Real and Complex Analysis (Rudin)

CHAPTER
SEVEN
DIFFERENTIATION
In elementary Calculus we learn that integration and differentiation are inverses
of each other. This fundamental relation persists to a large extent in the context
of the Lebesgue integral. We shall see that some of the most important facts
about differentiation of integrals and integration of derivatives can be obtained
with a minimum of effort by first studying derivatives of measures and the associated
maximal functions. The Radon-Nikodym theorem and the Lebesgue decomposition
will playa prominent role.
Derivatives of Measures
We begin with a simple, theorem whose main purpose is to motivate the definitions
that follow.
7.1 Theorem Suppose J.l is a complex Borel measure on Rl and
f(x) = J.l« - 00, x)) (1)
If x E Rl and A is a complex number, each of the following two statements
implies the other:
(a) fis differentiable at x andf'(x) = A.
(b) To every € > 0 corresponds a (j > 0 such that
J.l(I) -
I AI < €
m(I)
(2)
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