DR. W. H. YOUNG ON THE GENERAL THEORY OF INTEGRATION. 233 



summations for every conceivable mod? of division is the upper (lower) integral of the 

 semi-continuous function for the fundamental set S. 



Taking this as the definition of the upper (lower) integral in the case of an upper 

 (lower) semi-continuous function, the definition in the general case is equivalent to 

 the following . 



Theorem 10. The upper (lower) integral of any function ivith respect to any 

 measurable set S is the upper (lotver) integral of its associated upper (lower) semi- 

 continuous function. 



12. The division of the continuum adopted by DARBOUX is, as has been proved, a 

 special case of a more general division of the continuum into intervals, by means 

 of which we obtain an upper summation differing by as little as we please from the 

 upper integral. This division was such that the sum of the intervals was equal to 

 that of the segment, while each interval had to be less than a quantity which 

 depended only on the degree of approximation desired. 



When the fundamental set S is not the continuum, but merely any measurable 

 set, we can, given any small positive quantity e', find a set of intervals enclosing 

 every point of S, that is, having every point of 8 as an internal point, the content 

 of the intervals lying between S and S + e 7 . If we assign any small positive 

 quantity e, there will only be a finite number of the intervals which are not less 

 than e, since their content is finite. Each of these we can divide into a finite 

 number of parts less than e, or we can in any other way determine inside the 

 intervals a set of intervals enclosing all points of S, with the possible exception of a 

 component of S of zero content. 



In each of these intervals there is a measurable component of S of content less 

 than e, and the sum of all these components is S. This division of the set S will be 

 found to correspond very closely to the division of the continuum contemplated 

 above, which is, of course, a special case of such a division ; in particular it will be 

 shown to lend itself conveniently to form approximations to the upper integral. 



I shall, for convenience, refer to such a division as a division of S by means of 

 segments (e, e 1 ) ; and the upper summation ot any function over these components 

 I shall call the upper summation with respect to S over the intervals. 



13. Theorem 11. Given any small positive quantity i, we can determine a 

 l>,>*\tn-i- (/'Kintiti/ .',, tii(<-l, t!,.it, if /In funilinni-ntiil .s>7 S / iIir></,-t/ iii unii in-nunT 

 by means of segments (e, e'), then, provided only e S e t , and ef^e*, the upper 

 summation of the function with respect to S over these segments differs by less than e\ 

 from a, definite quantity I, the lower limit of nil such upper summations, when S is 

 divided by means of segments. 



For, I being a lower limit, we can determine a set of intervals enclosing S, such 

 that the upper summation with respect to S over these intervals lies between 

 I and I + a, where 



6a = *, . . , j .-..,. * ^ 4'< >J '. . (1). 



VOL. OCIV. A. 2 H 



