Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

63

Let us denote max I p1 - 2 1 by G, and let fn and gn be two
sequences which have as limits (Pi and 'P2 respectively. Define a
new function h, as follows:
h.(x) = fn(x) wherever Ifn(x) - gn(x) G,
(22) hn(x) = gn(x) + G wherever f,- gn > G,
hn(x) = gn(x) - G wherever gn - fn > G.
By this definition we have, as may be directly verified,
hn+l(x) 2 hn(x)
and also
lim hn(x) = p01(x).
7-=00
We have however by the definition (21):
I T[l - P2]1 = lim I T[f] - T[gn]
n=oo
= lim I T[hn - gn]
n=ca
which by (22) is c MG.
Our field as now extended is such that if (p, *, <pk belong to
it, any linear combination of them will belong to it, and will
satisfy the inequality (7). With this, we are in a position to
construct the Stieltjes integral.
The function which is unity in the sub-interval c < x < h,
and zero otherwise, is the limit of a sequence of continuous
functions fn(x), increasing with n. Hence the function
fed = 1, c < x - d (c > a),
(23)
= 0, otherwise,
is the difference of two such functions, and therefore a member
of the field of definition of the functional T. The function which
we denote by fad and define as:
fad = 1, a < x =- d,
(23')= 0 otherwise,
= 0, otherwise,

is the limit of a decreasing sequence, and is also of the field of T.