135 
We can therefore find a point « with positive coordinates so that 
for any point § in A 
| @ (é) — {ro dy |<g, 
E 
hence 
| De — friar (Sat? 
B, 
As |p—P|SP all points § in A for which ®(§) <q, satisfy the 
inequality 
Ren fron dn | < 89, 
R 
so that any point 3 in A has the property that the points § in B 
for which this unequality holds good, form a set with measure 
> (1 —hg)b. According to the first auxiliary proposition we have now 
appr. lim. p (§) = fre dy, 
E=0 
Hy. 
which was to be proved. 
A simple application of property 1 is e.g. : 
ee! MN 7 PE 
[f E indicates a measurable set with finite measure m, F (=) 
Ss 
the number of points of E with integer coordinates not lying in one 
of the coordinate planes, we have 
ld E 
appr. lim. — F (=) =m, 
F=0 © 
We may here replace r(z) by all the points on = with integer 
coordinates, among others when the intersection of E with each coor- 
dinate plane is limited. 
It appears from the first property that the integral of DenJor 
is at least as general as that of LrBrseuw; from the following pro- 
position (where n= 1 is assumed), the correctness will appear of 
Prof. DeNJoY’s supposition that the new integral notion is already 
more general for n = 1. 
Proposition 2: If f(—é)=—f(S, if (8) does not increase with 
ieneaseng. positive: 5, if f(6)=O0 for § >1 and ae Eye) =; 
= 
FEN) ts integrable (C) and the integral is 0. 
