In the book of analysis now solve the following exercisesSo

In the book of analysis now , solve the following exercises:

Solution

Ans-

t is fairly easy to show using integration by parts that if fL1fL1 is differentiable and fL1fL1 then under assumption that ff also vanishes at infinity one has supRf^()f1supR|f^()|f1 , this is evident using integration by parts

f^()=Reixf(x)dx=eixf(x)|+Rieixf(x)dx=if^()f^()=Reixf(x)dx=eixf(x)|+Rieixf(x)dx=if^()

the assumption that vanishes at infinity and still get this bound.we assume ,, are positive and finite measures. The Radon Nykodym theorem states that if , then there exists hL1(X)hL1(X) s.t.

(E)=Ehd(E)=Ehd

The construction is as follows: M={g|Eg(E)}M={g|Eg(E)} for any measurable set EE. And hh is chosen such that Xhd=sup{Xgd,gM}Xhd=sup{Xgd,gM}. We need to prove that Xhd=(X)Xhd=(X), which actually proves (E)=Ehd(E)=Ehd. However, I don\'t really see where we need