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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