Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

5

If the curve Ci may be shrunk down to zero without going
outside the region of definition, we shall have, for an additive
functional, the equation:
(6') y[C] = ffy'(M)do.
(C)
The formulae (4) to (7) may fail to hold in cases where the
derivative does not remain finite, or when the functional itself
depends in a special manner upon certain points. These important special cases will be treated, as need arises. But meanwhile it is perhaps desirable to point out sufficient conditions
under which formulae (4) to (7) may be deduced.
4. Existence of a Derivative, Additive Functionals vs. Functions of Point
Sets. An additive functional of a curve, y[C], is said to be a functional of
finite variation if the inequality
Zi | y[Ci] I < M
is satisfied, in which M is a constant, and the closed curves Ci are squares (or,
with equal generality, rectangles) mutually exterior (except for possible
common boundaries), finite or denumerably infinite in number, and all contained in a given finite region. The functional is absolutely continuous if the
quantity M can be made as small as desired, < e, by taking the sum of the
areas of the squares small enough, < 8, irrespective of their position.
It is also convenient to be able to speak of a restricted derivative, and
this we define as lim y[C]/(area inside C), the curve C being restricted to a
c=o
square. In some cases it may be desirable to restrict the curves to circles
instead of squares.
We notice at once the relation of the theory of additive functionals to that
of functions of point sets. In fact if we define a function f(w) of the points
in any cell of a square network as the value of y[C] for the contour of that
square, the value of the function of point sets f(e) will be defined for any
point set e which is measurable according to Borel, and if the frontier of e is
a standard curve C, we shall have f(e) = y[C]. To paraphrase a theorem of
De la Vallee-Poussin:*
Every additive functional which is continuous and of finite variation defines
a continuous additive function of point sets, for point sets measurable according
to Borel; the restricted derivatives of the functional and the function are the same
wherever they exist.
Hence it follows that an additive continuous functional of finite variation
has a finite derivative (in the restricted sense) at all points except possibly those
of a set of measure zero.
* De la Vallee-Poussin, Transactions of the American Mathematical Society,
vol. 16 (1915), p. 493.