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

 

Answer ALL the questions                                                                            (10 ´ 2 = 20 marks)

 

1. Let {A_n, n ≥ 1} be a sequence of subsets of a set W. Show that lim inf A_n C lim sup A_n.
2. Define minimal s – 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

 

Answer any FIVE questions.                                                                          (5 ´ 8 = 40 marks)

 

11. If {A_i, i ≥ 1} is a sequence of subsets of a set W then show that

(A_i ).

 

12. If D is a class of subsets of W and A C W, we denote D A the class {B A½B Î D}.  If the minimal s – field over D is    _W (D), Show that    _A (D  A) =     _W  (D)

 

13. Let ₀ be a field of subsets of W.  Let P be a probability measure on    ₀.  Let {A_n, n ≥ 1} and {B_n, n ≥ 1} be two increasing sequences of sets such that .        Then show that

 

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 m is a measure on (W, ) and A₁, A₂,… is a sequence of sets in    , Use Fatou’s lemma to show that
17. m
18. If m is finite, then show that m .

 

17. Define absolute continuity of measures. Show that l < < m if and only if  < < m.

 

18. State Radon – Nikodym theorem. Mention any two applications of this theorem to probability / statistics.

 

SECTION – C

 

Answer any TWO questions                                                                          (2 ´ 20 = 40 marks)

 

19. a) Let {x_n} be a sequence of real numbers, and let A_n = (-¥, x_n). What is the connection

between  sup x_n and   Similarly what is the connection between

inf  x_n and  inf A_n.

 

1. Show that every s – field is a field. Is the converse true?                        (8+12)

 

20. a) Let W be countably infinite set and let consist of all subsets of W.  Define

0       if A is finite

m (A) =     ¥     if A is infinite.

 

1. Show that m is finitely additive but not countably additive.
2. Show that W is the limit of an increasing sequence of sets A_n with

m (A_n) = 0 “_n but m (W) = .

 

1. b) Show that a s – field _s is a monotone class but the converse is not true.            (7+7+6)

 

21. a) State and establish Caratheodory extension theorem.

 

1. b) If exists and C Î IR then show that = .                           (12+8)

 

22. a) State and establish extended monotone convergence theorem.

 

1. b) State and establish Jordan – Hahn Decomposition theorem. (10+10)

 

 

 

 

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