文档介绍：Divulgaciones Matem´aticasVol. 8 No. 1 (2000), pp. 75–85
The Fundamental Theorem of
Calculus for Lebesgue Integral
El Teorema Fundamental del C´alculo
para la Integral de Lebesgue
Di´omedesB´arcenas(******@)
Departamento de Matem´ de Ciencias.
Universidad de los Andes. M´.
Abstract
In this paper we prove the Theorem announced in the title with-
out using Vitali’s Covering Lemma and have as a consequence of this
approach the equivalence of this theorem with that which states that
absolutely continuous functions with zero derivative almost everywhere
are constant. We also prove that the position of a bounded vari-
ation function is unique up to a constant.
Key words and phrases: Radon-Nikodym Theorem, Fundamental
Theorem of Calculus, Vitali’s covering Lemma.
Resumen
En este art´ıculose demuestra el Teorema Fundamental del C´alculo
para la integral de Lebesgue sin usar el Lema del cubrimiento de Vi-
tali, obteni´o consecuencia que dicho teorema es equivalente
al que afirma que toda funci´onabsolutamente continua con derivada
igual a cero en casi todo punto es constante. Tambi´ense prueba que la
posici´onde una funci´onde variaci´onacotada es ´unicaa menos
de una constante.
Palabras y frases clave: Teorema de Radon-Nikodym, Teorema Fun-
damental del C´alculo,Lema del cubrimiento de Vitali.
Received: 1999/08/18. Revised: 2000/02/24. Accepted: 2000/03/01.
MSC (1991): 26A24, 28A15.
Supported by - under project C-840-97.
76 Di´omedesB´arcenas
1 Introduction
The Fundamental Theorem of Calculus for Lebesgue Integral states that:
A function f : [a, b] R is absolutely continuous if and only if it is
→ 1
differentiable almost everywhere, its derivative f 0 L [a, b] and, for each
t [a, b], ∈
∈
t
f(t) = f(a) + f 0(s)ds.
Za
This theorem is extremely important in Lebesgue integration Theory and
several ways of proving it are found in classical Real Analy