MAT 708 Spring 2009
Quizzes
Name_____________________________________
Directions: Make sure to set up and end your proofs appropriately.
Date________________
1. Recall our proof of the Cantor-Bendixson Theorem (also called the CUP Theorem according to the
internet) and recall the function n :
   F 
F

\ F  1  


defined by n  x    n I n  F    x , in
which F is a closed set, I1 , I 2 , I3 ,... is an enumeration of the open intervals with rational endpoints, and
  F  is the least ordinal  for which F  \ F  1 . Prove that n is injective.
2. Prove that the Cantor Set is uncountable using the “middle-thirds interval” definition of the Cantor Set,
not the definition in terms of the ternary expansion.
3. Let
fn n 
be a sequence of Lebesgue measurable functions that converge in measure to f .
f n k  k 
Prove that a subsequence
exists that converges to f almost everywhere.
4. Let E be a set of finite outer measure contained in an open set O and let J be a Vitali covering of E
which contains only closed sets that are subsets of O . Pick an arbitrary   0 . Recall the definition of
kn , the construction of the pairwise disjoint I1 , I 2 , I 3 ,... J such that lh  I n 1  
chosen so that

 lh  I   5 . Prove E \
iN
 5I i  .
Ii 
i
i N
kn
and that N was
2
iN
5. Assume that f is an increasing real-valued function on  a, b . Let
Eu ,v   x : D  f  x   u  v  D f  x  . Prove m * Eu ,v  0 . Feel free to skip the construction of
I1 , I 2 , I3 ,..., I N which are pairwise disjoint elements of the Vitali Covering J and assume


m *  Eu ,v \ I i    ,
i N




m *  I iO   m *  Eu ,v    (**), and
 i N 
  f  x   f  x  h    h    m *  E    
N
i 1
N
i
i
i 1
i
u ,v
(*).
6. Assume that f is of bounded variation on  a, b . Prove that f is the difference of two monotone realvalued functions on  a, b .
MAT 708 Spring 2009
Quizzes
Name_____________________________________
Directions: Make sure to set up and end your proofs appropriately.
Date________________
7. Assume that f is integrable on  a, b and f  0 on the closed set F   a, b with positive measure.
Show that there exists x  a, b such that

x
a
f  0.
8. Lemma 9 (p106): If f is bounded and measurable on  a, b and F  x  
F '  x   f  x  for almost all x in  a, b .
 f t  dt  F  a  , then
x
a
9. Lemma 13 (p109): If f is absolutely continuous on  a, b and f '  x   0 a.e., then f is constant.