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)

 

 

Go To Main page