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Unformatted text preview: a 2 0 and a 2 2 we can go a step further and see that C is included in the set
E 2 0, 1 Þ 2 , 3 Þ 6 , 7 Þ 8 , 1
9
99
99
9 0 1
9 2
9 1
3 2
3 7
9 8
9 1 and, in general, if n is any positive integer then C is included in the set E n that is the union of 2 n closed
intervals of length 1/3 n and whose left endpoints are the numbers
n
j1 aj
3j where each number a j is either 0 or 2. It is not hard to show that
C Ý En.
n1 297 Ý
an
n1 3 n
Ý The value of the Cantor function at each member
Ý
n1 an
3n
n1 of the set C is defined by the equation
an .
2 n 1 Thus
0 0
Ý 1
n1 2
3n Ý
n1 2 1
2 n 1 1 2 1
2
3
3
1 2 1
4
9
9
7 8 3.
4
9
9
The function is a continuous strictly increasing function from C onto the interval 0, 1 . We now extend to
an increasing function from 0, 0 onto 0, 1 by making constant on every component interval of the open
set 0, 1
C. The graph of is shown in the following figure: Graph of the Cantor Function A General Discussion of the integral Þ x p d x
1 0 The purpose of this document is to explore some integrals of the form Þ x p d x where p is a positive
0
integer. Note that such an integral always exists because the integrand is an increasing function. However, it
is worth showing the existence of the integrals directly. We define f x x p for each x
0, 1 . For each
positive integer n we define P n to be the partition of 0, 1 whose points are the endpoints of the component
intervals of the set E n . For example,
P 2 0, 1 , 2 , 1 , 2 , 7 , 8 , 1 .
993399
1 0 1
9 2
9 1
3 2
3 For each n, if the partition P n is expressed as
P x 0 , x 1 , , x 2 n1
then we define two step functions s n and S n on 0, 1 by defining
p
sn xj Sn xj xj
for every j 0, 1, 2, , 2 n and by defining
p sn x xj 1 and
p Sn x xj
whenever 298 7
9 8
9 1 xj xj xj−1 0 x xj 1 1 We observe that s n f S n . Now given any two consecutive points x j 1 and x j of the partition P n there are
two possibilities: Either the interval x j 1 , x j is a component interval of the set E n ; in which case
var , x j 1 , x j 1n
2
or the interval x j 1 x j is a gap between two component intervals of E n ; in which case
var , x j 1 , x j 0.
When x j 1 , x j is a component interval of E n we have
p
p
p1
p2
p3
p
p1
x j x j 1 1n x j x j x 1 1 x j x 2 1 x j 1 n
j
j
3
3
and so
2n Þ0 Sn
1 p s n d xj 1
2n p xj 1 j1 2n
j1 1 xp
3n j 1 p2 p3 p1
1 xj x1 1 xj x2 1 xj
j
j 1
2n p
n.
3
Since the latter expression approaches 0 as n Ý we know that the pair of sequences s n and S n squeezes
f with respect to and so f is RiemannStieltjes integrable with respect to on the interval 0, 1 . We deduce
that Þ 0 s n d Þ 0 x p d x
1 lim nÝ 1 We now look at the step functions s n more closely. We write the 2 n left endpoints of the component intervals
of E n as
c 1 , c 2 , c 3 , , c 2 n
n we express c in the form
and for each k 1, 2, 3, , 2
k
n a kj
.
3j ck
j1 Where each of the numbers a kj is eith...
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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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