DR. W. H. YOUNG ON THE GENERAL THEORY OF INTEGRATION. 235 



15. Theorem 12. The quantity I is the upper integral of the function vrith 

 respect to S. 



The result of the preceding section, together with the tact that the upper integral of a 

 function is the same as that of its upper limiting function, show that it is only necessary 

 to prove the present theorem for upper semi-continuous functions. 



Let /' be an upper semi-continuous function, and let us suppose the fundamental 

 set S divided up into a finite or countably infinite number of sets of points, EI, Et . . . so 

 that the corresponding upper summation 



differs from the upper integral X by less than some assigned small positive quantity e. 

 Round every point of EI we can describe a small interval, in which the maximum 

 of f with respect to S differs from the value of f at P by a quantity less than, say 

 Ci, where 



= e 



and this interval may be decreased indefinitely. That is, we have a set of tiles,* each 

 of which may be chipped as much as we please, and their points of attachment fill up EI. 

 Applying the Tile Theorem, we obtain a finite or countably infinite set of the 

 tiles, each less than e, covering up every point of EI, and the sum of the tiles is less 

 than Ej + e',, where ^ ^ _ ^ 



In like manner we get a set of tiles from each set E,. Applying the Tile Theorem 

 to the set of all these tiles, since their points of attachment fill up S, we obtain a 

 finite or couutably infinite set of them, covering every point of S, and the sum of all 

 is less than S + e 7 . 



To each of these tiles d p we make correspond the lowest integer t such that its 

 point of attachment P belongs to E,- ; the sum of the tiles corresponding to any 

 particular integer .t will then be less than E, + e',, and the maximum of/ in each will 

 be less than or equal to yj+e,-. 



Now, if we include the boundary points of any interval, the set of intervals 

 consisting of (1) the simple parts and (2) the overlapping parts of the tiles, leaves 

 no points of S over, and gives us therefore a division of S by means of segments 

 (e, e'). Of these, the sum of the overlapping parts is less than e', since the content 

 of the tiles is not less than S, and their sum not greater than S+V ; thus the 

 contribution of the overlapping parts is numerically less than Me 7 . The contribution 

 of the simple parts, on the other hand, is less than the upper summation over the 

 tiles themselves, that is, less than 



Thus the upper summation over these segments (e, e 1 ) is less than 



X+e+Se+Me'+ee'+Me'. 



* Cp. " The Tile Theorem," ' Proc. Lond. Math. Soc.,' Ser. 2, vol. 2, Part I., p. 67. 



2 H 2 



