2

BRIAN S. THOMSON

solute continuity (the ingredients of their problem). Thus one might try for

an integral of the form

t

f(x)dxs

= l i m £ / ( £ ) ( * . - - x^y (3)

Ja

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),

|/|-o,xe/°

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.