MATH 6337: Homework 10 Solutions
8.11. If fk f in Lp , 1 p < , gk g pointwise, and |gk | M for all k , prove that
fk gk f g in Lp .
Solution. By Minkowskis inequality,
|fk gk f g |p |fk gk f gk |p + |f gk f g |p M |fk f |p + |f gk f g |p .
Since gk g poin
Homework #6 (Math 6337, due 3/8 in class)
1. #16 in page 148 of text.
2. #19 in page 148 of text.
3. #22 (a) in page 149 of text.
4. #24 in page 150 of text.
5. Let fn (x) (n = 1P; ) be monotone increasing functions on [a; b] and
;2
1
suppose that the ser
Homework #5 (Math 6337, due 2/27 in class)
1.
2.
3.
4.
5.
#4 in page 91 of text.
#7 in page 91 of text.
#17 in page 93 of text.
#19 in page 93 of text.
#24 in page 95 of text.
1
Homework #4 (Math 6337, due 2/15 in class)
1.
2.
3.
4.
5.
on E
#6 in page 91 of text.
#8 in page 91 of text.
#10 in page 91 of text.
#12 in page 92 of text.
1
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Homework #3 (Math 6337, due 2/6 in class)
1.
2.
3.
4.
5.
#17
#23
#27
#32
#33
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Math 6337 : Real Analysis I
Mid-term Exam 1
23 September 2011
Instructions: Answer all of the problems.
1. Recall that the outer measure of E Rn is dened to be
vol(Ij ) : cfw_Ij is a cover of E by closed intervals in Rn
|E |e = inf
j
Dene a closely relat
Math 6337 : Real Analysis I
Mid-term Exam 2
04 November 2011
Instructions: Answer all of the problems.
1. Use Fubinis Theorem to prove that
A
lim
A
Solution: Write
0
1
=
x
sin x
dx =
x
2
ext dt,
0
and then consider the following integral,
A
0
sin x
dx =
x
Math 6337 : Real Analysis I
Final Exam
14 December 2011
Instructions: Answer any 5 of the problems.
1. Prove that if 0 < < 1, there is no measurable subset E of R that satises
<
|E I |
<1
|I |
for every interval I in R.
Solution: We proceed by contradicti
MATH 6337: Homework 9 Solutions
8.1. For complex-valued, measurable f , f = f1 + if2 with f1 and f2 real-valued and measur
able, we have E f = E f1 + i E f2 . Prove that E f is nite if and only if E |f | is nite, and
2
2 1/2
f E |f |. [Hint: Note that E f
MATH 6337: Homework 8 Solutions
6.1.
(a) Let E be a measurable subset of R2 such that for almost every x R, cfw_y : (x, y ) E
has R-measure zero. Show that E has measure zero and that for almost every y R,
cfw_x : (x, y ) E has measure zero.
(b) Let f (
MATH 6337: Homework 7 Solutions
5.12. Give an example of a bounded, continuous function f on (0, ) such that lim f (x) =
x
0 but f Lp (0, ) for any p > 0.
/
Solution. An example is
f ( x) =
1,
1
,
ln(x)
xe
x e.
This function clearly satises the conditions
MATH 6337: Homework 6 Solutions
5.1. If f is a simple measurable function (not necessarily positive) taking values aj on Ej ,
j = 1, 2, ., N , show that
N
f=
aj |Ej | .
E
j =1
[Hint: Use (5.24). ]
Solution. Using the hint, we have
N
f=
E
N
N
f=
Ej
j =1
aj
MATH 6337: Homework 5 Solutions
4.14. Let f (x, y ) be as in Exercise 13. Show that, given > 0, there exists a closed set
F E with |E \ F | < such that f (x, y ) converges uniformly for x F to f (x) as y 0.
[Hint: Follow the proof of Egorovs theorem, usin
MATH 6337: Homework 4 Solutions
4.1. Prove the following
(i) Corollary 4.2: If f is measurable, then cfw_f > , cfw_f < +, cfw_f = +, cfw_a
f b, cfw_f = a, etc. are all measurable. Moreover, f is measurable if and only if
cfw_a < f < + is measurable for e
MATH 6337: Homework 3 Solutions
3.10. If E1 and E2 are measurable, show that |E1 E2 | + |E1 E2 | = |E1 | + |E2 |.
Solution. We may assume that both |E1 | , |E2 | < +, or else the result is trivially true.
Otherwise, since
(E1 E2 ) \ E2 = E1 \ (E1 E2 ),
E2
MATH 6337: Homework 2 Solutions
3.3. Construct a two-dimensional Cantor set in the unit square [0, 1] [0, 1] as follows:
subdivide the square into nine equal parts and keep on the four closed corner squares,
removing the remaining region (which forms a cr
MATH 6337: Homework 1
1.1. Prove the following facts, which were left as exercises.
(a) For a sequence of sets cfw_Ek , lim sup Ek consists of those points which belong to innitely
many Ek , and lim inf Ek consists of those points which belong to all Ek f
Homework #2 (Math 6337, due 1/25 in class)
1. An alternative denition of measurability is as follows: E
mesurable if for any set T Rn ;
m (T ) = m (T \ E ) + m (T
Show
2.
3.
4.
5.
6.
E) :
that this denition is equivalent with the one given in the text.
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Homework #1 (Math 6337, due 1/18 in class)
1. Prove the following facts mentioned in class.
(a) For a sequence of sets Ek , lim sup Ek consists of those points which belong
to innitely many Ek , and lim inf Ek consists of those points which belong to
all
MATH 6337: Homework 5 Solutions
4.14. Let f (x, y ) be as in Exercise 13. Show that, given > 0, there exists a closed set
F E with |E \ F | < such that f (x, y ) converges uniformly for x F to f (x) as y 0.
[Hint: Follow the proof of Egorovs theorem, usin
MATH 6337: Homework 4 Solutions
4.1. Prove the following
(i) Corollary 4.2: If f is measurable, then cfw_f > , cfw_f < +, cfw_f = +, cfw_a
f b, cfw_f = a, etc. are all measurable. Moreover, f is measurable if and only if
cfw_a < f < + is measurable for e
MATH 6337: Homework 3 Solutions
3.10. If E1 and E2 are measurable, show that |E1 E2 | + |E1 E2 | = |E1 | + |E2 |.
Solution. We may assume that both |E1 | , |E2 | < +, or else the result is trivially true.
Otherwise, since
(E1 E2 ) \ E2 = E1 \ (E1 E2 ),
E2
MATH 6337: Homework 2 Solutions
3.3. Construct a two-dimensional Cantor set in the unit square [0, 1] [0, 1] as follows:
subdivide the square into nine equal parts and keep on the four closed corner squares,
removing the remaining region (which forms a cr