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