1904.] 



The General Theory of Integration. 



447 



set. A particular form of division of S, analogous to that by means of 

 intervals of the same content as the fundamental segment, is shown to 

 lead infallibly to the upper and lower integrals of any function with 

 respect to S ; this mode of division is called division of S by means of 

 segments (e, e), it is such that each component lies inside a correspond- 

 ing interval of length less than e, the content of these intervals being 

 less than S + <?', and the points of S which are not internal to the 

 intervals forming a set of zero content. 



Based on this division of the fundamental set, we have an alternative 

 definition of upper and lower integration with respect to a fundamental 

 set, which is more nearly allied to the Darboux definitions for the case 

 when the fundamental set is a finite segment. This is as follows : — 



Let the fundamental set, excluding at most a set of points of zero 

 content, be enclosed in or on the borders of a set of non-over- 

 lapping segments each less than e, and of content less than S + e. 

 Then let the content of that component of S in any segment be 

 multiplied by the upper (lower) limit of the values of the function 

 at points of that component, and let the summation be formed 

 of all such products. Then it may be shown that this summation 

 has a definite limit when e is indefinitely decreased, independent 

 of the mode of construction of the segments and the mode in 

 which e approaches the value zero. This limit is called the upper 

 (lower) integral of the function with respect to the fundamental 

 set S. 



In the case when the upper and lower integrals coincide, the func- 

 tion is said to be integrable with respect to S, and the condition of 

 integrability is found in a form agreeing completely with Riemann's 

 condition in the case when S is a segment. To prove this the theorem 

 is required that the sum of any finite number of upper integrals of tipper 

 semi-continuous functions with respect to a fundamental set S is the upper 

 integral of their sum, and the proof of this theorem is given. 



It is then shown that, except in the case of upper (lower) semi- 

 continuous functions, the upper (lower) integral over the fundamental 

 set S is not necessarily equal to the sum of the upper (lower) integrals 

 over any set of components of S, but that this is the case when S is 

 divided by means of segments (e, e). 



A function which is integrable with respect to S is shown to have 

 the following properties : — 



(1) It is integrable over every component set of S. 



(2) The integral of the integrable function is equal to the sum of 



the integrals over every finite or countably infinite number of 

 components into which S may be divided. 



(3) The sum of the integrals of any finite number of integrable 



2 I 2 



