1 Introduction

In the study of the derivation properties of interval functions there are certain

arguments that reappear in many settings. It is our pupose here to present a

unified approach to some of these techniques. In order to focus our attention

we will present this material with a specific view to examining an interesting

and important study of Rogers and Taylor [16] characterizing those interval

functions which are in a sense absolutely continuous with respect to the

s-dimensional Hausdorff measure. We follow Rogers [15] in confining our

attention to the one-dimensional case although many of the arguments can

be used in higher dimensions where richer geometry and larger choice of

differentiation bases might obscure the natural simplicity of the ideas .

The problem is to determine the nature of continuous, nondecreasing

functions / on the interval [0,1] whose Lebesgue-Stieltjes measures Xj are

absolutely continuous with respect to the s-dimensional Hausdorff measure

on [0,1]. This problem leads naturally to an investigation of derivates of the

form

n*tf ^ r /(^) ~ f(y)

m

D{f,x)= hmsup — — (1)

y,z^x,yxz [Z — y ) s

which Besicovich [1] has called "Lipschitz numbers".

The discussion will be intimately related to the notion of a generalized

Riemann integral as defined by Henstock [7] and Kurzweil [11]. Recall that

the classical Riemann integral defined as a limit of Riemann sums

rb n

f(x)dx = \im^2 f(^i)(xi - xt-_i) (2)

has been shown to allow greater flexibility than appears at a first study. By

altering the definition of the limit in equation (2) one can obtain integrals

that express exactly the improper Riemann integral, the Lebesgue integral,

the Denjoy-Perron integral, the Denjoy-Khintchine integral, the approximate

Perron and the approximate symmetric Perron integrals. The derivation

theory for these integrals is particularly immediate for the same concept

(what we call a "full covering relation" in Section 2) expresses both the

integral and derivative, revealing a single unifying connection between these

two classical ideas.

In connection with the problem of Rogers and Taylor one might expect

a similar approach to relate Hausdorff measures, Lipschitz numbers and ab-

/

J a

1