DR. W. H. YOUNG ON THE GENERAL THEORY OF INTEGRATION. 243 



Condition of Inbyrability. T/ie necessary and sufficient condition fur the 

 integrability of a function f over a measurable set S is that the content L of the set 

 of points for which the maximum of the function is ^ k while the minimum S k, 

 should be a function of k whose value is zero, except for a set of lvalues of k of 

 content zero ; for each of these values <;/' k the points of discontinuity of the function 

 still form a set of content zero. 



Theorem 23. The content C of that component of the fundamental set whose points 

 are points of continuity at which an rntegrable function = k, is a function of k whose 

 value is zero, except for a set of values of k of content zero. 



Theorem 24. The content C of that component of the fundamental set at whose 

 ptrints a continuous function = k, is a function of k whose value is zero, except for a 

 set of values of k of content zero. 



24. We now return to the tentative definitions of 6, which we saw did not agree 

 with the usual definitions. On the other hand, the definitions we have since 

 constructed seem more artificial than these. It suggests itself, therefore, that the 

 most logical plan is to throw overboard the Riemann and Darboux definitions 

 altogether, and to define an integral as follows : 



Let the fundamental set be divided into measurable components in any conceivable 

 ivay, and let tlie content of each component be multiplied by the upper (lower) limit of 

 the values of the function at points of that component, and the sum of all such products 

 be formed ; then the outer (inner) measure of the integral is defined to be the lower 

 (upper) limit of all such summations. 



If we assume either that all sets are measurable, or that all functions are such that 

 the points for which the values of the function are 2: k (^ k) are measurable, or that 

 the functions with which we are concerned have this property, the argument of 21 

 still applies, and we can assert that the outer measure is equal to the inner measure 

 of the integral, and that each can be expressed in either of the two forms 



KS+Tlrft, 



.'K 



where I and J are the contents of those components of S, at every point of which the 

 function has values respectively i k and ^ k. 



It is not known whether any but measurable sets exist, or whether any functions 

 can be constructed not having the above property. M. LEBESUUE has, therefore, used 

 the term summable to denote the functions under consideration. 



Precisely as in 22 we can now prove the following theorem : 



Theorem. The content C of that component of the fundamental set at every point 

 of which a snimmtlilf function has the value k, is a function of k whose value is zero 

 except for a set of mines of k of content zero. 



In the case of a summable function, therefore, the outer and inner measures of the 

 integral agree, and we may call either the generalised integral of the function. As 



2 I 2 



