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



22. By 18 the condition that a function should be integrable with respect to a 

 measurable set S is that the upper integral of the associated oscillation function 

 should be zero. The latter function is upper semi-continuous, by the corollary to 

 Theorem 8, so that we can apply the preceding theorem, putting K = 0, since this 



function is always positive or zero. 



f K ' 

 Thus the condition of integrability is that I (k)dk should vanish, where (k) is the 



Jo 



content of that component of S at every point of which the oscillation of the given 

 function is k. This is only possible if (k) is zero, except possibly at a set of values 

 of k of zero content ; as however, if (k) were positive for any value of k, it would 

 be positive for every lesser value of k, and therefore for a set of values of content 

 greater than zero, it is clear that there can be no exception. Thus we get again the 

 condition of integrability obtained in 18 (Theorem 14). 



23. When the function is integrable, the upper and lower integrals are equal, 

 otherwise the upper is the greater, thus 



therefore 



- Jdk, 



K 



Now no point can be such that the maximum there is less than k while the minimum 

 is greater than k thus every point of S belongs to at least one of the two sets 

 I and J. Let L denote the content of the set of points common to both I and J, 

 then* 



Whence 



S + L)^>(K'-K)S, that is 



the sign of equality being allowable if, and only if, the function f is integrable. 



Now L is the content of the set of points at each of which the maximum ^ k, while 

 the minimum is S k ; these points consist of : 



(1.) Points of continuity at which f=k; 



(2.) Points of discontinuity at which either the maximum is >& and the 

 minimum S k, or the maximum = k and the minimum <&. 



But it is clear, as at the end of 22, that if for any value of k the content of the 

 set (2) were not zero, there would be a set of values of k of positive content, for each 

 of which the content of the set (2) would not be zero ; L would then not be a null 

 function and the given function would therefore not be integrable. 



Thus we get a new form of the condition of integrability, and also theorems 

 relating to the distribution of the points of continuity of an integrable function, and 

 the values of a continuous function : 



* ' Theory of Content,' p. 49, (5). 



