133 
Auxihary proposition 2: Suppose 0<q<1. If B represents a 
point with positive coordinates, if the non-negative function fy) is 
summable and has a sum <q’, the points § in B with y(§)2q 
form a set with measure < hbq, where h only depends on the choice 
of p and the net of cells. 
Proof: Let K be the part of R, that has positive coordinates. 
Let there be associated to each point & of &,' the function F (§) 
indicating the sum of the measures m; of all the cells G; of 
which the point 9; lies in X and each coordinate is more than p. 
If #(3) is positive, each coordinate of & is more than p. In accord- 
ance with the condition made for the net of cells, we can now 
determine a number A only dependent on the choice of p and the 
net of cells, so that the point 4 the coordinates of which are 
n 
Vh—1 times as much as those of &, has the property that the cells 
G; of which the sum of the measures has been called F'(§), all 
lie in Y. Hence F'(é) <(h—1)z. 
For any point « in B we have 
ER 4 Ff (a8) dd > ah FE jen las) ae zm fA beh 
R, (4;) 
where the sum has to be extended over all 7 for which each coor- 
dinate of 1; is more than p and where in the last integral each 
coordinate of & exceeds the corresponding coordinate of 4;. The 
second member is in this case 
E' m; 4 a= (= des, mi f (nis). 
() (2) 
A similar reasoning gives the analogous expressions for the other 
parts into which &, is divided by the coordinate planes and from 
this there follows through addition 
de ‘mp (& 
aw framen |S — 2 mi f (1/8) =| Als 
(«) (a) 
1 pi q° 
[res ds =-(7@ ds < 5 
R, B, 
there follows therefore 
(Ba | ne 
B, 
From 
@) 
