140 
In this case: 
y' 4 eb 1 
—log—.  & vOs Org 2S (67 --—Ojy: 
q q uZj rw 4 u>j>w 
= : PP Ou eV Puan L7G. ; 
The second member of (8) is then < (8+1+1+1)qv=69y, 
so that the set of the points ¢ between O and y for which F'(&) > q, 
has at the origin a measure < 6qy. The points § between 0 and y 
with #(é)<q form therefore a set with measure > (1—6q) y. 
This holds good for any q between O and 3; according to the first 
auxiliary proposition the non-negative function F'(8) has in this 
case for £0 an approximate limit 0. By the aid of this we shall 
prove that also ®(£) has an approximate limit 0 for §—~0O. From 
the aforesaid, 
ee ide 
Ff (05-1 §—1) > f (ng E—1) 2 f(Oj—1) 
and 
mjf (Bj 81) fre §—1) dn> mj f(6; 5-1) 
55 
follows 
appr. lim. je —f = for dn = OB NI) EM MO 
of 
The values € for which 5 lies on the boundary of one of the 
cells G;, form an enumerable set so that we need only consider the 
case that é lies inside a cell G;. If A; indicates the left extremity, 
g; the right extremity of this cell, we have owing to f(— €) = — f (6) 
j=l 
sE (ft dn=é [FeaE—V ay + Ef 018-1) dy = 
J : 
j 
El 2 I.E 
=i (y) dy + ik Jm) dn = J fm) dn. 
— pe! Pis! 
The difference of the positive numbers 1 — 7;& and g;£—1 is 
less than their sum &(9;— 4); f(y) is absolutely taken < f(g; §—1) 
+/(—aé) = f(0;é— 1) —f(i— AE), hence the absolute value of 
the latter integral is less than &(u;— 4, { f(g: — 1) —f (dié — 1) } and 
