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Unformatted text preview: m E 0 and if
E is any elementary set that includes A we have m E
1.
The set 0, 1
Q has the desired properties.
6. Given that E is an elementary set that is not closed and that F is a closed elementary subset of E, prove that
m E F 0. Solution: Choose a lower bound a and an upper bound b of the set E. Since the set E F is
elementary, if we want to show that m E F 0 then, from Exercise 2, all we have to show is that the set
E F cannot be finite. The fact that E F is not finite follows from the fact that finite sets are always 291 closed, that F is closed and that the set E, which isn’t closed is the union of the two sets F and E F. 7. Given that is an increasing function, that f is a step function, that E is an elementary set and that f x 0
whenever x R E, prove that
Ý ÞE fd Þ Ý fd. Solution: The desired equality follows at once from the definitions and the fact that
f f E .
8. Given that is an increasing function, that f and g are step functions, that E is an elementary set and that
fx
g x whenever x R, prove that ÞE fd ÞE gd.
Choose an interval a, b outside of which both of the functions f and g are zero. Since f E
follows from the nonnegativity property of integrals of step functions that g E , it ÞE fd Þ a f E d Þ a g E d ÞE gd
b b 9. Given that is an increasing function, that f is a nonnegative step function, that A and B are elementary sets
and that A B, prove that ÞA fd ÞB fd.
The desired inequality follows at once from the fact that f A f B . We choose an interval a, b that
includes the set B and use the nonnegativity property to obtain ÞA f Þ a f A Þ a f B ÞB f
b b 10. Given that is an increasing function, that f is a step function and that E is an elementary set, prove that ÞE fd ÞE f d. Hint: Use the fact that
f  E f E f  E . 11. Given that A and B are elementary sets, prove that ÞA B ÞB A m A B. Solution: Choose a lower bound a and an upper bound b of the set A Þ B. We see that
ÞA B d Þ a B A d Þ a A B d ÞB A d.
b b The fact that these expressions are equal to m A B follows at once from the fact that
AB A B.
For some additional exercises on the variation of a function on elementary sets click on the following icon.
. Additional Exercises on Elementary Sets and Infinite
Series
Suppose that is an increasing function. 292 1. Given that H is a closed elementary set and U n is a sequence of open elementary sets and that
Ý H Un,
n1 use this earlier exercise to deduce that, for some positive integer N we have
N var , H var , U n
n1 and deduce that
Ý var , H var , U n .
n1 Choose a positive integer N such that
N H Un.
n1 We have
var , U n
n1 Ý N N var , H var , U n
n1 var , U n .
n1 2. Given that E is an elementary set and that U n is a sequence of open elementary sets and that
Ý E Un,
n1 prove that
Ý var , U n . var , E
n1 Hint: Make use of the theorem on approximating by closed sets and open set...
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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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