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



its associated upper limiting function, and that its lower integral is equal to that ot 

 its associated lower limiting function,* we have the following theorems, corresponding 

 to that given at the end of 7. 



Theorem 6. To find the upper integral of any function we may proceed as follows; 

 divide the segment or region into any finite or countably infinite number of measurable 

 sets of points, multiply the content of each set by the upper (lower) limit of the 

 maxima t (minima) of the function at points of that set and sum all such products ; 

 then the lower (ujyper) limit of all such summations for every conceivable mode of 

 division is the upper (lower) integral of the function for the segment or region. 



9. We are now able to define upper and lower integration over a set of points, 

 instead of merely over an ordinary region ; we may, indeed, if we please, suppose the 

 function defined only for the set of points. The set will lie assumed to be measurable, 

 so that, like an ordinary region, it can be divided into component sets, each of which 

 is measurable. 



In the special case, in which the content of the set is zero, we define the upper 

 and lower integrals to be zero also. In the general case the definition will be as 

 follows : 



Divide the fundamental set S into any finite or countably infinite number of 

 measurable components, multiply the content of each component by the upper (lower) 

 limit of the maxima (minima) unth respect to S of the function at all points of that 

 component and sum all siich products; then the lower (upper) limit of all such 

 summations for every conceivable mode of division is the upper (lower) integral oj the 

 function for the fundamental set S. 



Further, when the upper and lower integrals for S are equal, the function may be 

 said to be integrable over that fundamental set S. 



Thus we have defined upper integration, lower integration, and integration over 

 any measurable set S in such a manner that, in the particular case when the set S is 

 a segment or region, we get the ordinary Riemann and Darboux integrals. 



10. Summing up our results so far, we saw that, though DAKBOUX'S form of the 

 definition was preferable to RIEMANN'S, it did not at once lend itself to generalisation. 

 I then showed how to modify it so that the number of intervals should not necessarily 

 be finite, provided that their content was equal to that of the segment. We then 

 saw how the introduction of the maximum (minimum) at a point, instead of the value 



* Cp. Upper and Lower Integration, ' Proc. Lond. Math. Soc.,' Ser. 2, vol. 2, Part I., p. 55, also 11, 

 below. 



t It should be carefully noted that the maximum at a point of a set is the lower limit of the tipper 

 limit of the values of the function in a small interval or region containing the point, when that interval 

 or region is indefinitely decreased. Similarly for the minimum. In the enunciation of the above 

 theorem, therefore, the word " maxima (minima) " has nothing to do with the particular set to which the 

 point belongs, the upper (lower) limit, however, is taken with respect to that set, that is, it is the upper 

 limit of the maxima corresponding to the various points of that set. Similarly in the definition of 9 the 

 set S takes the place of the segment or region. 



