Mathematics. — “On an Integral Notion of Densor.” By J. G. 
VAN DER CoRPUT. (Communicated by Prof. Arn. DenJoy). 
(Communicated at the meeting of April 30, 1921). 
In this paper small Roman letters represent real numbers, Greek 
letters points in n-dimensional space, #,, and Roman capitals n-dimen- 
sional sets of points; an exception is only made for the letters 
Ff, f,® and p, which will always represent functions. The letters 
a, 6, x indicate the products of the coordinates resp. of a, 8, & By 
En the point is indicated of which each coordinate is the product 
of the corresponding coordinates of § and 7; if # is formed by the 
points §, En represents the set of the points En. When the coordinates 
a. 
: Sly. 
of 7 are all different from zero, each coordinate of — is equal to 
7 
E 
the ratio of the corresponding coordinates of € and 7; — indicates 
i 
wis 
the set of the points —, where & is again an arbitrary element of 
7 
E of KE are different from 
£. When the coordinates of every point 
1 ie af ee 
zero, E represents the set of the points = Finally, when the coor- 
dinates of a, 8,§&, are positive, A, B, X, Y represent the sets of the 
points of which each coordinate is positive and less than the corre- 
sponding coordinate resp. of a, 8, &, 7. 
We form a net of cells of 2 dimensions, i.e. an enumerable 
sequence of separate, open, connected sets of points G;(i = 1, 2,....), 
measurable (J), so that every point of R, lies in one of the cells 
G; or on its boundary. Of the projection of G; on each coordinate- 
axis it is assumed here that when g; indicates the upper limit, 9'/, 
the lower limit of the distances from the origin to the points of 
! 
# EL approaches to zero at the same time as —. 
Ji Ji 
this projection, 
In each cell G; or on its boundary we choose further a point 7; 
and we shall indicate the measure of G; by m;. 
Let f(m) be defined for all 7 in R,; let p bean arbitrary number 
between O and 1 and let us put 
y (§) = « = mF (§ Hi) 
Q* 
