554 
MATHEMATICS: R, D. CARMICHAEL 
preceding paragraph and of the fact that U{x) is obviously equal to 
V(x) + /(x), where V{x) is the total variation of v{x) on (ax), it is 
obviously sufficient to prove this for the case when u{x) is a continuous 
function. Now, when u(x) is continuous, we have (Vallee Poussin, 
Cours Analyse, vol. 1, 3rd edn, p. 73) 
with a like relation when b is replaced by x and the interval (ab) by 
the interval (ax). Since no | u(xi) — u(Xi_i) \ is greater than the 
corresponding difference U(xi) — U(Xi_i) and since the sum of the 
latter differences, for i = 1, 2, . . . , n, is U(b), we see that 
n 
and that no bracketed term here is negative. Hence for every e there 
exists a di such that the term (i = 1, 2, . . . , n) oi the sum in (4) 
is less than e when d<8i. Hence, for such d, we have 
n 
0^2 (^i-^i)[U(Xi) - U(Xi_,) - I M(x,)-u(Xi_,) I ] < (M-m) € (5) 
i=l 
Hence, as b approaches zero the sum in (5) approaches zero as a limit. 
This result and relation (2) imply (1). But the latter is sufficient to 
the existence of the integral oif(x) as to U (x), and indeed as to u(x). 
We are thus led to the following theorem: 
Theorem 11. // u(x) is of bounded variation and f(x) is bounded on 
(ab), then a necessary and sufficient condition for the existence of the 
integral of f(x) as to u(x) is the existence of the integral of f(x) as to U(x). 
Since (1) and (2) are identical when u(x) = U(x) we now have readily 
the following theorems : 
Theorem HI. A necessary and suffilcient condition for the existence 
of the integral of the bounded function f(x) as to the function u(x) of 
bounded variation is that the upper and lower integrals of f(x) as to the 
total variation function U(x) of u(x) shall exist and be equal. 
Theorem IV. A necessary and sufficient condition for the existence of 
the integral from a to b of the bounded function f(x) as to the function 
u(x) of bounded variation is that the total oscillation of f(x) as to u(x) 
from a to b shall be zero, that is, that 
