14 
POINT SETS AND PROPER INTEGRALS 
2. To illustrate the necessity of making/>0 in 1), let us take 
2i to be the Pringsheim set of I, 740, 2, while f shall = — 1 in 21. 
Then 
f / = -1 
j/=°- 
XX /=0 ’ 
and the relation 1) does not hold here. 
On the other hand 
Hence 
Iterable Fields 
11. 1. There is a large class of limited point sets which do not 
have content and yet — 
21 = f £• (1 
i/33 
Any limited point set satisfying the relation 1) we call iterable, 
or more specifically iterable with respect to -23. 
Example 1. Let 2i consist of the rational points in the unit 
square. Obviously _ — 
21= | £ = I 23 = 1, 
i/e 
so that 2i is iterable both with respect to 23 and £. 
Example 2. Let 21 consist of the points x, y in the unit square 
defined thus: 
For rational a; let 0 < y < |. 
For irrational x let 0 < y < 1. 
Here 21=1- 
Thus 21 is iterable with respect to £ but not with respect to 23. 
Example 3 
fined thus: 
Here 21 = 
Hence 21 is 
Example ^ 
the rational 
Here 21 = 
and similar i 
to either 23 < 
Example 5 
Here 21 = 
Hence 2i i 
2. Every i 
its projection 
This folio - 
12. 1. A1 
ing a proper 
Example. 
unit square 
set 2i* is ite 
2. The re 
ing that if 5 
is also iteral 
For let 2t 
crete has cc 
iterable witl