Let be the lebesgue measure on , a signed measure is absolutely continous with respect to , be the Radon-Nikodym derivative of with respect to .
, and for any set sequence which shrink nicely to , we have
Prove that is the Radon-Nikodym derivative of with respect to .
and show that .
The Definition of "Shrink nicely" in Rudin's Real and Complex Analysis Book:
A sequence of measureable sets shrink nicely to x if there is a positive number and a sequence of Ball B(x,r_i) centered at with radias satisfying
(1) is contained in
(3) , for all i=1,2,....
I have proved the first problem.
for the second problem, Completely don't know where to start, Please help me out. Appreciation more than I can say!