1873_solutions

Equivalent a the function f is integrable on the

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Unformatted text preview: 1 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 with zero measure. 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 integrable on a, b . Solution: See the solution to Exercise 3. 3. Suppose that x n is a sequence in an interval a, b , that x n has only finitely many partial limits, and that f is a bounded function on a, b that is continuous at every member of a, b that does not belong to the range of the sequence x n . Prove that f is integrable on a, b . Solution: We shall write the set of partial limits of junior Lebesgue criterion, suppose that x n as y 1 , y 2 , , y k . To prove that f satisfies the  0. We define k U  yj 2k j1 , yj 2k and we observe that k k m mU yj j1 2k , yj 2k  j1 k . Since the set a, b U is closed and bounded and since the sequence x n has no partial limits in U. Therefore, if a, b U we know that x n cannot be frequently in the set a, b F  x n n  1, 2,  U then the set F is finite and so m F  0. We have thus found an elementary subset U Þ F of a, b such that m U Þ F and such that f is continuous at every number x a, b UÞF . 4. This exercise does not ask you for a proof. Suppose that x n is a 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 belong to the range of the sequence x n . Do you think that the function f has to be integrable on a, b ? What does your intuition tell you? Solution: The function f must be integrable. This fact will follow from the full version of the Lebesgue criterion for integrability that will appear in the chapter on sets of measure zero. Some Exercises on the Composition Theorem 1. Given two functions f and g defined on a set S, we define the functions f f gx  fx if f x gx if f x  g x and 276 gx g and f g as follows: f fx if f x gx gx  gx if f x  g x . Given Riemann integrable functions f and g on an interval a, b , make the observations f  g  |f g | f g 2 and f  g |f g | f g 2 and deduce that the functions f g and f g are also integrable on a, b . There really isn’t much to do in this exercise. The equation f x  g x  |f x g x | f gx 2 and f x  g x |f x g x | f gx 2 for each x follow at once when we consider the cases f x g x and f x  g x . 2. Given that f is a nonnegative integrable function on an interval a, b , explain why the function f is integrable. This exercise follows at once from the fact that the square root function is uniformly continuous on the range of f. 3. Given that f is integrable on an interval a, b , that f x 1 for every x a, b and that g x  log f x for every x a, b , explain why the function g must...
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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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