Real and Complex Analysis (Rudin)

140 REAL AND COMPLEX ANALYSIS
We shall now discuss these topics.
7.8 Theorem Suppose J.L is a complex Borel measure on Rk, and J.L ~ m. Let f be
the Radon-Nikodym derivative of J.L with respect to m. Then DJ.L = f a.e. Em],
and
(1)
for all Borel sets E c: Rk.
In other words, the Radon-Nikodym derivative can also be obtained as a
limit of the quotients Q, J.L.
PROOF The Radon-Nikodym theorem asserts that (1) holds withfin place of
DJ.L. At any Lebesgue point x off, it follows that
f(x) = lim _1_ r f dm = lim J.L(B(x, r)) .
,-0 m(B,) JB(X.,) ,-0 m(B(x, r))
(2)
Thus (DJ.L)(x) exists and equals f(x) at every Lebesgue point off, hence a.e.
~]. M
7.9 Nicely shrinking sets Suppose x e Rk. A sequence {Ei} of Borel sets in Rk is
said to shrink to x nicely if there is a number IX > 0 with the following property:
There is a sequence of balls B(x, ri), with lim ri = 0, such that Ei c: B(x, ri) and
m(Ei) :2= IX • m(B(x, ri)) (1)
for i = 1,2,3, ....
Note that it is not required that x e E i , nor even that x be in the closure of
Ei. Condition (1) is a quantitative version of the requirement that each Ei must
occupy a substantial portion of some spherical neighborhood of x. For example,
a nested sequence of k-cells whose longest edge is at most 1,000 times as long as
its shortest edge and whose diameter tends to 0 shrinks nicely. A nested sequence
of rectangles (in R2) whose edges have lengths Iii and (1/0 2 does not shrink
nicely.
7.10 Theorem Associate to each x e Rk a sequence {Ei(X)} that shrinks to x
nicely, and letfe IJ(Rk). Then
f(x) = lim 1 r f dm
i-oo m(Ei(X)) JEi(X)
(1)
at every Lebesgue point off, hence a.e. Em].