06.11.2003
1.00 - 4.00
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION - STATISTICS
FIRST SEMESTER – NOVEMBER 2003
ST-1801/S716 - MEASURE THEORY
Max:100 marks
SECTION-A
Answer ALL the questions.
(10x2=20 marks)
1. For a sequence {An} of sets, if An  A, show that AnC  A C .
2.
3.
4.
5.
Define a monotone increasing sequence of sets and give its limt.
Show that a  - field is a monotone class.
Define the indicator function of a set A.
Show that the set rational numbers is a Borel set.
6. If X is a simple function, show that e X is a simple function.
7. If X1 and X2 are measurable functions with respect to
prove that max { X1, X2} is
measurable w.r.t .
1
1
8. If  = {1,2,3,4},
is the power set of  , μ {} = 0, μ {1} = , μ {1,2} = ,
2
4
3
μ {1,2,3} = , μ () = 1, is μ a measure on (, )?
4
 




9. If μ is a measure, show that μ   An  ≤   ( An ). .
 n  1  n 1


10. If  = [0,1] and μ is the Lebesgue measure, write down the value of
 
C
 2  A   B d , where C is the set of rationals, A = [0, 3/4] and B = [1/2, 1].

SECTION-B
Answer any FIVE questions.
(5x8=40 marks)
11. Prove that there exists a unique and minimal -field on a given non - empty class of sets.
12. Define Borel  - field of subsets of real line. Show that the minimal  - field generated by
the class of all open intervals is a Borel  - field.
(2+6)
13. a) Define a finitely additive and a countably additive set functions.
b) Let  = {-3, -1, 0, 1, 3} and for A  , let  (A) =
K
with 1 = min (, O), show
K A
that  is not even finitely additive.
14. If  is an extended real valued  - additive set function on a ring  such that (A) > - 
for every A  , show that  is continuous at every set A  .
15. If X1 and X2 are measurable functions w.r.t
show that (X1 + X2) is also measurable
w.r.t. prove that lim inf Xn is measurable w.r.t
.
1
16. Define the Lebesgue - Stieltjes (LS) measure induced by a distribution function F on IR.
If μ is the LS measure induced by
F(x) = 1 - e-x if x > 0
0
if x ≤ 0,
then find (a) μ (0, 2) (b) μ [-1, +1] and (c) μ (A), where A = {0, 1, 2, 3, 4}.
17. Show that a measure on a  - field can be extended to a complete measure.
18. State and establish Fatou's lemma.
(2+6)
SECTION-C
Answer any TWO questions.
(2x20=40 marks)
19. a) Distinguish between (i) a ring and a field
(ii) a ring and a  - ring.
b) Define the minimal -field containing a given class of sets. Give an example.
c) Show that the inverse image of a -field is a -field.
20. a) Define (i) extension of a measure (ii) completion of a measure.
b) State and prove the Caratheodory extension theorem.
21. a) Prove that if 0 ≤ Xn  X, then
X

n
(6)
(14)

d  X d .
(8)

b) If X and Y are measurable functions on a measure space, show that
 ( X Y ) d   X d   Y d .


(12)

22. a) If X ≥ 0 is an integrable function, prove that  (A) =
 X d , A a measurable set,
A
defines a measure, which is absolutely continuous with respect to the measure . (10)
b) State and prove the Lebesgue "dominated" convergence theorem. Is the
“denominated" condition necessary? Justify your answer.
(10)

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