LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – MATHEMATICSSECOND SEMESTER – NOVEMBER 2012MT 2811 – MEASURE THEORY AND INTEGRATION
Date : 06/11/2012 Dept. No. Max. : 100 Marks Time : 1:00 – 4:00
ANSWER ALL QUESTIONS EACH QUESTION CARRIES 20 MARKS : 5 x 20 = 100
I. (a) Define outer measure and show that it is translation invariant (5)(OR)(b) Prove that B is the algebra generated by each of the following classes: the open intervals open sets, the sets, and the sets. (5)
(c) Prove that the outer measure of an interval equals its length (15)(OR)(d) Prove that Not every measurable set is a Borel set. (15)
II. (a) State and prove Lebesque Monotone Convergence theorem. (5)(OR) (b) Prove that if f is a non negative measurable function then there exists a sequence (5) of measurable monotonically increasing simple function such that .(c) State and prove Fatou’s Lemma for measurable functions. (15)(OR)(d) State and prove Lebesgue Dominated Convergence theorem. (15)
III. (a) Show that if is a sequence in a ring Âthen there is a sequence of disjoint sets of  such that for each i and for each N so that (5)(OR) (b) Prove that with a usual notations the outer measure on H(Â),and the outer measure outer measure defined by on S( Â) and on are the same. (5)
(c) Show that if is a measure on a -ring then the class of sets of the form for any sets E,N such that While N is contained in some set in of zero measure is a -ring and the set function defined by is a complete measure on . (15)(OR) (d) Prove that if is an outer measure on H(Â),. Let denote the class of Measurable sets then Prove that is a – ring and restricted to is a complete measure. (15)
IV. (a) Prove that space is a vector space for . (5)(OR)(b) State and prove Minkowski’s inequality. (5)
(c) State and prove Jensen’s inequality. Also prove that every function convex on an open interval is continuous. (15)(OR)(d) Prove that where is convex on (a, b) and . Also prove that a differentiable function is convex on (a, b) if and only if is a monotone increasing function. (15)V. (a) Define the following terms: total variation, absolutely continuous, and mutually singular with respect to signed measure. (5)(OR)(b) Let v be a signed measure on [X, S]. Construct the measures v+ and v- on [X, S] such that v = v+ – v- and v+ ┴ v-. (5)
(c) If , , and are – finite signed measure on [X, S] and « , « then prove that . Also prove that a countable union of positive sets with respect to a signed measure v is a positive set. (15)(OR)(d) State and prove Hahn decomposition theorem. (15)
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