# Rudin Analysis book：shrink nicely

• Dec 28th 2009, 06:16 AM
Shanks
Rudin Analysis book：shrink nicely
Let $\displaystyle m$ be the lebesgue measure on $\displaystyle R^k$, a signed measure $\displaystyle u$ is absolutely continous with respect to $\displaystyle m$,$\displaystyle f \in L^1(R^k)$ be the Radon-Nikodym derivative of $\displaystyle u$ with respect to $\displaystyle m$.
$\displaystyle x\in R^k$, and for any set sequence $\displaystyle \{E_i\}$ which shrink nicely to $\displaystyle x$, we have
$\displaystyle \limsup_{i\to \infty}\frac{u(E_i)}{m(E_i)}\leq f(x)$.
Prove that $\displaystyle -f$ is the Radon-Nikodym derivative of $\displaystyle -u$ with respect to $\displaystyle m$.
and show that $\displaystyle \liminf_{i\to \infty}\frac{u(E_i)}{m(E_i)}\geq f(x)$.
The Definition of "Shrink nicely" in Rudin's Real and Complex Analysis Book:
A sequence of measureable sets $\displaystyle {E_i}$ shrink nicely to x if there is a positive number $\displaystyle a$ and a sequence of Ball B(x,r_i) centered at $\displaystyle x$ with radias $\displaystyle r_i$ satisfying
(1)$\displaystyle E_i$ is contained in $\displaystyle B(x,r_i)$
(2)$\displaystyle \lim_{i\to \infty}r_i=0$
(3)$\displaystyle \left|u\right|(E_i)\geq am(B(x,r_i))$, 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!
• Dec 29th 2009, 02:04 AM
Moo
Hello,

Fatou's lemma ? (Worried)
• Dec 29th 2009, 06:40 AM
Shanks
Topic about differnetiation of measure in real analysis.
the "shink nicely" section in Rudin' Book "Real and Complex Analysis" in Chpter 7.
• Jan 6th 2010, 09:23 AM
Shanks
Solved. never mind, forget it!