1873_solutions

If the set c has measure zero the fact that c does

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Unformatted text preview: obtain a contradiction, suppose that E is an elementary set, that C  E, and that m E  1 3. Choose an open elementary set U such that E U and m U  1 3. Since m En  Ý  En  n1 U 1 Ý  En  UC U n1 it follows from the Cantor intersection theorem that, for some n, we have E n n  1 3. mU m E n   1 3  3 2 3 This is the desired contradiction. U and we deduce that d. Prove that the function  C  is Riemann integrable on 0, 1 if and only if   1/3. Solution: This assertion follows at once from part c and the fact that the set C  is the set of discontinuities of the function  C  . e. Given 0   1/3, prove that there is a strictly increasing continuous function from 0, 1 onto 0, 1 that sends the set C  onto the usual Cantor set C. Solution: For each number x C  we define the function x from Z  into 0, 2 as follows: Given any positive integer n, if the number x belongs to one of the 2 n component intervals I of E n 1  and if, after I is split into two subintervals by the removal of its centrally located open subinterval of I with length  1/3 n 1 , the number x lies in the left subinterval we define x n  0 and if x lies in the right subinterval we define x n  2. We call the function x the address of the number x. We now define Ý ux  n1 xn 3n for every number x C  and we observe that u is a strictly increasing continuous function from C  onto C 1/3 . We now extend u to be a function defined on the entire interval 0, 1 as follows: Given x 0, 1 C  , if p is the greatest member of C  that is less than x and q is the least member of 400 C  that is greater than x then we define ux up  uq q up p x p. f. Prove that if f   C where C is the usual Cantor set and if u is the function found in part e then, although u is continuous on 0, 1 and f is Riemann integrable on the range of u, the function f u is not Riemann integrable on 0, 1 . Solution: This assertion follows at once from the fact that f u  C  . 5. In this exercise we introduce a further extension of the notion of extended Riemann integrability that was introduced in some earlier optional reading. We shall say that a function f defined on an interval a, b is almost extended Riemann integrable on a, b if there exists a sequence f n of Riemann integrable functions and a number K such that the following two conditions hold For each n we have |f n x | For almost every x K for almost every x a, b we have f n x f x as n a, b . Ý. a. Given an almost extended Riemann integrable function f on an interval a, b , give a reasonable definition b of the integral Þ f. Take care to say why your definition makes sense. Which theorem are you using for a this purpose? b. Prove that the integration of almost extended Riemann integrable functions is linear, nonnegative and additive. c. Give an example of a function that is almost extended Riemann integrable on 0, 1 but is not Riemann integrable on 0, 1 . 6. An example of Sierpinski shows that it is possible to find a subset S of the...
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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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