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
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
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
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 dis
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 inter
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
perform
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 me
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
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  >
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
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.
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 s
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,
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]).
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
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 approxim
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
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)