Outline of a Theory of Functions of an Abstract Variable 69 
(1) i yigiy ie qf: fads 
(2) i seo - ‘foiids = J “fade 
(3) J eas = af sear 
(4) f ve + g(x)}dx =f roars fi gar. 
T.18. Let f(x) be a bounded integrable-R function and let uw 
be its upper bound. Then we have 
5 
if f(x)dx 
Ta 
T.19. If f,(x) is continuous on a uniform arc and if { fn(2) } 
converges uniformly to f(x) on the arc I',, then 
lim fi foeax = [tim fy(a)de = fi sear. 
Integrability-L. We have considered the integrability of 
functions which are defined for every point of a given anatase 
fact for a whole region containing the arc, and in this connection 
we have found that the class of continuous functions is integra- 
ble-R. It is possible, however, to extend the notion of integra- 
bility to the class of M-functions by the following broadening 
of our conception, which in its application reduces essentially to 
the Lesbegue-Stieltjes integral. 
If in the application of a system of nets to an arc we remove 
the restriction that the number of subdivisions is finite, and 
then form the sum (A) for a function defined over the arc, we 
May speak of the lower limit of the sum for all systems of nets; 
that is, for all modes of subdivision. If the function is defined 
only over a set of points S contained in the arc we omit all those 
intervals from (A) which do not contain points of S. On the 
other hand, we may consider the upper limit of the sum when we 
~ Omit all those intervals which are not contained in S. If these 
two limits exist and are equal, the function is said to be in- 
tegrable-Z, 
T.20. A function which is integrable-R is also integrable-L 
and the two integrals are equal. 
The proof of the next theorem follows that of T.16 very 
ly. 
S UY ab 
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