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(MATH301)2001-2008_f_final_MATH301_by_cs_xsx329.pdf
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Math 301 (Real Analysis) Fall 2008
Final Examination C (Duration: 120 minutes)
Directions: Thisisa closedbook exam. Works (including scratch works)mustbe shown legiblyto receive

credits. Answers alone are worth very little. Calculators are allowed.
Notations: R denotes the set of all real numbers. N denotes the set of allpositive integers. m(S)denotes
the Lebesgue measure of S. m .(S)denotes the outer measureof S.

Problems
.nx
1. (10 marks)For every n N, de.ne fn :[0, 1] R by fn(x)= nxe2 . Determine if
11
lim fn(x)dx = lim fn(x)dx
n n
00
by computingboth sides. Determine iffn(x)converges uniformly on [0, 1].
1
.(sin nx)lnx
2. (15 marks) Prove that lim dx =0.
n 2
1/n 1+ nx

3. (a) (5 marks) State the topological continuitytheorem.
(b) (10 marks)For each t R, let Jt be a nonemptyopen set onR suchthat R =
Jt. Let f :R R tR
be a function. For every t R, the function ft : Jt R de.ned by ft(x)= f(x)is continuous. Prove that f(x)is continuous onR.
4. Let X .R. We say X is a G-set in R if and only if there exists a sequence of open sets O1,O2,O3,... in R such that X = O1 O2 O3 .
(a)
(5 marks) Let f :R R be continuous. Prove that f.1({0})is a G-set in R.

(b)
(5 marks) State the de.nition of outer measure.

(c)
(15 marks) Let S be abounded measurable subset of R. Prove that there is a G-set X in R such that S .X and m(S)= m(X).


5.
(15 marks) Let f :R R satisfy the property that for all x, y R, wehave |f(x).f(y)|e|x|+|y||x .y|.

If E isabounded subset of R and m .(E)=0, then prove that m ..f(E).=0by using the structure theorem of open sets, where f(E)= {f(x): x E}.

6.
(a) (5 marks) Let f :[0, 1] R be measurable such that


|f|ln(1+ |f|)dm < +.
[0,1]
1
Prove that |f|dm < .
[0,1]
(b) (15 marks) Let g, h :[0, 1] R be Lebesgue integrable such that
gdm = h dm.
[0,1] [0,1]
Prove that either g = h almost everywhere on [0, 1] or there exists a measurable subset E of [0, 1] such
that E .gdm >
=[0, 1] and h dm.
EE
Notations: R denotes the set of all real numbers. m denotes the Lebesgue measure on R.
1. (20 marks) Consider the system of equations
x = cos(st)+t sin s and y = sin(st)+t cos s.

Show that near p =(s, t, x, y)=(0, 1, 1, 1), (s, t)canbe expressed asa di.erentiable functionof(x, y) .s.s.t .t
and .nd the value of ,, and at p.
.x.y.x .y
nx cos x
2. (a) (10 marks) Determine (with proof)whether fn(x)= converges uniformly on [0, 1] or not.
4+ n2x2
(b) (5 marks) State the Lebesgue Dominated Convergence Theorem.
1
nx cos x
(c) (15 marks) Determine lim
n
dx with proof. 4+ n2x2
0
3. Let E1,E2,E3,... be a sequence of nonempty measurable subsets of R such that every real number is

in only .nitely manyof the sets E1,E2,E3,... and m(Ek)< . For everypositive integer n, let
k=1
Hn = {x :x R and x is in at least n of the sets E1,E2,E3,...}.
(a)
(5 marks) Prove t