Exam Name : Real Analysis
For Applied Students
Department of Mathematics Ferdowsi University of Mashhad
Second Term of 1386-87
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 .