Colloquium publications.

116

THE CAMBRIDGE COLLOQUIUM.

The consequence of specialization.of the range of the general
variable, say by the introduction of concepts like that of distance,* is however the subject of fruitful study.
70. The Content of the Operation J. So far the operation J
has been defined by postulates, and not explicitly. In order to
get some idea of its generality for a given range of the general
variable, let us choose the range as the most familiar one, namely,
the one-dimensional continuum. Let us assume that the class 91)
contains (or can be extended by definition to contain) the totality
of continuous functions over this range, which we may take to
be the interval ab. In this case, the operation is merely a Stieltjes
integral:
(23) JtuK(st)r(u) = (t)dta(st).
In fact from the dominance property (Do) and the modular
property (M) it follows that if s is given a constant value So,
and |I (t) I < 1, then there is a constant Cso such that
I JtuK(sot)q(u) I < Cso.
Hence for every vq it follows by the linearity property (L) that
the inequality
b
(24) | JtuK(st)7(u) |< CS max I \r
is satisfied. This and the linearity property constitute the two
Riesz conditions of Art. 40, and our statement is therefore
proved.
We may express a(st) directly in terms of K(st). Either side of
(23) is a linear functional of a, which we may denote by
b
T[].
a
* Besides the references to Moore, see papers by Frechet, and Pitcher, e.
g., Pitcher, American Journal of Mathematics, vol. 36 (1914), pp. 261-266,
where the literature is given. The fundamental memoir on this subject is
Frechet's Thesis, already cited.