Vol. 8, 1922 
MATHEMATICS: C. A. FISCHER 
27 
shall be satisfied, and that there be an M independent of u which satisfies 
the inequality 
\L{u)\i.M .max\u\, (3) 
while if it is to be linear-w this inequality must be replaced by 
\L{u)\i.M^j'u2i^^)dx. (4) 
F. Riesz has proved that every linear functional can be put into the Stieltjes 
form 
L{u)= J u{x)da{x), (5) 
where a(x) is independent of m and has finite variation.^ Frechet has 
reduced this to the form 
L{u)= y^AM^.„) + j ^u{x)^{x)dx + j ^u{x)d\{x), (6) 
M = l 
where the A^s are constants, the x„'s are the points where a{x) is dis- 
continuous, ^{x) is the derivative of a{x) where the derivative exists, 
and \{x) is continuous, and has its derivative equal to zero excepting on 
a set of measure zero.^ The principal object of the present note is to 
prove that when L is linear-w equation (6) becomes simply 
L{u)= y u{x)^{x)dx. (7) 
The class \u{x)] will include all bounded, real-valued functions defined 
on (a, h), which are integrable with respect to a function of finite variation 
by Young's method of monotone sequences, beginning with the set of 
continuous functions.^ 
In the proof of equation (5) it was only assumed that L{u) is defined 
for continuous w's, and to make the theorem apply to the larger class, 
L must always be defined in vSuch a way that whenever a monotone se- 
quence approaches a discontinuous u{x), L{un) will approach 
L{u). When this is done equation (6) will also be satisfied for the larger 
domain [u{x)], since the limiting processes extending the definition of the 
Stieltjes integral (5) to the domain [u{x)] will make each term of equation 
(6) approach the proper value. On the other hand if such a monotone 
sequence approaches u{x), the equation 
limit C\u„-u 
n — >■ ao J a 
Ydx = 0 
must be satisfied.^ Then if L is linear-w inequality (4) implies that 
L{Un — u) will approach zero, and L(w„) must approach L{u). 
Since a uniformly convergent sequence must converge in the mean, 
a linear-w functional must also be linear. 
