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
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