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



upper integral, so defined, is greater than, or equal to, the lower integral ; (2) these 

 definitions agree with DARBOUX'S in all cases. 



It is easy to prove that (1) holds : 



Theorem, 4. The lower limit of the upper summations is not less than the upper 

 limit of the lower summations. 



For otherwise we could clearly take a quantity L lying between the upper and 

 lower integrals so defined, and find two divisions of the segment so that the upper 

 summation for the first division is less than L and the lower summation for the 

 second greater than L. If, now, we consider the division of the segment got by 

 combining these two divisions, that is, if we divide each of the former sets up into 

 the components which it has in common with each of the latter sets, the upper 

 siimmation is not increased nor the lower summation diminished, thus for this 

 division the upper summation is less than L and the lower summation greater than L, 

 which is impossible. This proves (1) to hold. 



In regard to (2) the following simple example shows that, in the general case, 

 there is no agreement between our tentative definitions and those of DARBOUX. By 

 dividing up into sets of points, instead of into intervals, we get a lower value for the 

 upper integral than that given by DARBOUX'S definition, and a greater value than 

 the lower integral. 



Example 5. Take the function which is 1 at all the rational points of the 

 segment (0, 1), and zero everywhere else. The Darboux upper integral, from to 1, 

 has the value 1 ; the lower limit of all possible upper summations is however 0, since 

 the rational points can be enclosed in a set of intervals whose content is as small as 

 we please. 



7. I now proceed to show that in the case of a function which is integrable, in 

 the ordinary sense of the word, so that the ordinary upper and lower integrals 

 coincide, the division into sets of points, instead of merely into a finite number of 

 intervals, leads to the same limit, viz., the integral of the function. To see this, we 

 have merely to remark that the upper and lower summations are respectively less 

 than the ordinary upper integral and greater than the ordinary lower integral, 

 except in the case of equality, and that, as shown in the preceding article, the 

 lower limit of the upper summations is not less than the upper limit of the lower 

 summations. 



Thus we have the theorem : 



Theorem 5. If a function be integrable in a given segment (region), the value of 

 the integral is equal to the limit obtained as follows : divide the segment (region) into 

 any finite or countably infinite number of measurable sets of points, multiply the 

 content of each set by the upper (loiver) limit of the values of the function for points of 

 that set, and sum all such products ; then the lower (upper) limit of such summations 

 for every conceivable mode of division is the integral of the function. 



8. I next proceed to show that in the case of an upper (lower) semi-continuous 



