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Exam Name : Real Analysis

For Applied Students

MSC Exam

Department of Mathematics Ferdowsi University of Mashhad

Second Term of 1386-87

Dr Sh.Hejazian

1. Define each of the following :
-algebra of sets, measure, finite measure, -finite measure, complete measure, measurable function.

2. Give an example of a finite measure and a -finite measure which is not finite.

3. State and prove the monotone convergence theorem.

4. Let E be a Lebesgue measurable set with . Show that E contains a nonmeasurable set.

5. Let a complete measure and f be a measurable function, If f=g a.e. show that g is measurable.

6. Let . Show that

7. If , and , show that .