Real Analysis 1386/09/06 Dr. H.R.E.vishki

Exam : Real Analysis

Date : 1386/09/06

Pro : Dr. H.R.E.vishki

Uni : Ferdowsi of Mashad

1. Define by

show that is finitely additive but it is not a measure on .

2. Let be a complete measure on X :

i ) if are two functions such that f=ga.e , then show that the measurability off is equivalent to that of g.

ii ) Let a.e, show that if is measurable for all then so is f.

3. i ) State and prove Dominated Convergence Theorem.

ii ) Letbe a sequence of bounded measurable functions such that on X , show that if Then

Through an example show that the hypothesis " " is essential.

4. i ) Show that is Banach.

ii ) Show that and for every finite measure . Is the same inclusion true when is infinite . Prove yours claim.