DR. \V. H. YOUNG ON THE GENERAL THEORY OF INTEGRATION. 227 



of intervals should be finite, the ' upper summation can always be made numerically 

 as small as we please. The same is true of the lower summation. 



Theorem 3. If, however, we make the resti-iction that the set of points complementary 

 to tlie intervals always has zero content, ur has a content which decreases without limit, 

 as d does so, the limits approached by the upper and lower summations are perfectly 

 definite, and are, of course, the upper and lower integrals. 



The proof of this statement is identical in character with that given in 3 for the 

 corresponding theorem regarding the integral of an integrable function, which theorem 

 is, of course, a special case of the above. 



We should naturally ask whether we cannot correct this discrepancy by adding to 

 the upper or lower summation over the set of intervals the product of the content of 

 the complementary set of points into the upper or lower limit of the function for 

 points of that set ; or, if this does not suffice, by dividing up the complementary set 

 itself into components and adding the sum of the corresponding products. That 

 neither of these corrections suffice is shown by the following example : 



Example 4. Take the same set as before at which the function has the value 1, and 

 inside the largest of its black intervals place a similar set G' of content I', at every 

 point of which internal to that black interval the function has the value 2. Every- 

 where else the function is to be zero. The upper integral is I + 21'. 



If now we merely subdivide the black intervals of G' whioh lie inside the largest 

 black interval of G, and subdivide the remaining black intervals of G, the product oi 

 the content of the complementary set of points into the upper limit of the function 

 for points of that set will be at least 2(1 + 1'), while the summation over the intervals 

 will be zero. Thus the addition of the term in question would not correct the 

 result. 



If, on the other hand, we subdivide the complementary set into components whose 

 contents are themselves less than the norm d, we could, on the principle which has 

 already been employed, insure that each component contained a point at which the 

 function had the value 2, and the result would be the same as before. In neither 

 case do we obtain the upper integral. 



6. As we have seen in 4 and 5, the Darboux definitions of upper and lower 

 integration require modifications, if they are to be generalised in the manner proposed. 



We are naturally led to define upper and lower integration tentatively as follows : 



Let the division of the segment into (measurable) sets be performed in any con- 

 ceivable way, and let the upper limit of the values of the function in each partial 

 set be multiplied by the content of that set, and let the sum of these products be 

 formed ; then the upper integral is defined to be the lower limit of all such sums. 



Similarly the lower integral might be defined, the words " upper " and " lower " 

 being throughout interchanged. There is clearly no logical reason to prevent our 

 considering these limits. 



The names upper and lower integrals will, however, not be suitable, unless (1) the 



2 Q 2 



