DERIVATES OF INTERVAL FUNCTIONS

3

Sections 7.4 and 7.5 return to the problem of Rogers and Taylor - deter-

mining the nature of functions that are absolutely continuous with repect

to the Hausdorff measures. Note that their study focuses on functions

of bounded variation (or more properly their Lebesgue-Stieltjes measures)

whereas here the methods apply directly to functions that are VBG* on a

set. Thus we are able to state their results in a bit more generality, but

more interestingly as an application of techniques that apply to general in-

terval functions. At the same time we are able to prove a parallel charac-

terization of the functions that are absolutely continuous with respect to the

5-dimensional packing measure of Tricot.

The only background required of the reader is a familiarity with such

standard parts of analysis as may be found in the well-known treatise of

Saks [17]. In the discussion of Hausdorff measures in Section 6.3 it is assumed

that the reader is familiar with the rudiments of that subject and in particular

the density theorems for those measures. The intention of this memoir is to

introduce a new perspective on certain applications of the Vitali covering

theorem. While specialists will see how this could be applied in a number

of different settings even the general reader may find the presentation of

the classical differentiation theory of real functions unusual and, perhaps,

interesting.

2 Covering relations

By a covering relation is meant a collection of pairs (J, x) where / is a closed

interval and x £ / . The language is taken from Federer [5, p. 151]. A number

of the constructions in classical analysis may be expressed using the notion

of a covering relation and many arguments can be simplified by an appeal to

this language.

We illustrate with an example. If F' — f everywhere how might proper-

ties of F be recovered from a knowledge only of the function / ? Of course if /

is integrable then the solution is transparent: the integral f* f(t) dt recovers

F up to an additive constant and fE \f(t)\ dt recovers the variation of F on a

measurable set E. If / is not integrable in either the Lebesgue or improper

Riemann senses it is less clear how to proceed. Such a function / would have

a dense set of intervals on which it is Lebesgue integrable; by using this fact

together with deeper properties of derivatives Denjoy in 1912 was able to