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:-

1. (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)

 

 

1. (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)                                                                                                                                                                                    

 

1. (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)

 

1. (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)

 

 

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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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