70 Transactions of the Academy of Science of St. Louis 
T.21. If f(x) is continuous and bounded on a set of points S 
on a uniform arc I’, it is integrable-L on Tap. 
The characteristic function #(x) for a set S is defined to be 
1 when x belongs to S and 0 otherwise. Accordingly, the meas- 
ure of a set S which lies on the arc I,, is then defined by the 
equation 
us) =f o(ayliax, 
where we assume that the right hand member is self-explana- 
tory. It is seen, for instance, that the measure of I, itself is 
Vab- 
The following theorems, analogous to those concerning func- 
tions continuous on the whole of I's, are proved in a similar 
manner. 
T.22. Let f(x) be a bounded integrable-L function, and let A 
be its least upper bound on a set S of a uniform arc Is. We 
then have b 
1 f(x)dx 
Ta 
T.23. If f(x) isa bounded M-function on a uniform arc and 
the sets of the sequence over which it is continuous are measur- 
able, it is integrable-Z on the arc. (The proof follows readily 
from the last theorems and the lemma: If S is the outer limit 
of 15.) then w(.S) =lim w(S,) if S, is measurable for every 7). 
The integral of @tm,x"’. The calculation of this integral must 
be carried out in a different manner from that ordinarily em- 
ployed, for here it must be done directly. 
The integral of @tmx’. Let Tas be a uniform arc in Ey); we 
shall show that 
Tin i xtdx = Sint (br+! — grt?) 
Poe ee 
S du(S) 
Let 
n+l 
(1) Snes = Sime. ef hes ( Hi 2s) 
‘ 0 
where x9 =, X,41=0, and s is a positive integer <r; then it 
will first be shown that 
(2) lim S,,0o=lim S,,, for each s. 
We have 
V1 = % “+ (ign ge X;) 
: 5 
ai = > ( ; ance ie. 
0 
and 
