224 DR. \V. H. YOUNG ON THE GENERAL THEORY OF INTEGRATION. 



maximum value by a quantity as small as we please, and there is nothing to prevent 

 our taking precisely this point as that at which the value of the function is to be 

 taken in forming the summation ; similarly for the minimum value, and it is clear 

 that the limits obtained in these two cases could only agree when the function is a 

 constant. 



It is plain, moreover, that there is no room for discussion of the range of possible 

 values of limits corresponding to the various conceivable modes of division. Speaking 

 generally, the range will be from SM to Sm, where S is the length of the segment, 

 or more generally tt-dimeusional volume of the region, and M and m are the upper and 

 lower limits of the values of the function. Thus the Riemann definition completely 

 breaks down when we attempt to generalise it in this direction. 



2. There is another direction, however, in which an important generalisation of 

 the Riemauu definition is possible. If we change the words " finite number of 

 intervals" into "set of intervals" and add "such that the content of the external 

 points is zero," the definition still holds good ; we get a perfectly definite limit, which 

 is, of course, the Riemann integral. 



To prove this, we notice first that the content of the set of intervals is in tins case, 

 and in this case alone, the same as that of the segment (or region) under discussion, 

 say S. 



Let the intervals be arranged in any way in countable order d\ t d.,... Then, since 

 the d's are all positive, their sum is an absolutely convergent series; therefore the 

 same is true when the content of each d r is multiplied by a quantity f, which, for all 

 values off, lies between finite upper a'ud lower limits, say between M, as is the case 

 in forming our summations.* 



If we consider n of these intervals in order d l} c/ 2 . . . d,,, these leave over a finite 

 number of complementary intervals, say d\, d' a . . . d', a , and we can so choose n that 



the sum of these latter intervals d' r is less than ^7, while the contribution to our 



summation over the remaining intervals c/ /H .,, cZ, l+2 ... is numerically less than ^e. 



If now we form the summation in RIEMANN'S way over the finite number of 

 intervals di, d. 2 . . . d u , d\, d'. 2 . . . d' M , and compare it with the corresponding summation 

 over the set of intervals d lt d^ . . . ad inf., we see that the difference between the two 

 .summations is less than c. Since c is at our disposal, and we can insure that both 

 the intervals c/, and d' r are less than any assigned norm, this proves the statement 

 embodied again in the following theorem : 



* I take this first opportunity of emphasising the fact that, though it is convenient, indeed necessary, 

 iu forming the sum of an infinite number of terms to arrange them in some sort of order, in doing so here 

 we do not introduce the idea of order into the concept of integration. Indeed, from the definitions it is 

 evident that the concept of integration no more of itself involves the idea of order than do the concepts of 

 length, area, and volume. The distinction of the two notions has, perhaps, not always been present to 

 the mind of some writers. 



