0"a/3 — 21/, aß — &aß- 
LS 
GENERAL THEORY 
55 
integral; similar 
L) is convergent, 
Thus, 
4 
(11 
(3 
But ff=fg-i> 
«/<g 
(4 
by 44, l. Thus 2) follows on using 5), 11) and taking a <cr\ 
a". 
57. If the integral i / converges and is an unmixed part of 
(5 
21 such that 23u = 21 as u = 0, then 
lim f f = Jf 
(1 
b such that 
M=0 i/51 
(« 
For if we set 2i = 23« + G£ M , the last set is an unmixed part of 21 
and (E M = 0. Now 
notone increasing 
ff = f + f • 
Passing to the limit, we get 1) on using 56. 
58. 1. Let ® a 0 = hv(2l/o|3i 2l ?a|8 , 2i /+g , ia ^). 
If 1°, the upper contents of 
shows that 
fa0 = ^/a0 — ®a0 1 Qa0 = <^0 a/ 3 ~ T) aj3 , fy a/3 = 21 f+g, a(3 — T> a 0 
(1 
h 
= 0 as a, /3 = oo , 
1 
a 
¿mcZ if 2°, the upper integrals off g,f-\-g are convergent, then 
— _ 
(2 
(8 
//*1° ÄoZds, «/, 3°, the lower integrals off, g,f-\-g are conver- 
gent, then 
(9 
X /+ X^-X (/+i,) - 
(3 
Let us prove 2); the relation 3) is similarly established. 
Let 
(10 
Z>., 0 be a cubical division of space. Let (g a|3 denote the points of 
£) a ß lying in cells of L aß , containing no point of the sets 1). 
Let