LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034SUPPLEMENTARY SEMESTER EXAMINATION – JUN 2006 M.Sc. DEGREE EXAMINATIONMT 2801 – MEASURE AND INTEGRATION
Date & Time : 27/06/2006/9.00 – 12.00 Dept. No. Max. : 100 Marks
ANSWER ALL :- MARKSI a (1) Define outer measure and show that it is translation invariant (8) (OR)
(2) State and prove countable sub additive theorem for outer measures.
b (1) Prove that there exists a non measurable set (17) (OR)
(2) Show that the following statements are equivalent for a set E : (i) E is measurable(i) 0 , G an open set ,G E, such that m(G – E) , (ii) G, Gδ -set, G E, such that m (G – E) = 0(iii) 0 , F a closed set, F E, such that m (E – F) ,(iv) F, an Fσ–set, F E, such that m (E – F) = 0 .
II. a. (1) If is a measurable simple function ,then in the usual notations prove (8) (i) dx = aį m ( A 0 for any measurable set E.
(ii) dx = dx + dx for any disjoint measurable sets A and B.
(iii) a dx = a dx if a 0.
(OR)
(2) Let f and g be non negative measurable functions.Then prove f dx + g dx = (f + g) dx
b. (1) State and prove Fatou’s Lemma for measurable functions (17) (OR)
(2) Show that if f is a non negative measurable function., then a sequence of measurable simple functions such that (x) f (x) . III a (1) Show that with a usual notations the outer measure on H(),and the (8) outer measure defined by on S( and on S are the same.
(OR)
(2) Let ‘s’ be a non negative measurable simple function defined on a measure space (X, S , ) Define (E) = s d then is a measure on (X, S ) and if ‘t’ is another non negative measurable simple function defined on a measure space (X, S, ) then prove that (s + t) d = s d + t d . . b (1) State and prove Holder’s’s inequality for convex functions (17)
(OR)
(2) (i) State and prove Jensen’s inequality for convex functions (8+9)
(ii) If f, g LP (, are complex numbers then prove that,
(fg) LP ( and (fg) d = f d + g d IV. a (1) Show that if be a sequence of sets in a ring R then there exisists a sequence of disjoint sets of R such that Bi Ai for each i and A = B for each N ,so that A i = Bi . (OR)
(2) State and prove ‘Egorov’s theorem for almost uniform convergence.
b. (1) State and prove ‘Completeness theorem’ for convergence in measure. (17) . (OR)
(2) (i) State and prove Reisz-Fisher’s theorem (8+9)
(ii) State and prove Jordan’s lemma.