IGRALS 
GENERAL THEORY 
41 
■p», and in d L . o)[ > o) t . 
îe increasing function 
3 integrable in 2t aj3 . 
some 7 in 53- Then 
it is in 53 by I, 700, 3. 
ace f is integrable in 51 
e points of 0Ï lying 
points of 53 at which 
\er integral in any part 
Inch f is limited, and if 
convergent, f is integra 
tor let 
exist. Since 
necessarily 
exists and 1), 2) 
f / = I'm f f 
equal. 
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(2 
42. 1. In studying the function/it is sometimes convenient to 
introduce two auxiliary functions defined as follows: 
g=f where/>0, 
= 0 where /<0. 
h = —f where f < 0, 
= 0 where/>0. 
Thus g, h are both > 0 and 
f = g-h 
\f\ = g + h. 
We call them the associated non-negative functions. 
2. As usual let 5f a 0 denote the points of 51 at which — a </ < /3. 
Let 5fp denote the points where </ < /3, and 5l a the points where h < a. 
Then — — 
I g = lim I g, 
i/w „ „ 
For 
a,0= 
J 7¿ = lim y h. 
31 a,p=»J% afi 
f 9= ( by 5, 4. 
s)lf ^ 9T « 
Letting a, /8 == oo, this last gives 1). 
A similar demonstration establishes 2). 
3. We cannot say always 
) g = lim I g ; I h = lim I h, 
5131 a ,0=xil3l o)3 - ft=r„ „ 
a, /3=co 
as the following example shows. 
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