132 
in the points § with positive coordinates where the right hand 
member has meaning; the sum extends over all 7 for which each 
coordinate of 9; taken absolutely is more than p. As an extension 
and a slight alteration of the integral-definition (C) given by Prof. 
A. Densoy') I shall say that f(m) is integrable (C’) and that the 
integral has the value 7 when the approximate limit of  (<) exists 
and has a value /, independent of the choice of p, the net of cells 
and the points 1; We write this: 
appr. lim. p (8) = 1; 
5=0 
this means that an enumerable set V ean be found for which 
lin. p (8) =, and the metric density of V at the origin is 1 (here 
£—0 
each coordinate of every element of V is considered to be positive). 
We say that V has a metric density d at the origin, if for each 
positive number e<{d a point @ with positive coordinates can be 
found so that all the points 8 in A have the property that the 
measure of the subsets of V in B lies between (d—e)b and (d+e)b. 
Hence OS d< 1. The following auxiliary theorem is of importance, 
of which the proof, given by Prof. Drengoy *) for linear sets, can be 
extended at once to more dimensional sets. 
Auwiliary proposition 1: Lf for every positive q <1 a point « with 
positive coordinates and a measurable set V can be found for every 
point S of which \p(&)—l) <q, while for every point Bin A the subset 
of V in B has a measure > (1—q)b, we have 
appr. lim. p ($) = /. 
gesl) 
For „== 1 Prof. Drnioy has found the proof of 
Proposition 1: Every summable function is integrable (C) and the 
integrals are identical. 
We shall precede the proof of this theorem for an arbitrary n by 
an auxiliary proposition. 
1) Sur Vintegration riemannienne; Comptes Rendus, 169, (1919); p. 219—221. 
Prof. Densoy gives the definition only for n = 1. The alteration in the two defi- 
nitions for “= 1 consists chiefly of this, that in the originai definition the length of 
each interval Gi is supposed to be less than 1, whereas we assume only that 
this length divided by the distance from Gi to the origin, approaches to zero for 
io Prof. Densoy gives at the same time two more integral definitions, which 
he indicates by (A) and (B). Mr. T. J. Boks studies in his thesis for the doctorate 
(not yet published) the integral notion (B) and derives the two properties of them, 
‘corresponding to the properties 1 and 2 of this paper. Probably this thesis (written 
at Utrecht) will appear in the Rendiconti di Palermo (1921). 
2) Sur les fonctions dérivées sommables, Bulletin de la Societé Mathématique 
de France, tome 43 (1915); p. 161—248; cf. p. 165—168. 
