1873_solutions

That x x if x 0 x 1 if x 0 prove that the integral

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Unformatted text preview: ck on the icon 310 . Positive Integrable Functions Have Positive Integrals Suppose that  is an increasing function, that f is a nonnegative function that is integrable with respect to  on an interval a, b . Suppose that var , a, b  0 and that Þ a fd  0. b 1. Prove that for every number  0 there exists an elementary set E such that var , 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 2. Prove that if, for every positive integer n, we choose an elementary set E n such that var , a, b var , E n  2n and such that var , a, b x a, b fx En 2n then for every elementary E satisfying Ý E  En n1 we have var , E  var , a, b . 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 E  En n1 then the special exercises on elementary sets guarantee that Ý Ý var , E n  var , E n1 n1 var , a, b 2n  var , a, b . 3. Prove that if the sets E n are defined as in Exercise 2, the set Ý a, b  En 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 var , a, b  var , a, b . 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 var , a, b fx  2n 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: a. For every number satisfying 311  0 there exists a sequence E n of Ý x  En a, b n1 we have f x  0. b. For every elementary set E satisfying Ý E  En n1 we have var , E  . Suppose that  0. We choose a sequence E n of elementary sets such that the conditions var , E n  n 2 and x a, b fx En 2n hold for each n. Continue as above. Of course, we don’t have a guarantee that a, b nonempty unless  b a. Ý  n1 E n is Some Exercises on the Junior Lebesgue Criterion 1. True or false: Every step function satisfies the junior Lebesgue criterion. Of course this statement is true. The set of discontinuities of a step function, being finite, is an elementary set. If we express this set as E  t j j  1, 2, , n then n J , t j . var , E  j1 2. Suppose that x n is a convergent sequence in an interval a, b and that f is a bounded function on a, b that is continuous at every member of a, b that does not lie in the range of the sequence x n . Prove that f is Riemann integrable on a, b . Solution: See the solution to Exercise 4. 3. Suppose that x n is a convergent sequence in an interval a, b that  is an increasing function and that f is a bounded function on a, b that is continuous at every member of a, b that does not lie in the range of the sequence x n . Prove that f is Riemann-Sti...
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