Dfc. W. H. YOUNG ON THK GENERAL THEOKY OP INTEGRATION. 251 



lower summations without being K-ss tlum the lower limit of the upper summations; 

 neither can it be greater than the lower limit of the upper summations without being 

 greater than the upper limit of the lower summations. Thus I must be actually 

 equal to the upper limit of the lower summations as well as to the lower limit of the 

 upj>er summations in the case of either X" or X' ; that is to say, I is the generalised 

 integral of either X" or X', and both these functions are summable. [Q.E.D.] 



Corollary 1. Back ordinate section of a measurable set beiny moved on its 

 ordinate in such a manner that the (linear) content of the section is unaltered, and 

 that the whole set remains measurable, the content of the whole set is unaltered. 



Corollary 2. At each point of a set of points of content A draw an onlinate, 

 and on it take any set of points of (linear) content B, the content of the whole set 

 so formed is AB. 



Here, as elsewhere, the fundamental set need not be a linear set, but may have a 



content of any number of dimensions. 



ft? 

 Idk as the content of any 

 



measurable set (provided the set got by closing it has finite content), here 1 is the 

 content of the set of points of the fundamental set at which the inner (or the outer) 

 measure of the content is i k. This, together with the preceding section, give the 

 solution of the problem alluded to in 26, viz., the reduction of the calculation of 

 n-dimeusional content to that of (n 1 )-dimensional content, and so ultimately to 

 that of linear content. 



Bearing in mind the definition of a generalised integral, we have the following rule 

 for finding the content of an u-dimeusional set : Take any hyperplane section of the 

 set, project the set on to this hyperplaue, and take any measurable set containing 

 this projection as the fundamental set S. Divide S up in any way into a finite or 

 countably infinite set of measurable components, and multiply the content of each 

 component by the upper (lower) limit of the values of the (linear) inner or outer 

 content of the corresponding ordinate sections of the given set ; summing all 

 such products, the lower (upper) limit of all such summations is the content of the 

 given set. 



32. I have explained the geometrical representation of generalised integration 

 with respect to a set. It is of interest to note the corresponding representations of 

 what I call ordinary upper and lower integration with respect to a set. 



Consider first the case where the fundamental set S is a segment, and form the set 

 of blocks corresponding to the generalised integral of the function. If the function 

 is everywhere positive, the geometrical representation of the ordinary upper integral 

 is obtained by closing the plane set of points constituted by the blocks. If the 

 function is not everywhere positive, we can make it so by adding a constant, which 

 is geometrically equivalent to adding a rectangle to the representative set, part of 

 which, viz., E.,, was to be considered as negative, and must be subtracted from the 



2 K 2 



