1873_solutions

P in other words the number 2 must be in one of the

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Unformatted text preview: m E  0 and if E is any elementary set that includes A we have m E 1. The set 0, 1 Q has the desired properties. 6. Given that E is an elementary set that is not closed and that F is a closed elementary subset of E, prove that m E F  0. Solution: Choose a lower bound a and an upper bound b of the set E. Since the set E F is elementary, if we want to show that m E F  0 then, from Exercise 2, all we have to show is that the set E F cannot be finite. The fact that E F is not finite follows from the fact that finite sets are always 291 closed, that F is closed and that the set E, which isn’t closed is the union of the two sets F and E F. 7. Given that  is an increasing function, that f is a step function, that E is an elementary set and that f x  0 whenever x R E, prove that Ý ÞE fd  Þ Ý fd. Solution: The desired equality follows at once from the definitions and the fact that f  f E . 8. Given that  is an increasing function, that f and g are step functions, that E is an elementary set and that fx g x whenever x R, prove that ÞE fd ÞE gd. Choose an interval a, b outside of which both of the functions f and g are zero. Since f E follows from the nonnegativity property of integrals of step functions that g E , it ÞE fd  Þ a f E d Þ a g E d  ÞE gd b b 9. Given that  is an increasing function, that f is a nonnegative step function, that A and B are elementary sets and that A B, prove that ÞA fd ÞB fd. The desired inequality follows at once from the fact that f A f B . We choose an interval a, b that includes the set B and use the nonnegativity property to obtain ÞA f  Þ a f A Þ a f B  ÞB f b b 10. Given that  is an increasing function, that f is a step function and that E is an elementary set, prove that ÞE fd ÞE |f |d. Hint: Use the fact that |f | E f E |f | E . 11. Given that A and B are elementary sets, prove that ÞA  B  ÞB  A  m A B. Solution: Choose a lower bound a and an upper bound b of the set A Þ B. We see that ÞA  B d  Þ a  B  A d  Þ a  A  B d  ÞB  A d. b b The fact that these expressions are equal to m A B follows at once from the fact that AB  A B. For some additional exercises on the variation of a function  on elementary sets click on the following icon. . Additional Exercises on Elementary Sets and Infinite Series Suppose that  is an increasing function. 292 1. Given that H is a closed elementary set and U n is a sequence of open elementary sets and that Ý H  Un, n1 use this earlier exercise to deduce that, for some positive integer N we have N var , H var , U n n1 and deduce that Ý var , H var , U n . n1 Choose a positive integer N such that N H  Un. n1 We have var ,  U n n1 Ý N N var , H var , U n n1 var , U n . n1 2. Given that E is an elementary set and that U n is a sequence of open elementary sets and that Ý E  Un, n1 prove that Ý var , U n . var , E n1 Hint: Make use of the theorem on approximating by closed sets and open set...
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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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