### Functions of Limited Variation and Lebesgue Integrals

The Annals of Mathematics, 2nd Ser., Vol. 20, No. 1 (Sept 1918), 1-8

Introduction

A fundamental theorem in the theory of Lebesgue integrals is that the four derivates of a function continuous and of limited variation are summable and all equal except over a set of measure zero. This theorem is proved by Lebesgue by actually calculating the variation of the function and by Vallée Poussin by using two auxiliary functions, called majorating and minorating functions, which, with their derivatives, satisfy certain conditions of inequality.

The purpose of this paper is to show how the proof of this theorem can be simplified by the study of a very simple monotone function, which we shall name *measure function*, preliminary to proving directly Lebesgue's theorem that a continuous function with a bounded derivate has a derivative except for a set of measure zero, and to the deduction of known existence theorems for derivatives independently of the theory of majorating functions.