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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.
 Fall '08
 STAFF
 Math, Calculus

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