Ý  n...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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