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

(2)

(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!