LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – NOVEMBER 2003

ST-1801/S716 – MEASURE THEORY

06.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL the questions.                                                                              (10×2=20 marks)

 

1. For a sequence {A_n} of sets, if A_n A, show that .
2. Define a monotone increasing sequence of sets and give its limt.
3. Show that a s – field is a monotone class.
4. Define the indicator function of a set A.
5. Show that the set rational numbers is a Borel set.
6. If X is a simple function, show that is a simple function.
7. If X₁ and X₂ are measurable functions with respect to prove that max { X_1, X₂} is measurable w.r.t     .
8. If = {1,2,3,4},      is the power set of , μ {f} = 0, μ {1} = , μ {1,2} = ,

μ {1,2,3} =  μ (W) = 1, is μ a measure on (W,     )?

9. If μ is a measure, show that μ ≤ .
10. If = [0,1] and μ is the Lebesgue measure, write down the value of , where C is the set of rationals, A = [0, 3/4] and B = [1/2, 1].

 

SECTION-B

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

 

11. Prove that there exists a unique and minimal s-field on a given non – empty class of sets.
12. Define Borel s – field of subsets of real line. Show that the minimal s – field generated by the class of all open intervals is a Borel s – field.          (2+6)
13. a) Define a finitely additive and a countably additive set functions.
14. b) Let W = {-3, -1, 0, 1, 3} and for A W, let l (A) = with l¹ = min (l, O), show that l is not even finitely additive.
15. If l is an extended real valued s – additive set function on a ring  such that l(A) > – for every A Î Â, show that l is continuous at every set A Î Â.
16. If X₁ and X₂ are measurable functions w.r.t show that (X₁ + X₂) is also measurable w.r.t.    prove that lim inf X_n is measurable w.r.t      .
17. 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

 

• if x ≤ 0,

then find (a) μ (0, 2) (b) μ [-1, +1] and (c) μ (A), where A = {0, 1, 2, 3, 4}.                 (2+6)

17. Show that a measure on a s – field can be extended to a complete measure.
18. State and establish Fatou’s lemma.

 

SECTION-C

 

Answer any TWO questions.                                                                           (2×20=40 marks)

 

19. a) Distinguish between (i) a ring and a field (ii) a ring and a s – ring.
20. b) Define the minimal s-field containing a given class of sets. Give an example.
21. c) Show that the inverse image of a s-field is a s-field.
22. a) Define (i) extension of a measure (ii) completion of a measure. (6)
23. b) State and prove the Caratheodory extension theorem. (14)
24. a) Prove that if 0 ≤ X_n X, then . (8)
25. b) If X and Y are measurable functions on a measure space, show that

.                                                                                (12)

22. a) If X ≥ 0 is an integrable function, prove that j (A) = A a measurable set,

defines a measure, which is absolutely continuous with respect to the measure m.  (10)

1. b) State and prove the Lebesgue “dominated” convergence theorem. Is the

“denominated” condition necessary?  Justify your answer.                                     (10)

 

 

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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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