MAMT08
MAMT08
Examination December 2011
M.A. (Final)
Mathematics
First Paper
Measure and Integration
Time : 3 Hrs.
M.M. : 70
Note: This Question Paper consists of 4 Sections. Read instructions carefully
before attempting the Question.
(Section–A)
All Questions are Compulsory.
101=10
1.
Define Measurable space.
2.
Define outer measure.
3.
Define Limit superior and Limit Inferior and give examples.
4.
State Fubini’s theorem and give example.
5.
State “Minikowski’s inequality” ?
6.
Define the conjugate number of the LP- Space?
7.
State Random Nikodym Theorem.
8.
Define the Lebesgue Measurable function and its properties.
9.
State Hahn Decomposition Theorem.
10. Define Lebesgue integral of a function f  x  over E.
Section–B
Attempt any Ten Question from the following.
102=20
11. To prove that an outer measure is monotonic and   sub additive.
12. If S1 and S2 are measurable sets and if S1  S2   , then
m  S1  S2   m  S1   m  S2  .
13.
14.
15.
16.
To prove that every monotonic sequence of sets is convergent.
If f and g are measurable functions defined over a measurable set E and if
g vanishing nowhere on the set E, then the quotient function f/g is
measurable over E.
State and prove the “Lebesgue decomposition theorem”?
If f and g are bounded measurable functions on a set E of finite measure,
then prove that
  af  bg   a
E
17.
(2)
E
f  b g
E
To prove that a sequence  f n  of functions belonging to an LP- Space
has atmost one limit.
(1)
MAMT08
18.
19.
To prove that if the outer measure of a set is zero, then the set is
measurable.
If f  L2 0,1 , then show that

1
0
20.
21.
22.
MAMT08
f  x  dx     f  x  2 
 0

1
Show that if f  x  
x
1
3
E
29.
, 0  n  1 and f  0  0
Is Lebesgue integral over 0,1 .
23.
24.

f  x  dx    f  x  dx.
k 1
Ek
If f and g be measurable functions on  a, b . If f and g are square integral,
then fg  L  a, b and 

b
a
fg 

b
a
nx
for 0  x  1 .
1  n2 x2
x p 1
1
1
1
Let p  0, q  0 then show that 
dx  

 ....
0 1  x2
p p  q p  2q
and deduce that
1 1 1
   ...
2 3 4

1 1 1
 1     ...
(ii).
4
3 5 7
(i). log 2  1 
Section–C
Attempt any Two Question from the following.
2×10=20
26.
If f be a bounded measurable function defined over
a measurable set E such that
f  x  0
ii.
 f  x  dx  0,
E
then prove that f  x   0 almost every where in E.
27.

Show that LP , d
 is a metric space.
(2)
f dx
   g dx 
1
2
b
2
1
2
a
State and prove the Fundamental theorem of Integral calculus?
Section-D
Attempt any One Question from the following
31. State and prove Lebesgue Convergence theorem.
32. State and prove Riesz- Fischer theorem.
1
i.
2
30.
State and prove Lebesgue monotone convergence theorem.
Show that the theorem of bounded convergence is applicable to
fn  x  
25.
Let f be a bounded measurable function defined over a measurable set E
and let E be the union of pair wise disjoint set Ek. Show that

1
2
To show that an open set in a metric space is measurable space.
If f and g are real valued measurable functions on a measurable space X.
Prove that f  g and f  g are measurable functions.
1
28.
(3)
1×20=20