Exam Name : Real Analysis1
MSC Final Exam
first Term of 1389-90
Dr Ali Hatam
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)
Good luck ♠