141 
the approximate limit of this is according to the auxiliary propo- 
sition 3, equal to zero. As appears from (9) also the approximate 
limit of D(E) is in this case equal to zero for E00 and herewith 
proposition 2 has been proved. 
This property gives measurable functions that are integrable (C) 
but not summable. These functions are all unlimited at the upper 
side as well as at the lower side. That this is also necessary 
appears from 
Proposition 3: If a measurable function f (§) is integrable (C) 
and limited at the upper or at the lower side, this function is summable. 
Proof: Assume for instance that /(§) is limited at the lower side; 
let us put f:(§) = (5) or =t according to whether /(§)< or >t. 
The limited function /,(§) is summable, hence integrable (C), and 
the integrals are equal. Let this ¢ommon value be called s,. For 
increasing ¢ s, does not decrease and from /;,(&)</(§) follows that 
s, is not more than the integral (C) of f(£). The integrals 5, are 
therefore limited, hence /(§) is summable. 
Some properties holding good for the integrals of Lesescur, remain 
valid, others do not. From proposition 2, for instance, appears that 
the following two properties of the integrals of Lrpescue are lost: 
When a function is summable, its absolute value is also summable. 
When a function is summable in a set, it is also summable in any 
measurable subset. 
Finally we shall discuss three more properties that remain valid. 
Proposition 4: When a function f(§) ts integrable (C), the points 
for which f(§) is infinite, form a set with measure zero. 
Proof: We shall show that a function with the property that the 
points § for which the coordinates are positive and /(&) is infinite, 
form a set É with positive exterior measure, is not integrable (C); 
we can confine ourselves to a very simple net of cells, namely to 
the net of cells of which every cell consists of an n-dimensional cube 
of which the sides have a length 1 and are parallel to the coordi- 
nate-axes, and of which the centre coincides with a point with integer 
coordinates. In an analogous way each of the 2”—1 other parts 
into which £#, is divided by the coordinate-axes, may be treated. 
The exterior measure of EZ being positive, there exists a beam 
of which the coordinates of the angular points are positive and the 
sides are parallel to the coordinate-axes, while a subset D of F 
in M at a positive distance from the boundary, has a positive exterior 
measure. By enlarging the beam, if necessary, we can attain that 
if a and 2 represent the angular points of H, resp. with the least 
and with the largest coordinates, the coordinates of 8 are twice those 
