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Unformatted text preview: ck on the icon 310 . Positive Integrable Functions Have Positive Integrals
Suppose that is an increasing function, that f is a nonnegative function that is integrable with respect to on
an interval a, b . Suppose that var , a, b 0 and that Þ a fd 0.
b 1. Prove that for every number 0 there exists an elementary set E such that var , E
x
a, b
fx
E. and such that This exercise is a duplicate of the last exercise in the exercises on exercises on integrability
2. Prove that if, for every positive integer n, we choose an elementary set E n such that
var , a, b
var , E n
2n
and such that
var , a, b
x
a, b
fx
En
2n
then for every elementary E satisfying
Ý E En
n1 we have
var , E var , a, b .
To obtain this proof you will need to make use of the special group of exercises on elementary sets that can . be reached by clicking on the icon The existence of the sets E n follows from Exercise 1. Now if
Ý E En
n1 then the special exercises on elementary sets guarantee that
Ý Ý var , E n var , E
n1 n1 var , a, b
2n var , a, b . 3. Prove that if the sets E n are defined as in Exercise 2, the set
Ý a, b En
n1 must be nonempty and deduce that there must exist a number x
a, b such that f x 0.
Ý
The elementary set a, b can’t be a subset of n1 E n because we do not have
var , a, b var , a, b .
Thus there must be numbers x in a, b that do not belong to any of the sets E n and since any such
number x must satisfy the inequality
var , a, b
fx
2n
for every positive integer n we have f x 0 for such numbers x.
4. Improve on the preceding exercises by proving that for every number
elementary sets such that the following two conditions hold:
a. For every number satisfying 311 0 there exists a sequence E n of Ý x En a, b n1 we have f x 0.
b. For every elementary set E satisfying
Ý E En
n1 we have var , E .
Suppose that 0. We choose a sequence E n of elementary sets such that the conditions
var , E n n
2 and
x a, b fx En
2n
hold for each n. Continue as above. Of course, we don’t have a guarantee that a, b
nonempty unless b a. Ý
n1 E n is Some Exercises on the Junior Lebesgue Criterion
1. True or false: Every step function satisfies the junior Lebesgue criterion.
Of course this statement is true. The set of discontinuities of a step function, being finite, is an
elementary set. If we express this set as
E t j j 1, 2, , n
then
n J , t j . var , E
j1 2. Suppose that x n is a convergent sequence in an interval a, b and that f is a bounded function on a, b that
is continuous at every member of a, b that does not lie in the range of the sequence x n . Prove that f is
Riemann integrable on a, b . Solution: See the solution to Exercise 4.
3. Suppose that x n is a convergent sequence in an interval a, b that is an increasing function and that f is a
bounded function on a, b that is continuous at every member of a, b that does not lie in the range of the
sequence x n . Prove that f is RiemannSti...
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 Fall '08
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 Math, Calculus

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