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(MATH301)[2001-2010](f)final~cyyuaa^_47550.pdf
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Directions: Thisisa closedbook exam. Correct detailed worksmustbe shown legibly
to receive credits. Answers alone are worth very little. Calculators are allowed. Notations: R denotes the set of all real numbers. Q denotes the set of all rational numbers. m(S)denotes the Lebesgue measure ofS. m.(S)denotes the outer measure of
S.
Problems
1. (20 marks) Consider the system of equations
cos(wy)=1+sin(xz) and x +y= cos(wz).

Show that near p=(w,x,y,z)=(1,.1,2,.2), (x,z)canbe expressed asa di.eren-.x .z
tiable function of(w,y)and .nd thevalueof and at(w,y)=(1,2).
.y .y
2. (a) (5 marks) State the Lebesgue dominated convergence theorem.
2n
.1
(b) (15 marks) Determine the value of lim sin.x.dx with proof.
n+ 1+ 1 x2 n


3.
(20 marks) Let P be the subset of [0,1] consisting of all numbers x suchthat (in base 10) every digit of x after the decimalpointisa primenumber (i.e. x =0.d1d2d3 ..., where the digits d1,d2,d3,... are all prime numbers). Prove that P is a measurable set and compute m(P)in details.

1 .

4.
Let f :R [.1,1]be measurable suchthat for every a R, lim fdm


h0+ 2h
[a.h,a+h]
is equal to a real number. De.ne g:R R by
1 .

g(x)= lim f dm.
h0+ 2h
[x.h,x+h]
(a)
(5 marks) If f is a continuous function, then prove that gis a measurable function.

(b)
(15 marks) If f is a discontinuous and measurable function, then prove that g is a measurable function.


5. (20 marks) Let E beabounded subsetofR suchthat m .(E)> 0. Prove that if c R and0 <c<m .(E), then there exists a subset F of E suchthat m .(F)= c.
CEnd ofPaperC
Math 301 (Real Analysis) Fall 2009
Final Examination C (Duration: 120 minutes)
Directions: Thisisa closedbook exam. Correct detailed worksmustbe 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
1. (20 marks) Consider the system of equations
wxyz =4 and cos(xy)+sin(wz)=1.

Show that near p =(w,x,y,z)=(1,.1,2,.2), (x,z)can be expressed as a di.erentiable function of .x .z
(w,y)and .nd thevalueofand at(w,y)=(1,2).
.y .y
2. (a)(5 marks)For all n N and x [0,+), let
.1if x [1/n,n] ne.x cos(x/n)
gn(x)= and fn(x)= gn(x).
0if x . [1/n,n] x +n
Determine whether fn converges uniformly on [0,+)or not.

n
.ne.x cos(x/n)
(b) (15 marks) Determine the value of lim dx with proof.
n+ x +n
1/n
3. (15 marks) Let f :[0,+)[0,1]be measurable. Prove that the set

S = .a : a [0,+) and .f(a +i)R.
i=1
is measurable.
4. (10 marks) Let f : R R be Lebesgue integrable. Let I be an interval such that m(I) > 0 and 1 . ..
a = f dm. If E = {x :x I and f(x)>a}, then prove that |f .a|dm =2 (f .a)dm.
m(I)
I IE
5.
(10 marks) Let f :R [0,1]bea function and de.ne g :R R by