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



Also, by the definition of content, we can determine a set of intervals containing 

 S, whose content lies between S and S + &, where 



a ........... (2), 



M being any quantity greater than the greatest numerical value of the function in 

 question for points of S. 



The set of intervals consisting of the common parts of these two sets will contain 

 S, have content lying l)etween S and S + &, and also be such that the upper summation 

 with respect to S over it has a value lying between I and I + a. From among the 

 intervals of this set, we can determine a finite number n, such that the sum of the 

 remaining intervals is less than 6. The content of the component of S which is not 

 external to these intervals is not less than S b. The upper summation over the 

 remaining intervals is, by 2, numerically less than a ; therefore the upper summation 

 over these n intervals is less than I + 2a. 



Let us now consider any division of S by means of segments (e, e'). There cannot 

 be more than 2 of these which lie partly inside and partly outside the n intervals. 

 The contribution of these to the upper summation will therefore be numerically less 

 than 2nMe. 



Since, by what has been shown, the segments which are not entirely external to 

 the n intervals, enclose a component of S of content not less than S b, the sum 

 of those segments which are external to the n intervals is not greater than 

 S + e' (S b), that is e' + b ; their contribution therefore to the upper summation is 

 less than M (e + b) numerically. 



Finally the contribution made by those segments which lie each inside one of the n 

 intervals cannot be greater than the upper summation over those intervals, that is 

 cannot be greater than I + 2a. Hence the upper summation over these segments is 

 less than I-f 3a + M (e' + 2ne). Using (1), it follows that we have only to take 



so that the upper summation over the segments may differ from I by less than e a ; 

 this proves the theorem. 



14. It is clear from the above proof that, when we divide S by means of 

 segments (e, e), it is immaterial whether, in estimating the upper limit with respect 

 to S over any segment, we include the end-points of that segment (supposing them 

 to belong to S), or not ; the potency of the points of S not included will in either 

 case be the same, and, as the theorem shows, in either case the upper summation will 

 be within d of the limit I, provided only e and e' are sufficiently small. 



Suppose, then, we exclude the end-points of every segment. Then the upper limit 

 of the values of the function for points of S inside any segment being the same as 

 that of the associated upper limiting function, we have the following result : The 

 quantity I is the same, for a function and its upper limiting function, 



