DR. W. If. YOUNG ON THE GENERAL THEORY OF INTEGRATION. 239 



This follows from the definition at the end of 1 1 ; for if we divide the funda- 

 mental set S into (for instance) two measurable components S, and S_., and then 

 subdivide all three by means of segments, the definition shows that, in the case of an 

 upper semi-continuous function, the sum of the upper summations over the segments 

 with respect to Si and Sj is not less than the upper integral. But in each segment 

 the sum of the contents of the parts of Si and S s is the content of the corresponding 

 part of S, while the upper limit corresponding to S is the greater of the upper limits 

 corresponding to S! and S, so that the sum of the upper summations for Si and S s is 

 not greater than the upper summation for S. Thus the sum of the upper summations 

 with respect to Si and Sj lies between the upper integral with respect to S and the 

 upper summation with respect to S. And since, by properly constructing the 

 segments, we can make the upper summations differ by as little as we please from 

 the corresponding upper integrals, this proves the theorem. 



The above method of proof shows at the same time why it is that the theorem does 

 not hold for every function, for, if we form the associated upper limiting functions with 

 respect to Si and S, the values at the different points of S t are not always the same. 



20. If in finding upper (lower) integrals we wish to divide the fundamental set 

 up into convenient components, we must first replace the function by its associated 

 upper (lower) limiting function. 



Example 6 is a particularly instructive one ; the function is integrable over each 

 of the two component sets into which the segment which is the range of variation is 

 divided, and is not integrable over the segment itself. We easily see, however, that 

 the following theorems hold : 



Theorem, 18. If a, function be integrable over the fundamental set S, it is integrable 

 over every component set of S. From Theorem 1 7 it now follows that 



Theorem 19. The integral of an integrable function over its fundamental set S is 

 equal to the sum of its inta/rals over every finite or countably infinite number of 

 components into which S may be divided. 



For, as has been shown in the preceding section, the upper integral of a function 

 over the fundamental set S cannot be less than the sum of the upper integrals over 

 the component sets ; and the lower integral of the function over S cannot ta greater 

 than the sum of the lower integrals over the component sets. Since, however, the 

 function is integrable over S, it is also integrable over each of the comjxments. 

 Therefore the integral over S cannot be less nor greater than the sum of the integrals 

 over the components, so that it must be equal to this sum. [Q.E.D.] 



In this connection, it should be noted that the integration of an integrable 

 function (which has finite upper and lower limits) involves nowhere the idea of order, 

 even when, for convenience, we determine it as the sum of a countably infinite 

 number of integrals. The series of such integrals is an absolutely convergent one, 

 and it has the same sum however it be arranged. 



Theorem 20. The sum of the integrals of any finite number of integrable functions 



