PROCEEDINGS 
OF THE 
NATIONAL ACADEMY OF SCIENCES 
Volume 3 • NOVEMBER 15.1917 Number 1 1 
A NECESSARY AND SUFFICIENT CONDITION FOR THE EXISTENCE 
OF A STIELTJES INTEGRAL 
By Gilbert Ames Bliss 
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CHICAGO 
Communicated September 12, 1917 
The purpose of this note is to suggest a proof of the following theorem : 
// u(x) is of limited variation andf{x) bounded on the interval a Sb, 
then a necessary and sujjictent condition for the existence of the Stieltjes 
integral 
n 
fdu (1) ' 
is that the total variation of u(x) on the set D of discontinuities of f(x) be 
zero. If U{x) is the total variation of u{x) on the interval ax, and a an 
interval with end points Xi, X2, then by definition 
U(a) = U{X2) - U{x,) 
and the total variation of u{x) on D is defined to be the greatest lower bound 
of the sums 
for denumerable sequences of intervals [ak] containing the points of D as 
interior points of the set which the intervals define. 
Let the interval ab be subdivided by points = 0, 1, . . ^) 
with Xo = a, x„ = 6, 0 < x» — Xi-i < 8 (i = 1, 2, . . ., n), and let 5 
denote the sum 
n n 
i=i t=i 
where the values Xi are arbitrarily selected in the intervals Xi-iXi. 
633 
