UNIVERSITY OF MALTA UNIVERSITY OF MALTA

FACULTY OF SCIENCE

Department of Mathematics

B.Sc. Part II

May/June Session 1996


MA224 Integration 27 May 1996

1045-1200


Answer TWO questions

1. Let I be an interval. Describe how the set L(I) of Lebesgue integrable functions is constructed and how the Lebesgue integral
ó
õ

I 
 f
for f Î L(I) is defined starting from the integral of a step function. State clearly all definitions you use and all results which are needed to ensure well-definition of the integral.

Let f:[0,1]® \Bbb R be defined by

f(x) = ì
ï
ï
í
ï
ï
î
1
2
x  is irrational
1
x  is rational
Show that f is Lebesgue integrable but not Riemann integrable on [0,1].

[You may assume, without proof, the fact that the set of rationals in [0,1] has measure zero. Also, results about Riemann integrability in terms of upper and lower Riemann sums may be used without proof. However, the use of any of these results must be clearly explained.]



2. Let f:I® \Bbb R where I = [a,b] and suppose f Î L(I). For any x Î [a,b] let Ix = [a,x] and define F0:[a,b]®\Bbb R by

F0(x) = ó
õ

Ix 
 f.
Suppose f is continuous at p Î [a,b]. Prove that F0¢(p) = f(p). State clearly the basic properties of the integral which are required in order to obtain this result.

Show also that if f is continuous on I and F is any primitive of f on I, then

ó
õ

I 
 f = F(b) - F(a).

For x > 0, the function l is defined by

l(x) = ó
õ 		x

1 
 1
t
 dt.
By differentiating xl(x)-x, find
ó
õ 		a

1 
 l
in terms of l(a), where a > 0.



Note. In the following question, all integrals are Lebesgue integrals.

3. State carefully, but without proof, Levi's Monotone Convergence Theorem and Lebesgue's Dominated Convergence Theorem for sequences of Lebesgue integrable functions.

Let f,g be defined on I = ]0,¥[ by f(x) = e-xxa-1 and g(x) = e-x/2, where a is a constant greater than 0.

[Hint for (iii): Since e-x/2xa-1 ® 0 as x®¥, it is bounded on [1,¥[.]

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