HW 12
1. Let (fn : R ![0; +1)1
n=1 be a sequence in L1 (R) such that fn (x) !
f (x) a.e. in R and
Z
Z
fn dx !
f dx as n ! 1 and f 2 L1 (R) .
R
R
Prove that for every Lebesgue measurable set E R,
Z
Z
f (x) dx !
f (x) dx as n ! 1.
E
E
Hint: Use the followin

HW 101
1. Prove the following version of Egoros theorem which includes the
case when (E) = +1:
Let E be a non-empty Lebesgue measurable set (possibly of innite measure), (fn : E ! R)1
n=1 be a sequence of Lebesgue measurable functions
and f : E ! R be suc

HW 111
1. Sec. 4.3:17. We have:
Z
f dx =
sup
h2L0 (E);0 h f =+1
E
because (E) = 0 implies that
in R, and E f 2 L0 (E).
R
E
hdx =
Z
R
hdx = 0
E
E f dx
R
=0(
Ef
= 0 a.e.
2. Sec. 4.3:19. Let In = n1 ; 1 , n = 1; 2; : . Let fn = In f , n 2 N. We
readily see t

HW 81
1. Let 'C : [0; 1] ! [0; 1] be Cantors function.
(a) Prove directly that 'C does not satisfy Lipschitzs condition (see
this hw for the denition of the Lipschitzs condition) on the closed
interval [0; 1].
(b) A function f : [a; b] ! R is said to sati

HW91
1. State and proof Egoros theorem for the case when fn (x) ! f (x) =
+1 a.e. in a Lebesgue measurable set E of nite measure.
A
Remark: fn
+1 as n ! +1 if 8 > 0 9N 2 N : n
N )
1
8x 2 A.
fn (x)
Solution. First we state Egoros theorem when f (x) = +1.
T

HW71
Problems #1, #2 and #3 deal with the metric outer measure.
Denition 1 Let (X; ) be a metric space and let
: 2X ! [0; +1] be an
outer measure on X. Then
is called the metric outer measure if for every
X
two sets E; F 2 2 such that
dist (E; F ) =
inf
x

1. Let (fn : R ![0; +1)1
n=1 be a sequence in L1 (R) such that fn (x) !
f (x) a.e. in R and
Z
Z
fn dx !
f dx as n ! 1 and f 2 L1 (R) .
R
R
Prove that for every Lebesgue measurable set E R,
Z
Z
f (x) dx !
f (x) dx as n ! 1.
E
E
Hint: Use the following well

HW 10
1. Prove the following version of Egoros theorem which includes the
case when (E) = +1:
Let E be a non-empty Lebesgue measurable set (possibly of innite measure), (fn : E ! R)1
n=1 be a sequence of Lebesgue measurable functions
and f : E ! R be such

1. Sec. 4.3:17.
2. Sec. 4.3:19.
3. Sec. 4.3: 23.
4. Sec. 4.3: 24.
5. Sec. 4.3: 25.
6. Show that the strict inequality in Fatous lemma is possible.
Hint. Consider the sequence (fn )1
n=1 , fn = [n;n+1] .
7. Sec. 4.3: 26.
8. Sec. 4.3: 27.
9. Sec. 4.6: 40.
1

1. State and proof Egoros theorem for the case when fn (x) ! f (x) =
+1 a.e. in a Lebesgue measurable set E of nite measure.
2. (a) Given n 2 N prove that there are unique k 2 N and m 2 0; 1; 2; :; 2k
such that
n = 2k + m:
(you can use the well-known fact

HW8
1. Let 'C : [0; 1] ! [0; 1] be Cantors function.
(a) Prove directly that 'C does not satisfy Lipschitzs condition on the
closed interval [0; 1].
(b) A function f : [a; b] ! R is said to satisfy Hlders condition with
exponent ; 0 < < 1, and constant M

Problems #1, #2 and #3 deal with the metric outer measure.
Denition 1 Let (X; ) be a metric space and let
: 2X ! [0; +1] be an
outer measure on X. Then
is called the metric outer measure if for every
X
two sets E; F 2 2 such that
dist (E; F ) =
inf
x2E;y2

HW4
1. Let X be an uncountable set and R 2X be the family of all sets in X
which are either countable or their complements (in X) are countable.
Prove that R is a ring.
2. Describe the minimal ring generated by F in the following cases:
Describe the minim

HW2
1. Prove that any set of triangles in R2 whose vertices have rational coordinates is countable.
Note: Let x1 ; x2 ; x3 2 R2 . Then the triangle T with vertices x1 ; x2 ; x3
is the union of points of the segments [x1 ; x2 ] ; [x2 ; x3 ] and [x1 ; x3 ].

1. Let X be a set and
Prove that A X is
F A0 .
: 2X ! [0; +1] be an outer measure on X.
-measurable if and only if for every E A and
(E [ F ) =
(E) +
(F ) .
2. Let X be a set,
: 2X ! [0; +1] be an outer measure on X. Let
E; F 2 2X and let one of these set

onto
1. (a) Let (A; ) be a well-ordered set, B
A and f : A ! B be an
order isomorphism of (A; ) onto (B; ). Prove that for every a 2 A,
a f (a).
(b) Let (A; ) be a well-ordered set and a 2 A. Recall that the set
Ia = fx 2 A j x
ag
is called an initial seg