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Unformatted text preview: ry number 0 there exist functions g and h that are RiemannStieltjes integrable with respect to
on the interval a, b such that g f h and Þa h
b g d . To show that condition a implies condition b we assume that condition a holds. In other words, we
assume that f is integrable with respect to on a, b . Suppose that 0. Using the first criterion
for integrability we choose a partition P of a, b such that Þ a w P, f d
b . We define S u P, f and s l P, f and observe that the functions s and S have the desired
properties.
The proof that condition a implies condition c is very similar. This time, the partition P is chosen
using the second criterion for integrability.
To prove that condition b implies condition a we assume that condition b holds. What we shall
show is that the first criterion for integrability holds. Suppose that 0. Using condition b we
choose step functions s and S on the interval a, b such that s f S and Þa S
b s d . Choose a partition P of a, b such that both s and S step within P and observe that since 308 s l P, f u P, f S we have Þ a w P, f d Þ a u P, f
b b l P, f d Þa S
b s d . The proof that condition c implies condition a is very similar. This time we show that f is integrable
by showing that the second criterion for integrability holds.
4. Suppose that is an increasing function, that f is a bounded function on an interval a, b and that for every
number 0 there exists an elementary subset E of a, b such that var , E and such that the function
f 1 E is integrable with respect to on a, b . Prove that f must be integrable with respect to on the
interval a, b . Solution: In order to show that f is integrable on a, b we shall show that f satisfies the second criterion for integrability. Suppose that 0. Using the given property of f we choose an elementary subset A of a, b such that var , A /2 and such
that the function f 1 A is integrable with respect to on a, b . Choose a partition P 1 of a, b such that
the function A steps within P. Now, using the fact that f 1 A satisfies the second criterion for
integrability we choose a partition P 2 of a, b such that if we definè
B x
a, b
w P, f 1 A x
.
then var , B /2. We now define P to be the common refinement of P 1 and P 2 and we express P as
P x 0 , x 1 , , x n .
For each j 1, 2, , n, if the open interval x j 1 , x j is not included in A Þ B then, since the functions f
and f 1 A agree in the interval x j 1 , x j the condition
w P, f x w P, f 1 A x
x j 1 , x j . Since
var , A Þ B
var , A var , B
2
2
we have succeeded in showing that f satisfies the second criterion for integrability. must hold whenever x 5.
a. Suppose that is an increasing function, that f is a nonnegative function defined on an interval a, b and
that for every number 0, the set
x
a, b
fx
is finite. Prove that f must be integrable with respect to on a, b and that Þ fd 0.
a
We shall show that f satisfies the first criter...
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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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