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Ý a 8. Given that f is a step function and that and are increasing functions, prove that
Ý Þ Ý fd Ý Ý Þ Ý fd Þ Ý fd. We begin by choosing an interval a, b and a partition
P x 0 , x 1 , , x n
of a, b such that f x 0 whenever a number x lies outside the interval a, b and such that f steps
within the partition P. If the constant value of f in each interval x j 1 , x j is j then
Ý Þ Ý fd n n f x j J , x j
j0 xj j n
1 n f xj
j0
Ý xj j1 J , x j J , x j j xj xj xj 1 xj 1 j1
Ý Þ Ý fd Þ Ý fd. 9. Given that f is a step function and that is an increasing function and that c is a nonnegative number, prove
that
Ý Þ Ý fd c cÞ Ý
Ý fd. We begin by choosing an interval a, b and a partition
P x 0 , x 1 , , x n
of a, b such that f x 0 whenever a number x lies outside the interval a, b and such that f steps
within the partition P. If the constant value of f in each interval x j 1 , x j is j then 290 Ý Þ Ý fd c n n f x j J c, x j
j0
n c x j n f x j J , x j c c c x j j 1 j1 j0 j xj xj 1 j1 cÞ Ý
Ý fd. Exercises on Elementary Sets
1. Given that A and B are elementary sets and is an increasing function, prove that
var , A Þ B var , A var , B var , A B . Solution: We begin by choosing a lower bound a and an upper bound b of the set A Þ B. By looking
at the different cases we can see easily that whenever x
a, b we have
A ÞB A B A B .
Therefore
var , A Þ B
Þ a AÞB d Þ a A B
b b A B d Þ a A d Þ a B d Þ a A B d
b b var , A var , B b var , A B. 2. Prove that if E is an elementary set and m E 0 then E must be finite.
Choose an interval a, b such that the function E is zero outside a, b and a partition
P x 0 , x 1 , , x n
of a, b such that E steps within P. Since E is nonnegative and its sum over P is zero, the
constant value of E in each interval x j 1 x j must be zero. Therefore E can be nonzero only at
points of P, in other words, every member of E is a point of P and we conclude that the set E is
finite.
3. Explain why the set of all rational numbers in the interval 0, 1 is not elementary.
The desired result will be clear when we have proved the stronger assertion that is made in
Exercise 4.
4. Prove that if E is an elementary subset of 0, 1 and if every rational number in the interval belongs to E then
the set 0, 1
E must be finite.
We assume that E is an elementary subset of 0, 1 and that every rational number in 0, 1 belongs
to E. Choose a partition
P x 0 , x 1 , , x n
of a, b such that E steps within P. For each j, since the interval x j 1 , x j contains some rational
numbers, the constant value of E in x j 1 , x j must be 1. Since the numbers in 0, 1 that do not
belong to E have to be points of P, the set 0, 1
E must be finite.
5. Give an example of a set A of numbers such that if E is any elementary subset of A we have...
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 Fall '08
 STAFF
 Math, Calculus

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