DR. W. H. YOUNG ON THE GENERAL THEORY OF INTEGRATION. 231 



at a point, permitted of division of the segment into sets of points, instead of merely 

 into sets of intervals ; and, finally, that in this form the definition is applicable to 

 integration over a set of points. 



There is, however, another mode by which we can define integration with respect 

 to a set of points in such a manner as to get the Darboux integral when the 

 fundamental set reduces to a segment or ordinary region. This new generalised 

 definition is in some respects more closely analogous to the original Dartxmx definition 

 and brings out, more clearly than the one just given, the distinction between an 

 interval and a set of points in general, throwing light, as it does, on the question why 

 it is that the limit of the upper (lower) summations over a segment is different when 

 the segment is divided into intervals from what it is when the division is into set of 

 points. Like the Darboux definition, it concerns itself with the actual value of the 

 function at a point instead of the maximum (minimum) there, while it divides the 

 fundamental set up into components, closely analogous to the intervals in the Darboux 

 definition. 



It will be convenient, and conduce to clearness, to give first a few preliminary 

 explanations and theorems. We require to define and give one or two properties of 

 semi-continuous functions, when the region of existence is a set of points. It will be 

 found that the introduction of these functions materially simplifies the treatment of 

 the subject. 



11. The definition of an upper (lower) semi-continuous function, defined for any 

 fundamental measurable set, does not differ from the usual definition for a segment, 

 or ordinary region, the maximum (minimum) in each case is to be estimated with 

 respect to the fundamental set alone, in the usual case the fundamental set being that 

 segment or region, and in the general case that measurable set. 



Theorem 7. If a function, defined unth respect to a measurable set of points S, be 

 an upper (lower) semi-continuous function, the points at which the value of the 

 function is i k (s k) form a measurable set. 



Complete the set of points S, i.e., form the smallest closed set H of which S is a 

 component. Attribute to the function at the points of H which are not points of S 

 the upper (lower) limit of the values of the original function at points of S in a small 

 neighbourhood of the point, when that neighbourhood is indefinitely decreased. We 

 thus get a function which is upper (lower) semi-continuous with respect to a closed 

 set H. By an argument of precisely the same nature as that used for the case when 

 H is a segment, it follows that the set of points at which the new function is 

 i k (^ k) is a closed set, say Q. 



Since both Q and S are measurable, the same is true of their common component, 

 which is none other than the set of points at which the given function, upper (lower) 

 semi-continuous with respect to S, is 2: k(^k). [Q.E.D.] 



Definition. If at every point of the fundamental set S we take, as the value of a 

 new function at any point, the maximum (minimum) with respect to S of a given 



