Homework 7 Hints
7.6
Check the derivative of x , x1 , satises 7.29.
7.10
Note the fact that the image of [ai , bi ] is an interval of length at most V (bi ) V (ai ). Then for a set Z
with measure zero, we have for any > 0, Z G where G  < . Use the fact
Homework 4 Hints
4.15
Proof 1:
Let En = cfw_x E : k, fk (x) n, then En
. Since E0 = E is of nite measure, so
limn En  =  limn En  = 0, so for any > 0, there is N with EN  < 2 , then on E\EN we have
fk (x) < N for any k, since EN is measurable, s
Name
M381C Exam 2
Instructions: Do as many problems as you can. Complete solutions (except for minor aws)
to 23 problems will be considered a good performance.
1. Suppose is a realvalued function on R such that
1
1
f (x) dx
(f (x) dx
0
0
for every real
Homework 3 Hints
25
We will follow the hint on the book. We rst construct a Cantortype set E1 with positive measure
1 1 the same manner as (5) in HW2. Then the removed set is a countable union of disjoint open
intervals, on each subinterval(we may includ
Homework 2 Hints
4
Perfect set: We note that at each stage, the end points of the closed intervals are kept. Then we note
that the points in the set, they are either (a) an end point of a closed interval that is kept at some
stage or (b) falls in a closed
Name
M381C Final Exam
Instructions: Do as many problems as you can in 3 hours. No notes, books, googling,
etc. Complete solutions (except for minor aws) to 4 problems will be considered a good
performance.
Please place your nal under my oce door (RLM 9.15
Name
M381C Exam 1
Instructions: Do as many problems as you can. Complete solutions (except for minor aws)
to 4 problems will be considered a good performance.
1. Let E R be a set with positive nite measure. Show that for any number 0 < c < 1
there exists
Homework 1 Hints
1
(k)
(i)
Let K be a compact set.
compact closed : We will show K c is open. Consider a point z K c , for each point x K, we
can chose two open balls Ux centered at z and Vx centered at x with radius d(z,x) . All such Vx forms
3
a open co
Homework 6 Hints
7.1
Since m = m(cfw_f  > 0) > 0, so there is > 0 such that m(cfw_f  > ) > m . Then there is r such
2
that m(cfw_f  > cfw_x < r) > m and we note that cfw_f >cfw_x<r f  > m . For any x, a ball
4
4
centered at x with radius x
Homework 10 Hints
6.1
(a) Apply Tonellis to E (x, y).
(b) Apply Tonellis to f , we will see all the three integrals are nite.
6.5
(a) If (y) = for some y > 0, then E f = 0 (y)dy = . Otherwise (y) is a welldened decreasing function on (0, ), so it has at m
Homework 8 Hints
8.4
Holders inequality:  f g f p gp . When f p = gp = 1.

f g
f g
gp
f p
+
p
p
E
f p gp
=
Equality holds in the rst inequality i f g is always of the same sign. In the second one, equality
p
p
f
g
holds i f p =
Homework 13 Hints
1
(b)(c): This is easily seen using the fact is a probability measure.
(a)(b): Let fk (x) = maxcfw_0, 1 k d(x, F ). So we gave a sequence of functions fk Cc (X) and
fk
F . Then lim supn n (F ) lim supn fk dn = fk d (F ).
1
1
(b),(c)(a):
Homework 12 Hints
8.8
Following the hint in the book,
p1
p
f (x, y)dx dy
=
f (x, y)dx dy
f (z, y)dz
f (z, y)dz
p1
f (x, y)dy dx
1/p
(p1)p
f (z, y)dz
=
f (z, y)dz
11/p
p
Divide both side by
f (z, y)dz
p
dy
11/p
1/p
f (x, y)p dy
dy
1/p
f (x, y)p dy
M381C Practice for the nal
1. Let f L1 ([0, 1]). Prove that
1/p
1
p
f  dm
lim
p0
1
log f  dm
= exp
0
0
where, by denition, exp() = 0. To simplify the problem, you may assume log f 
L1 ([0, 1]). Hint: rewrite the left hand side in a form to which yo
Homework 10 Hints
1
N
Ak = (N )Ak ( +1 Ak ). By nitely additivity, we have (Ak ) = k=1 (Ak ) + (BN ),
k=1
k=N
where BN = +1 Ak . BN
, so (BN ) 0. Thus we have the countably additivity.
k=N
2
1
(i) = 1 (), so A.
1
1
(ii) If A A, then for some n and B measu
Homework 14 Hints
1
Follow the hint.
2
First we note that the span of cfw_en is dense in C([0, 1]). Then we can approximate each en by its
Taylor series expansion at x = 0 which is a uniform approximation on [0, 1]. Thus we can show
polynomial is dense i
Homework 5 Hints
10
From Lp convergence and 5.9 in the last homework, we see there is a subsequence fkj of fk converges
to f a.e. in E. Then using Fatous Lemma, we can show that E f p lim inf E fkj p M .
11
For 0 < p < 1, 1/x Lp (0, 1), for p > 1, 1/x
Homework 9 Hints
2
f measurable so cfw_x : f (x) > a measurable for any a. Since cfw_x : g(x) > a diers from cfw_x : f (x) > a
by a set of measure zero and the measure space is complete, cfw_x : g(x) > a is also measurable. For
incomplete measure space, i