23(5 DR. W. H. YOUNG ON THE GENERAL THEORY OF INTEGRATION. 



Choose e and e' so small that 1) this is less than X + e, and 2) it (lifters from I by less 

 than e (by Theorem 11), e being an assigned small positive quantity as small as we 

 please, which proves that in this case X and I must be identical, and so, as already 

 pointed out, proves the theorem. 



16. It is now possible to give the promised alternative definition of integration ; 

 the preceding theorems prove the equivalence of the two definitions : 



Definition. 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-overlapping 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 is 

 indefinitely decreased, independent of the mode of construction of the segments and 

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

 (loiver) limit of the function with respect to the fundamental set S. 



In the special case when the upper and lower integrals agree, the common value is 

 called the integral of the function. 



17. To find the condition of integrability of a function with respect to a funda- 

 mental set, we require the following theorem : 



Theorem 13. The upper (lower) integral of the sum of any finite number of upper 

 (lower) semi-continuous functions with respect to a fundamental set S is the sum of the 

 upper integrals of those functions. 



Let F be the sum of two upper semi-continuous functions /i and/ a ; then F is itself 

 upper semi-continuous. Then, by Theorem 11, 61 being assigned, we can determine e s 

 so that if S be divided by means of segments (e, e') and the corresponding upper 

 summations formed, then provided only e and e' are each less than e*, the summations, 



. - ..' v ,. .'. . (2), 



i ,.'/>. , . (3), 



differ from the corresponding upper integrals by less than e^. 



Since fi, fz, and F are all upper semi-continuous with respect to S, we can, to each 

 point P of S, determine an interval round P, such that the maxima of /i, / 2 , and F in 

 that interval differ from the corresponding values of f h ft, and F, by less than some 

 assigned .small positive quantity e 3 , and this interval may be indefinitely decreased. 



Applying the Tile Theorem, we see that every point of S is covered by one at 

 least of a countable number of these tiles, each less than e, no point of attachment 

 being covered by any other tile, and the sum of the tiles differing from S by less 

 than e'. Thus we may take as our segments (e, e') those respectively covered by the 

 simple and overlapping portions of these tiles, if we include the boundary points of 

 each such segment in it. 



