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
Previous Page Next Page