DR. W. H. YOUNG ON THE GENERAL THEORY OF INTEGRATION. 237 



The upper summation over the overlapping parts is in each of the three cases 

 numerically less than Me', where M is a quantity greater than the greatest numerical 

 values of /i, / s , and F. The simple part , of any tile rf p> contains the point of attach- 

 ment P,, therefore the upper limit in that part differs from the value at P by less 

 than ej. Thus the upper summation over the segments lies between 



and 



Thus, denoting by 6 with any distinguishing suffix or accent a quantity numerically 

 less than 1, 



with similar equations for the other two functions. 

 Now the summation 



is the sum of the corresponding summations for f\ and/ ; whence it follows that the 

 upper integral \ Yds differs from the sum of the upper integrals of f v and f* with 

 respect to S by a quantity which is smaller than any assigned positive quantity, 

 which proves the theorem. 



18. Denoting by f h f*, and F the associated upper, lower, and oscillation 

 functions of a given function /, the three functions f lt ft, and F are all upper semi- 

 continuous with respect to the fundamental set, and F is the sum of fi andyj. 



The upper and lower integrals of f with respect to the fundamental set are, by 

 Tlieorem 10, the upper integral of fi and the lower integral of f t respectively, and 

 the latter is minus the upper integral of f+ Thus the excess of the upper over the 

 lower integral of f with respect to the fundamental set is the sum of the upper 

 integrals offi &ndf t , that is, by Theorem 13, the upper integral of F with respect to 

 the fundamental set. 



Thus f will be integrable with respect to the fundamental set if, and only if, the 

 upper integral of its associated oscillation function be zero. 



Let GI denote the component of the fundamental set at which F has a value greater 

 than, or equal to, k; by Tlieorem 7, G* is a measurable set, let its content be I*. 

 Then if, omitting at most a component of zero content, we enclose the fundamental 

 set by means of segments (e, e 1 ), since the common part of two measurable sets is 

 measurable, there will be at most a component of Gj of zero content not included in 

 the segments, and the remaining component of G* will have content I*. Thus the 

 content of the components of the fundamental set in segments containing points of 

 G* will not be less than l t , and the upper limit of F in each will not be less than k. 

 Hence the upper summation will not be less than 1 . / . 



Thus it is clear that the upper integral of F cannot vanish unless, for all positive 

 values of k, It is zero. 



Theorem 14. Thus tJie necessary and sufficient conditions that a function f should 



