28 
MATHEMATICS: C. A. FISCHER 
Proc. N. a. S. 
If L{u) is linear- w, and u{x) is defined as identically zero excepting at 
Xi where it is equal to unity, equation (6) becomes L{u) =Ai, and inequality 
(4) implies that Ai = 0. Similarly ^42 = ^3= =0. In other words 
a{x) must be continuous. 
Since a{x) also has finite variation, it can be put into the form a{x) = 
oLiix) — a2{x) , where ai and are continuous, have finite variation, and 
are monotone increasing. The derivatives of ai and will be called 
lSi and A set E of measure zero will contain all points where either 
|8 fails to exist. Since no derivative number of a monotone increasing 
function can be negative, the functions Xv {x), defined by the equations 
\i{x) = ai{x) - p ^i{x)dx, 2) (8) 
must be monotone increasing,'^ and their derivatives will vanish excepting 
on E. If a fei> 0 is now given, there must be a sequence of non-overlapping 
00 
open intervals, Bi^^^h containing each point of E as an interior 
« = i 
point of some I„, such that the measure of Bi is less than K Then the 
variation of X/ on Bi is equal to its variation on (a, 6),^ which is X^ (6) - 
X/(a). The lower semicontinuous functions Ui{x), U2{x), will now 
be defined by the equations 
^^(^) = 1 on the interior of the intervals h, h, /«, 
Unix)=0 for all other values of x, 
and consequently 
Variation of X, on {i = l, 2; n=l, 2,. . .). 
1 
The limit Vi{x) of this monotone sequence of w's, which vanishes excepting 
on Bi, must satisfy the equations 
^\i{x)dhM=\{b)-\{a), {i=l, 2) 
If a sequence h>k2> — > 0 is chosen, beginning with the above 
ku the sequences of intervals B2, Bs, can be chosen in such a way 
that each contains E, every point of each is a point of the preceding, and 
the measure of Bj is less that kj. Then the sequence Viix), v-six), , 
determined as Vi{x) was, will be monotone decreasing, and its limit v{x) 
will vanish excepting on a set of measure zero, and satisfy the equations 
'\{x)dUx)=\{b)-\i{a). 2) (9) 
j: 
Since v{x) vanishes excepting on a set of measure zero, the equations 
J\{x)^i{x)dx = 0, {i=l, 2) 
