LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION - STATISTICS
FIRST SEMESTER – NOVEMBER 2004
ST 1902 - MEASURE THEORY
03.11.2004
Max:100 marks
9.00 - 12.00 Noon
SECTION - A
(10  2 = 20 marks)
Answer ALL the questions
1. Let {An, n ≥ 1} be a sequence of subsets of a set . Show that lim inf An C lim sup An.
2. Define minimal  - field.
3. What is a set function.?
4. Give an example of a counting measure.
5. Show that any interval is a Borel set but Borel set need not be an interval.
6. Define an Outer measure.
7. Define Lebesgue - Stieltjes measure.
8. Show that a composition of measurable functions is measurable.
9. Define a simple function with an example.
10. State Borel-Cantelli lemma.
SECTION - B
(5  8 = 40 marks)
Answer any FIVE questions.
11. If {Ai, i ≥ 1} is a sequence of subsets of a set  then show that


i 1
i 1
 Ai   (Ai  Ai 1C  .... A1C ).
12. If D is a class of subsets of  and A C , we denote D  A the class {B  AB  D}.
If the minimal  - field over D is  (D), Show that A (D  A) =  (D)  A.
13. Let 0 be a field of subsets of . Let P be a probability measure on 0. Let {An, n ≥ 1}
and {Bn, n ≥ 1} be two increasing sequences of sets such that lim An  lim Bn .
n 
n 
Then show that lim P ( An )  lim P ( Bn ).
n 
n 
14. State and establish monotone class theorem.
15. If h and g are IB - measurable functions, then show that max {f, g} and min {f, g} are
also IB - measurable functions.
16. If  is a measure on (, ) and A1, A2,... is a sequence of sets in
show that
, Use Fatou's lemma to
i)


  lim inf An   lim inf μ (A n )
 n 
 n 
ii)


If  is finite, then show that   lim sup An   lim sup μ (A n ) .
 n 
 n 
1
17. Define absolute continuity of measures. Show that  < <  if and only if λ < < .
18. State Radon - Nikodym theorem. Mention any two applications of this theorem to
probability / statistics.
SECTION - C
(2  20 = 40 marks)
Answer any TWO questions
19. a) Let {xn} be a sequence of real numbers, and let An = (-, xn). What is the connection
between lim sup xn and lim sup An ? Similarly what is the connection between
n 
n
lim inf xn and lim inf An.
n
n
b) Show that every  - field is a field. Is the converse true? Justify.
(8+12)
20. a) Let  be countably infinite set and let consist of all subsets of . Define
0
if A is finite
 (A) =  if A is infinite.
i)
ii)
Show that  is finitely additive but not countably additive.
Show that  is the limit of an increasing sequence of sets An with
 (An) = 0 n but  () =  .
b) Show that a  - field

is a monotone class but the converse is not true.
(7+7+6)
21. a) State and establish Caratheodory extension theorem.
b) If  hdμ exists and C  IR then show that

 Chdμ = C  hdμ .

(12+8)

22. a) State and establish extended monotone convergence theorem.
b) State and establish Jordan - Hahn Decomposition theorem.
(10+10)
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