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ST 1801N3 - 06.11.2003 1.00 4.00 LOYOLA COLLEGE(AUTONOMOUS...

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Unformatted text preview: 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. 1. 2. 3. 4. 5. 6. C For a sequence {An} of sets, if An → A, show that An → AC . (10x2=20 marks) Define a monotone increasing sequence of sets and give its limt. Show that a σ - field is a monotone class. Define the indicator function of a set A. Show that the set rational numbers is a Borel set. 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 . 8. If Ω = {1,2,3,4}, is the power set of , ì {φ} = 0, ì {1} = , ì {1,2} = , ì {1,2,3} = ì (Ω) = 1, is ì a measure on (Ω, )? 9. If ì is a measure, show that ì d” . 10. If = [0,1] and ì 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. (5x8=40 marks) 11. Prove that there exists a unique and minimal σ-field on a given non - empty class of sets. 12. Define Borel σ - field of subsets of real line. Show that the minimal σ - field generated by the class of all open intervals is a Borel σ - field. (2+6) 13. a) Define a finitely additive and a countably additive set functions. b) Let Ω = {-3, -1, 0, 1, 3} and for A Ω, let λ (A) = with λ1 = min (λ, O), show that λ is not even finitely additive. 14. If λ is an extended real valued σ - additive set function on a ring ℜ such that λ(A) > - for every A ∈ ℜ, show that λ is continuous at every set A ∈ ℜ. 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 . 16. Define the Lebesgue - Stieltjes (LS) measure induced by a distribution function F on IR. If ì is the LS measure induced by F(x) = 1 - e-x if x > 0 0 if x d” 0, then find (a) ì (0, 2) (b) ì [-1, +1] and (c) ì (A), where A = {0, 1, 2, 3, 4}. (2+6) 17. Show that a measure on a σ - field can be extended to a complete measure. 18. State and establish Fatou’s lemma. SECTION-C Answer any TWO questions. (2x20=40 marks) 19. a) Distinguish between (i) a ring and a field (ii) a ring and a σ - ring. b) Define the minimal σ-field containing a given class of sets. Give an example. c) Show that the inverse image of a σ-field is a σ-field. 20. a) Define (i) extension of a measure (ii) completion of a measure. (6) b) State and prove the Caratheodory extension theorem. (14) 21. a) Prove that if 0 d” Xn X, then . (8) b) If X and Y are measurable functions on a measure space, show that . (12) 22. a) If X e” 0 is an integrable function, prove that ϕ (A) = A a measurable set, defines a measure, which is absolutely continuous with respect to the measure µ. (10) b) State and prove the Lebesgue “dominated” convergence theorem. Is the “denominated” condition necessary? Justify your answer. (10) ...
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