King Saud university
final exam Math 580 (Measure Theory ) .
College of Sciences
Second semester 1432-1433H Time : 3 hours . (50) marks
Department of Mathematics (Dr.M.DAMLAKHI) .
Wednesday 25/6/1433H .
Answer only five from six of the following exercises :
Exercise 1 a) Let (X ,  ,  ) be a measure space and f , g  S  () ,where   , such
that f (x )  g (x ) for all x   . prove that  f (x )d    g (x )d  .


b) Let f  L 0  () and E   such that E   . prove that
f d   f
X E d

E
Exercise 2 : Let (X ,  ,  ) be a measure space and (A n ) be a sequence of subsets of X
contained in  .
1) Prove that  (
A k )  inf (  (A k ) .
k n
k n
A k and bn  inf  (A k ) Prove that B n  B n 1 and bn  bn 1 .
2) If B n 
k n
k n
3) Deduce that  (( (
n
A k ))  supinf  (A k ) .
k n
n
k n
4) Deduce that  (liminf An )  liminf  (An ) . (This is the Fatou’s Lemma for sequences of
subsets .)
Exercise 3 : a) Let (X ,  ) be a measurable space and X 

X n ,where X n 

. If
n 1
n  B X n ; B  
for each n . show that
n
is a   algebra .
b) Show that if  and  tow measure on  where (X ,  ) is a
measurable space , such that    and    prove that  is identically zero .

Exercise 4 a) Compute the following limit : nlim
 
a
n  sin x
dx
1  n 2x 2
for a  0 and a  0 (Hint :
put y  nx ) .
b) We consider Lebesgue measurable space ( , M , m ) .

 2x ;x  2
f (x )  

0 ; x  2
Let
x 2 ; x  0
.
g (x )  
0 ; x  0
and
 (E )   g (x )dx
We define two measure :  (E )   f (x )dx and
where E M .
E
E
Find the Lebesgue decomposition of  respect to  .
Exercise 5 : a) Let (X , S ,  ) an (Y , , ) be two   finite measure spaces and we define
the product measures  X  on  - algebra SX  by
( X  )(V )    (V x ) d     (V y ) d , where V Sx  . Prove that  X  is
X
Y
a measure on SX  .
b) we consider Lebesgue measure ( , M , m ) and we define a function
xy

; (x , y )  (0, 0)
 2
f on  1,1 x  1,1 by f (x , y )   (x  y 2 ) 2
0 ; (x , y )  (0, 0)

.
Show that the iterated
integrals of f over  1,1 x  1,1 are equals but f is not integrable ( Hint: prove that
1
1
0
0
 ( f (x , y )dy )dx
does not exist . ) .
Exercise 6: a) Let (X ,  ,  ) be measure space and p  1 . Prove that the set of
bonded functions on X is dense in L p (X ,  ) .
b) Let (f n )  L p (X ,  ) where  (X )   and p  1 .Show that if f n  f in
L p (X ,  ) then f n  f in L p  (X ,  ) ,where 1  p   p .
c) Let m be Lebesgue measure on X  0,1 and (f n ) a sequence of
3
measurable functions on X satisfying lim  f n (x ) dm  0 . Show that nlim
 
n 
X
X
f n (x )
dm  0 .
x