Ml.'. \\. H. YOUNG ON THK (JKNKI.'AI. THKol.'Y <M [NTBGRATION. 223 



the L.M.S. ah-eady cited, and I apply one of the results of this paper, viz., that the 

 content of any closed n dimensional set may IKJ expressed both as an ordinary integral 

 in terms of an n 1 dimensional content, or as the upj>er integral of the content of 

 sets of lower dimensions, to obtain the corresponding theorem for any measurable set 

 (contained in a closed set of finite content). 



PART 1. OF FINITE PROPER INTEGRALS. 



I. The necessary and sufficient condition that a function should satisfy the 

 requirements of J{IKM \\N'a definition is simply that the content of the j)oints of 

 continuity of the function should be equal to the length of the segment (, 6), (or, 

 in the case of a multiple integral, to the content of the region over which the 

 integration is extended), that is to say, when this condition is satisfied, the sum- 

 mations referred to in the Uiemann definition have a definite limit, independent of 

 the mode of division, &c. Is this still true when for " interval " the expression '' set 

 of points" is substituted,* and for " length of interval" the "content of the set of 

 points " ( The following example shows that, when these substitutions have been 

 made, the definition, as it stands, ceases to have any meaning, even in the case of 

 continuous functions. 



Example 1. Take y x as the function. The Kiemann integral I xdx has the 



value . If, however, after dividing the segment (0, 1) into n equal intervals, we 

 abstract the u points 



f.i-'^> 



and add them singly to what remains of the intervals, we obtain n measurable sets of 

 points, each of the same content as before, viz., - ; in each, however, there is a point 



iv 



at which the function has a value greater than ^, except possibly in one of the sets 

 in which there is a point at which the function has the value J. The summation is 

 therefore always greater than J, so that it is clear that we do not get the same limit 

 as before when n is indefinitely increased. 



The principle of this example shows that in the general case also, except in the 

 .single case when the function is a constant, different modes of division of the segment 

 into a finite number of sets of points each of content less than a given norm d, and 

 different modes of proceeding to the limit, will certainly not always give the same 

 limiting value of the summations. Thus, suppose the function to be continuous at its 

 upper limit, then we can arrange that the mode of division is such that to every 

 partial set at every stage a point belongs for which the function differs from its 



* I shall always, except when the contrary is stated, suppose that the seta employed are measurable Bets. 

 so that the sum of two non-overlapping sets has for content the sum of their separate contents. 



