134 
If (8) and (a) represent the measures of the sets, formed 
by the points with the property  (&) 2 q lying resp. in B and A, then 
P (a)<a, and if B—A indicates the set of the points lying in B 
but not in A, then 
1(® (8) vo fo @agcnf a 6 
B—A B—A 
ENE bg’ 
<u (Pag cay Ze 
Kd a 
(e) 
hence, if we assume a = bq 
q == hb. 
b 
P (8) << a+ (hl). 
Proof of proposition 1: It is known’), that it is possible to 
find a limited set # with an exterior measure (J) for every number 
q between O and 1 and for every summable function f(x), so that 
fn) is limited and continuous on # and the integral of | /(%)| 
extended over the complement of #, is less than q°. Let p be an 
arbitrary point with Aaen coordinates. If on £ we assume 
F(y) = 0, outside L, f(y) =| f(ú)| and further 
DP E == 8 S mil (nis), 
the points § in B with the property ®(§)>q form according to the 
afore mentioned auxiliary proposition a set with measure < hdq. 
The points § in B with the property d(8) <q form therefore a 
set with measure > (1—hg)b. 
Let us further assume 
D (§) = «2 Smif (n:5), 
extended over all 2 for which 4;§ lies on £. According to the con- 
dition made for the net of cells, the dimensions of the cells G, for 
which 7:§ lies on the limited set #, approach together with & to 
zero. As on the limited set A with exterior measure (J), f (1) is limited 
and continuous, f(y) is integrable on # according to Riemann and 
in accordance with the definition of the integrals of RirMaNN we 
have in this case 
lim. D (8) = froo dy. 
é=0 os 
1) Cf. e.g. C. Caratutopory, Vorlesungen über reelle Funktionen, 1918, p. 
469, proposition 12. 
