solute continuity (the ingredients of their problem). Thus one might try for
an integral of the form
= l i m £ / ( £ ) ( * . - - x^y (3)
t= i
for 0 s 1 under an appropriate interpretation of the limit process.
The integral defined in equation (3) cannot be defined in exactly the same
manner as that in (2); thus the integration theory that is available for such
integrals does not immediately apply. Nonetheless there are a number of
appropriate interpretations of (3) that appear to be useful and which have
connections with the theory of Hausdorff measures, illuminating some of
the notions that arise there. In particular the integral of Besicovitch in [1]
permits such a realization.
We begin with a general theory for the derivation and integration of
interval functions, centered on the study of limits of the form
limsup h(I)/k(I),
where the limit involves approach to x by intervals always containing x as an
interior point. Essentially then we have a study of this particular differentia-
tion basis. Other differentiation bases could certainly have been chosen. This
one has particularly smooth properties in many regards and is directed at
the study of Lipschitz numbers. General notions of limit, derivative, continu-
ity, variation, integral, absolute continuity and singularity are studied in this
setting. The main theme throughout could be said to be a systematization
of Vitali arguments.
We confine our applications to the study of functions of a real variable. In
Section 6 we discuss some classical measures on the real line. In particular
we show how the Lebesgue measure, the Lebesgue-Stieltjes measures and
the Hausdorff measures on the line can be studied in this setting. We then
turn to some classical properties of real functions, especially derivation and
variation. The familiar properties of monotonic functions in Section 7.1 come
as applications of the general differentiation results for interval functions.
The material of Denjoy, Lusin and Saks on the structure of VBG* functions
is developed in Section 7.2; a number of the characterizations given appear
to be new.
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