Real Analysis1 Final Exam - Dr Hatam - Amirkabir University - 1389/10/14
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Exam Name : Real Analysis1
MSC Final Exam
Amirkabir University of Tehran
first Term of 1389-90
Date: 1389/10/14
Dr Ali Hatam
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Over all the following questions 
and 
are positive mesases on 
-algebra 
in a set X.
1. Show that if is infinite, then is uncountable.
2. If 
for 
, apply the Fubini's Theorem 8.8 to deduce
. 
3. Let X be a topological space and 
be a family of lower semicontinuous function on X. Show that the function 
is also lower semicontinuous.
4. For 
define 
. Prove that d is a metric on 
.(Hint: Use decreasing function 
.)
5. Let 
be a measurable function, if 
, set 
, use Theorem 3.3 to prove that:
.
6. Prove that is -finite if and only if there exists a positive function 
.
7. If 
, then show that ,
(a) 
holds for every 
.
(b) If is finite measure then 
.
(c) If is finite measure and if 
, then 
.
8. Let L be a nonzero continuous linear functional on H and 
. Prove that 
is a vector space of dimension 1. (Hint: Use Theorem 4.11)
9. Let 
be a linear operator defined by 
. Show that T is bounded and 
. (Hint: Use function f=1 to obtain that 
.)
10. If and are finite measures and 
holds for every measurable set 
and if 
, then prove that there exists a unique measurable 
with respect to such that 
. (Hint: Define linear functional 
on Hilbert space 
and use both theorems 1.40 and 4.12)
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Good luck ♠