8 POINT SETS AND PROPER INTEGRALS 
To prove 3), we use the relation 
-|/|</<|/|. 
Hence r r 
J -|/|<J f<j |/|, 
•'a 
For 
But 
from which 3) follows on using 3, 3. 
The demonstration of 4) is similar. 
To prove 5), we observe that 
^MA< M'Zdv 
and obviously 
3. The read 
true if 51 is noi 
5. 1. Let f> 0 be limited in the limited fields 59, (5. Let 51 be 
the aggregate formed of the points in either 59 or &. Then 
Example. I 
59. 
Let 
(l 
Then 
This is obvious since the sums 
33 <£ 
4. Let 51 be 
51. Let g—fi 
may have terms in common. Such terms are therefore counted 
twice on the right of 1) and only once on the left, before passing 
to the limit. 
Remark. The relation 1) may not hold when/is not > V. 
Example. Let 51=(0, 1), 59 = rational points, and (£ = irra 
tional points in 51- Let/= 1 in 59, and — 1 in (5. Then 
For let Mu 1 
and 1) does not now hold. 
Passing to tl 
To prove 2) 
2. Let 51 be an unmixed partial aggregate of the limited field 59- 
Let (5 = 59 - 21. If 
ff=f in 51 
= 0 in (£, 
then 7» 
6. 1. Letf 
be an unmixed ] 
f/=D/ 
then