SOLUTIONS TO HOMEWORK 3.
Given > 2, prove that the set of all real numbers x, that satisfy
p
1
x
q
q
for innitely many distinct rational numbers p/q , has measure zero.
Lets call E the set of all real numbers x that satisfy |x p/q | 1/q for
innitely many
Homework 6,
M381C, 56415, Fall 2012
Oct 3 Oct 10
0. Read the next page.
h Lq with h
q
1
p
+ 1 = 1. Given f Lp , show that there exists a function
q
= 1, such that f h = f p .
1. Let 1 < p < + and
Remark. When combined with Hlders inequality (2), the above
Homework 7,
M381C, 56415, Fall 2012
Oct 10 Oct 17
0. (Optional) Let E Rn be measurable, with |E | < +. If f L1 (E ) show that
f
p
f
as p .
1. Let E be measurable, with |E | = 0, and let 1 q < p +. Prove the following.
(a) If |E | < + then Lp (E ) Lq (E ),
M381C, 56415, Fall 2012
Homework 8,
Oct 17 Oct 24
1. ( Prelim Jan 2006) Let I = [0, 1] and f L1 (I ). Assume that there exists C > 0
such that
|f | C |E |1/2
E
for any measurable set E I . Prove that f Lp (I ) for 1 p < 2, but that this fails
in general f
Homework 9,
M381C, 56415, Fall 2012
Oct 24 Oct 31
1. ( Wheeden-Zygmund 7.11) Prove the following result concerning changes of variable. Let g be monotone increasing and absolutely continuous on [, ], and let f be
bounded and measurable on [a, b] = [g (),
Homework 10,
M381C, 56415, Fall 2012
Oct 31 Nov 7
1. Let be a continuous linear functional on Lp (I ), where I = [a, b] and 1 p < .
(a) Show that F (x) = ([a,x] ) denes an absolutely continuous function F on I .
(b) Let q = (p 1)/p. Show that there exists
Homework 11,
M381C, 56415, Fall 2012
Nov 7 Nov 14
1. Let C be a real symmetric n n matrix whose eigenvalues are all positive, and dene
g (x) = (2 )n/2 [det(C )]1/2 exp 1 x (C 1 x) ,
2
x Rn .
Prove the following. (See the next page for notation etc.)
(a) d
Homework 12,
M381C, 56415, Fall 2012
Nov 14 Nov 21
0. (Optional, Wheeden-Zygmund 10.9) The symmetric dierence of two sets A and
B is dened as A B = (A \ B ) (B \ A). Let (X, A, ) be a measure space and
identify two sets A, B A if (A B ) = 0. On the result
Homework 13,
M381C, 56415, Fall 2012
Nov 21 Nov 30
1. (Wheeden-Zygmund 10.10) If is an additive set function whose Jordan decomposition is = V V , dene
f d =
E
f dV
E
f dV ,
E
provided not both integrals on the right side are innite with the same sign. I
SOLUTIONS TO HOMEWORK 1.
( Wheeden-Zygmund 1.1.n) Prove the (corrected) Theorem 1.14. Correction: The sequence cfw_xk should be in E \ cfw_x0 .
We just prove the rst half of the Theorem for the lim sup:
M = lim supxx0 ;xE f (x) if and only if (i) there i
Homework 5,
M381C, 56415, Fall 2012
Sep 26 Oct 3
1. Let f L(Rn ). The indenite integral F : R of f is dened by F (E ) =
The Hardy-Littlewood maximal function f for f is dened by
f (x) = sup
Q
1
| Q|
E
f.
|f | ,
Q
where the sup ranges over all cubes Q = x
Homework 4,
M381C, 56415, Fall 2012
Sep 19 Sep 26
0. (Optional) If E0 and E1 are disjoint closed subsets of Rn , show that the following
function h is continuous on Rn :
h(x) =
dist(x, E0 )
.
dist(x, E0 ) + dist(x, E1 )
1. Prove that if f is a measurable
Homework 3,
M381C, 56415, Fall 2012
Sep 12 Sep 19
1. Given > 2, prove that the set of all real numbers x, that satisfy
x
p
1
q
q
for innitely many distinct rational numbers p/q , has measure zero.
2. (Prelim Aug 2008) Let be a measure on the Borel subsets
SOLUTIONS TO HOMEWORK 4.
(Optional) If E0 and E1 are disjoint closed subsets of Rn , show that the
following function h is continuous on Rn :
dist(x, E0 )
h(x) =
.
dist(x, E0 ) + dist(x, E1 )
The distance function with respect to a closed set E is Lipschi
SOLUTIONS TO HOMEWORK 5.
Let f L(Rn ). The indenite integral F : R of f is dened by
F (E ) = E f . The Hardy-Littlewood maximal function f for f is dened
by
1
|f | ,
f (x) = sup
Q |Q| Q
where the sup ranges over all cubes Q = x + [r, r]n with r > 0.
(a) S
SOLUTIONS TO HOMEWORK 5.
1
Let 1 < p < + and p + 1 = 1. Given f Lp , show that there exists a
q
q with h
function h L
f h = f p.
q = 1, such that
For non negative functions f and h and conjugate exponents p and q ,
Hlders inequality
o
1/p
1/q
fp
fh
hq
,
SOLUTIONS TO HOMEWORK 7.
(Optional) Let E Rn be measurable, with |E | < +. If f L1 (E )
show that f p f as p .
Let M such that |cfw_f > M E | > 0. Then by Chebyshevs,
f
p
M |cfw_f > M E |1/p .
Sending p we obtain, lim inf p f
f
p
M . Now we use that
=
SOLUTIONS TO HOMEWORK 8.
( Prelim Jan 2006) Let I = [0, 1] and f L1 (I ). Assume that there
exists C > 0 such that
|f | C |E |1/2
E
for any measurable set E I . Prove that f Lp (I ) for 1 p < 2, but that
this fails in general for p = 2 (give a counterexam
Midterm Exam 1
10032012
Real Analysis, M381C 56415
Solve 4 of the following 5 problems.
1. Let E, E1 , E2 , . . . Rn be measurable, with Ek E and |E \ Ek | 0 as k . If
f : E R is a function whose restriction to each Ek is measurable, show that f is
measur
Midterm Exam 2
11052012
Real Analysis, M381C 56415
Try to solve 4 of the following 5 problems.
1. Let f be measurable on I = [0, 1] and nite almost everywhere. Dene F (x, y ) =
f (x) f (y ). If F belongs to Lp (I I ), with 1 p < , show that f Lp (I ).
2.
Homework 1,
M381C, 56415, Fall 2012
Aug 29 Sep 5
1. (Wheeden-Zygmund 1.1.n) Prove the (corrected) Theorem 1.14.
Correction: The sequence cfw_xk should be in E \ cfw_x0 .
2. ( Wheeden-Zygmund 1.3) For sets E1 , E2 , . . . X , show that
(a) X \ lim sup Ek
Homework 2,
M381C, 56415, Fall 2012
Sep 5 Sep 12
1. ( Wheeden-Zygmund 3.9) Prove the following version of the Borel-Cantelli Lemma:
If (E1 , E2 , . . .) is a sequence of sets with
outer measure zero.
k
|Ek |e < +, show that lim sup Ek has
2. (Wheeden-Zygm
SOLUTIONS TO HOMEWORK 2.
( Wheeden-Zygmund 3.9) Prove the following version of the BorelCantelli Lemma: If (E1 , E2 , . . .) is a sequence of sets with k |Ek |e < +,
show that lim sup Ek has outer measure zero.
Given > 0 the convergence of the series impl