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



be integrable with respect to a measurable set S, is that the content of that component 

 of S at every point of which the oscillation of f is greater than, or equal to, k should, 

 for every value ofk, be zero. 



Remembering that the outer limiting set of measurable sets is measurable, and has 

 for content the limit of the contents of the defining components, this gives us the 

 following alternative statement : 



The necessary and sufficient condition that a function sJundd be integrable with 

 respect to a measurable set S is that the content of that component of S, at every point 

 of which the function is continuous with respect to S, should be equal to the content 



of 8. 



19. With the ordinary definition of upper (lower) integration, or of integration, it 

 was at once evident that if the segment or region of integration were divided into 

 two parts (segments or regions), the sum of the upper (lower) integrals over the 

 separate parts was the upper (lower) integral over the whole segment or region. 



Theorem 15. More generally, it is evident from Theorem 1 that the sum oj 

 the upper (lower) integrals over any set of non-overlapping segments or regions is 

 equal to the upper (lower) integral over the whole segment or region, provided only 

 the content of those segments or regions is the same as that of the fundamental 

 segment or region. 



That this is not so for the general case when the fundamental set of points, whether 

 a segment or not, is divided into component sets is shown by the following simple 

 example : 



Example 6. Let f be zero everywhere except at the rational points of the 

 segment (0, 1), and let f have the value unity at the rational points, and consider 

 the integrals over the rational and irrational points separately ; both of these are 

 zero. The upper integral over the whole segment is however unity, and the lower 

 integral is zero. 



The alternative definition of upper (lower) integration, given in 11, shows that 

 when the two component sets consist of parts of S obtained by means of segments, 

 the sum of the upper (lower) integrals over the two components is the upper (lower) 

 integral over the whole fundamental set. 



Theorem 16. More generally, the sum of the upper (lower) integrals over any 

 finite or countably infinite number of non-overlapping components of S obtained by 

 means of segments (e, e'), is the same as the upper (lower) integral over the whole 

 fundamental set, provided only the content of the components is equal to that of the 

 fundamental set. 



An upper (lower) semi-continuous function stands here again in an exceptional 

 position. We have, in fact, the following theorem : 



Theorem 17. The upper (lower) integral of an upper (lower) semi-continuous 

 function over any fundamental set S is equal to the sum of its integrals over every 

 finite or countably infinite number of sets into which S may be divided. 



