Real Analysis 1386/09/06 Dr. H.R.E.vishki
رای دهی: 5 / 5
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 ) Let
be 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.
