DR. W. II. YOUNG ON THE GENERAL THEORY OF INTEGRATION. 225 



Theorem 1. If the interval (a, 6) be divided into a set of intervals and a set of 

 paint* of zero content, the length of each interval being lens than some assigned norm d; 

 and if the product of the length of each interval by the value of a given 

 function t an i/ fiaint of that interval \te. formed and the summation of all 

 products calculated, this summation has a definite limit when d is indefinitely 

 diminished; this limit is, of course, the integral of the function. 



Corollary. The value of the integral of an integrable function is unaltered if, at 

 the points of a set of zero content, we arbitrarily change the values of the function. 



In other words, if we add to the function an integrable null function, we leave the 

 integral unaltered. 



It will be convenient to prove the following theorem, which is, in practice, 

 indispensable ; from it the theorem on which DARBOUX'S definition of upper (lower) 

 integration is based, can be at once deduced : 



3. Theorem 2. Given any small positive quantity e h we can determine a positive 

 quantity e, such that, if the segment S be divided up in any manner into a finite 

 number of non-overlapping intervals, then, provided only the length of each interval is 

 less than e, the upper (lower) summation of any function over these intervals differs 

 by less than e^ from a definite limiting value, the upper (lower) integral. 



The following is the proof for the case of the upper integral ; with slight modifi- 

 cation it holds for the lower integral. 



Let I be the lower limit of all such summations ; then we can determine a division 

 of S into a finite numl>er n of intervals, such that the upper summation over these 

 intervals lies between I and 1 + fa . . . . 



Let e be chosen to satisfy the following equation 



= 



where M is any quantity greater than the greatest value of the function, and let us 

 consider any division whatever, into a finite number of non-overlapping intervals each 

 less than e. The number of such intervals which do not lie entirely in one of the 

 n intervals previously determined, is at most n, so that the sum of the terms 

 corresponding to these intervals is less than nMe, that is fa. 



In each of the remaining intervals the upper limit of f is not greater than the 

 upper limit of f in that one of the n intervals in which it lies, so that the upper 

 summation over the remaining intervals is not greater than that over the n intervals. 

 Hence the summation over our intervals, each being less than e, is less than I + e,. 



[Q.E.D.] 



4. We have now to discuss the Dartaux form of the definition of integration, 

 that is, in the first instance, to consider the effect of the modifications proposed on 

 DARBOUX'S definitions of upper and lower integration. 



The following example* shows that the theorems stated in these definitions no 

 * Example 1, of course, shows this in the case of an integrable function. 



VOL. OCIV. A. 2 O 



