MATHEMATICS: R. D. CARMICBAEL 
551 
CONDITIONS NECESSARY AND SUFFICIENT FOR THE 
EXISTENCE OF A STIELTJES INTEGRAL 
By R. D. Carmichael 
Department of Mathematics, University of Illinois 
Communicated by E. H. Moore, October 8, 1919 
The purpose of this note is to suggest a method for deriving a neces- 
sary and sufficient condition for the existence of the Stieltjes integral of 
f{x) as to u{x) from to 6 in each of several forms generaHzing those 
frequently employed (see Encyclopedie des sciences mathematiques , IIi, 
pp. 171-174) in the special case of Cauchy-Riemann integration — 
where u{x) is of bounded variation and f{x) is bounded on the interval 
{ah), M^f(x) ^m. Bliss (Proceedings 3, 1917, pp. 633-637) has ob- 
tained one of these forms, perhaps the most satisfying of any; this 
note closes with his theorem, of which we give a new demonstration. 
Some of the other theorems stated (though here derived otherwise) 
are immediate consequences of the one due to Bliss or are otherwise 
intimately related to it, as the reader will readily see. Since it is 
believed that the present treatment will be found useful in connection 
with that of Bliss, our notation has been made to conform to his; more- 
over, reference to his paper is given for such isolated steps in the proof 
as may readily be supplied from it. 
For a given partition tt of {ah) due to the points Xq = a, Xi,X2, . . . , 
^n-h Xn = h,0 < Xi — Xi_i < d, form the sums 
n n n 
i = l i = l i=i 
where A,-w = u{xi) — u{Xi_i), Xi is any point of the interval {Xi_i, Xi), and 
M;[wJ is the least upper bound [the greatest lower bound] off{x) on 
(x,._i, Xi). If the limit of 5^ exists as 8 approaches zero, this limit 
is the Stieltjes integral oif{x) as to u{x) from a to h. In case u{x) is 
a monotonic non-decreasing function, the limit, if existing, of ^tt^tt]? 
as 8 approaches zero, is here called the upper [lower] Stieltjes integral 
oif{x) as to u{x) from a to h. 
Let us determine conditions under which the upper integral shall 
certainly exist when u{x) is monotonic non-decreasing. If we form a 
repartition tt' of {ah) as to tt by taking the points forming tt and certain 
additional points, it is clear that S^> ^ S^^. Moreover, there is ob- 
