MAR ATHANASIUS COLLEGE
MODEL QUESTION PAPER
PC3-MT01C03 MEASURETHEORY AND INTEGRATION
PART A Short Answer Questions( Answer any 5 questions. Each question has weight 1
.)
1) Define Lebesgue Outer measure of a set A. Show that [0,1] uncountable.
2) Define measurable function and show that the composition of a continuous and measurable function is measurable.
3) If ‘ f ‘is non –ve integrable function and F is defined by F(x) = x


. Prove that F is continuous
on R.
4) Show that if ‘ f ‘ is integrable then f is finite valued a.e .
5) Define signed measure. Show that
 
E
 fd

is a signed measure.
6) Define outermeasure
 
induced by measure

on an algebra A .Prove that if A

A then
 
( )
 
A .
7) Show that if f

L
1
  1
  1
( ) and g L ( ) then fg L (
 
) .
8) Define convergence in measure and almost uniform convergence. Prove that almost uniform
convergence imply convergence in measure.
PART B Short Essay ( Answer any 5 questions. Each question has weight 2
.)
9) Prove that the outer measure of an interval is its length.
10) Prove that there exist measurable set which is not a Borel set.
11) Let f and g are bounded measurable functions and f=g a.e then
E
 fdx
 gdx
E

.Prove. Is the
converse true.Justify.
12) State and prove Lebesgue Dominated convergence theorem for measurable functions.
13) Let E be a measurable set such that 0
 
( )
 
.Then there is a +ve set A contained in E with

( )

0 .Prove

14) Let ( f n
) be a sequence of non –ve measurable functions that converges a. e on a E to a function f
then

E fd
  lim
E
 f d n

15) State and prove Egorov’s Theorem.
16) By integrating e
 xy sin 2 y with respect to x and y show that


0 e
 y sin 2 y dy y
 tan 2 .
PART C Essay ( Answer any 3 questions. Each question has weight 5
.)
17) Define measurable set and prove the following. a)If

( )

0 then A is measurable. b)The class M of Lebesgue measurable sets is an algebra. c) Every interval is measurable. d) If E and E are measurable then
1 2
(
1

E
2
)

(
1

E
2
)

(
1
)

(
2
) e)Show that there exist a non measurable set.
18) A) If f and g are integrable functions show that a) af integrable and
 afdx
  b) f+g integrable and
 f
 g dx
  fdx
  gdx c) f integrable and
 fdx
  f dx .
B) Let f be an increasing real valued function on the interval [a,b].Then show that f is differentiable a.e and the derivative f’mesasurable and
 b a
'
( )

( )

( ) .
19) A)State and Prove Bounded convergence theorem.
B) Let f be a bounded function defined on a measurable set E with m(E)<

.Show that f is measurable iff f inf
 

E
 x dx
 sup
  f

E

( ) .
20)State and Prove a)Hahn decomposition theorem for signed measure.

b) State and Prove Lebesgue decomposition theorem.
21) Let
 be a measure defined on algebra A and
 
the outermeasure induced by

. Then the restriction

of
 
to the
 
measurable sets is an exention of

to the
  algebra containing A.
If
 algebra containing A.

.If
   only measure on the smallest
 
22) Let [

   
be

-finite measure space.For V x
 
( V x
 y
 
( V y
) for x

X and y
  
  write measurable and
 
X
     
Y

.Also show that the

-finite condition cannot be dropped.