137 
omitting the value or values of 7 for which the origin lies in G; 
; freee 1 
or on its boundary) the distances of which to — are more than 
Vi 
4q measure PER According to the preceding auxiliary proposition 
i 
(where the points d; indicate the endpoints of the sets of points —, if 
G; 
1 
necessary after reversal of the direction of the €-axis), the complement 
of E has at the origin a metric density <q, hence / itself has a metric 
| DB peer 1 
density >(1—q). For any Son E we have | &— —| 2 1 measure =~, 
7 | nT 2 Gi; 
hence according to (3) 
1 
measure— 
A mis i AE 
Em; f (ni §—-1) =— (81) f (mi €—1) > 0. 
IS gi 
Ni G, : a 
As this holds good for any g between O and 1 the theorem in 
question follows from this in connection with the auxiliary pro- 
position 1. 
Proof of proposition 2: The relation to be proved is: 
ene dn te) eM rd il ien ea wine (4) 
As f is zero to the left of the point — 1, 7 takes in this sum 
only values for which the whole or part of G; lies to the right of 
the origin. If w is an arbitrary point to the right of the origin, 
every term in (4) with such 7 that the whole or part of Gj lies to 
the left of w, approaches to zero with € and the number of these 
terms is limited, so that in (4) we need only take into account 
those values of 7 for which the cells of G; lie to the right of @. 
We shall indicate those cells, arranged from left to right, by 
0,10; (j= 1,2,...). As lim 
joer Yj—1 
= 1, we can choose w in such 
6; | 
a way that always aT <2. According to the preceding auxiliary 
jl 
proposition, for §—0O the approximate limit of the term or the 
Wee cae 
terms in (d) with such 7 that E lies in G; or on its boundary, 
is zero. We can therefore skip this value or these values of 7 and 
we shall indicate this by an accent to =, so that it is sufficient 
to prove 
appr. lim. D(E) =0, where ®(§)=6& BSL mj f (ns §—1). 
§=0 gl 
