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1f&1IfJT1f l ' i ' ~ \ I I l ! E P ~ *- tP x. :f;... ~ Copyright ReservedThe Chinese University of Hong Kong

..::. 0 - * -- .j:-& r ~ : ! i J 3 f+ ;;If Course Examination 2nd Term, 2010-11

*I- E l . M Q ~ & , t ~ ~ Course Code & Title: MATH40S0 Real a l y s i s _ +* 7t-iiTime allowed 2 hours 0 minutes - -=-- - ~ ~Student LD. No. Seat No. :

Answer all questions.

1. (a) Define the outer measure m* on the family 2lR. of all subsets of R(b) Show that m* is countably subadditive.(c) Show that m*(I ) equals the length l(I) if I is an intervaL(d) Show that there exist a countable family {Pn : n E N} of pairwise disjoint

subsets of IR such that m' (Q Pn ) # ~ m(Pn ).2. (a) State Fatou's Lemma and the Monotone Convergence Theorem; show that they

are equivalent (in the sense that if you assume one of them then the other oneholds).

(b) Let f be an integrable function on R and suppose that f ~ O. Show that, forany E > 0, there exists 8 > 0 such that Ii f I< E whenever A is of measuresmaller than O.

(c) Can the integrability or the non-negativity assumption for f in (b) be dropped?Substantiate your answer (provide a proof for a positive answer, and provide acounter-example for a negative answer).

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Course Code f+ El ,*" : MATH40S'O' .. ~ .=.. ~ (* . = . . ~ ) Page 2 of 2

3. (a) Show that a normed vector space (V, 1111) is a Banach space if every absolutelyconvergent series in V converges (that is, the partial sums of a series L0 Vn in

n= l

V converges whenever L0 Ilvnll < +00).n= l

(b) Let 1 < p < 00, and let E be a measurable set. Show that Lp(E) is a Banachspace (You may assume the Minkowski inequality but you are required to explainthe notation Lp (E).)

4. (a) Let f : JR ---+ JR be Riemann integrable over each bounded closed interval in[0, +00) such that

bA:= lim (R) r f(x)dx exists in JR.b-+oo JoProgressively for the following cases, show that f is Lebesgue integrable on[0, +00):(i) f ~ aon [0, +00)

b(ii) lim (n ) r If(x)ldx exists in Rb-+oo Jo(You may assume the knowledge that if f is Riemann integrable on a boundedclosed interval I then it is integrable on I.)

(b) Guess the limitI n (1 +;) n e-4x dx

and prove by Lebesgue theory that your guess is correct.

r- v End of Examination r-v