# Thread: Rudin Analysis book：shrink nicely

1. ## Rudin Analysis book：shrink nicely

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

2. Hello,

Fatou's lemma ?

3. Topic about differnetiation of measure in real analysis.
the "shink nicely" section in Rudin' Book "Real and Complex Analysis" in Chpter 7.

4. Solved. never mind, forget it!