LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS
SECOND SEMESTER – APRIL 2012
MT 2811 – MEASURE THEORY AND INTEGRATION
Date : 19-04-2012 Dept. No. Max. : 100 Marks
Time : 9:00 – 12:00
ANSWER ALL QUESTIONS:-
- (a) State and prove countable sub additive theorem for outer measures. (5)
(OR)
(b) Prove that every interval is measurable. (5)
(c) Prove that there exists a non measurable set. (15)
(OR)
(d) Show that Lebesgue measure is regular. (15)
- (a) Let f and g be non negative measurable functions. Then prove ò f dx + ò g dx = ò (f + g) dx . (5)
(OR)
(b) Prove that if the sequence is a sequence of non-negative measurable function
then . (5)
(c) State and prove Lebesgue Dominated Convergence theorem. (15)
(OR)
(d) If f is Riemann integrable and bounded over the finite interval [a,b] then prove that f
is integrable and . (15)
- (a) Show that with a usual notations the outer measure m* on H(Â),and the (5)
outer measure defined by on S( Â) and on contains are the same.
(OR)
(b) Prove that if m* is an outer measure on H(Â), defined by m on H(Â) then contains
, the -ring generated by Â. (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)
- (a) State and prove Holder’s inequality. (5)
(OR)
(b) Define the following terms: convergence in measure, almost uniform convergence and uniform convergence almost everywhere. (5)
(c) Let [X, S, ] be a measure space with . If is convex on (a, b) where and f is a measurable function such that , for all x, prove that . When does equality occur? (15)
(OR)
(d) State and prove completeness theorem for convergence in measure. Show that if almost uniform then in measure and almost everywhere. (15)
- (a) Define a positive set and show that a countable union of positive sets with respect to a
signed measure v is a positive set. (5)
(OR)
(b) Let v be a signed measure and let be measure on [X, S] such that are – finite, «, « then prove that . (5)
(c) Let v be a signed measure on [X, S]. (i) Let S and . Can you construct a positive set A with respect to v, such that and ? Justify your answer. (ii) Construct a positive set A and a negative set B such that . (15)
(OR)
(d) State and prove Lebesgue decomposition theorem. (15)
Latest Govt Job & Exam Updates: