448 



Dr. W. H. Young, 



[Apr. 23. 



functions over S is equal to the integral of the sum of those 

 functions over S. 



In § 21 the calculation of upper and lower integrals with respect to 

 any fundamental set S is reduced to a problem of ordinary integra- 

 tion. The formulae, which are similar in form to those already given 

 Iry the author for the case when S is a finite segment, in a paper 

 presented to the London Mathematical Society, are as follows : — 



The upper integral of any function with respect to a measurable set 



where K is any quantity not greater than the lower limit, and K' 

 not less than the upper limit of the function for points of S, I being 

 the content of that component of S at every point of which the 

 maximum of the function is greater than or equal to h 

 The lower integral is 



J being the content of that component of S at every point of which 

 the minimum of the function is less than or equal to k. 



These formulae lead to certain theorems with respect to the distribu- 

 tion of the values of an ordinary continuous function and of an 

 integrable function. 



The remainder of the paper is devoted to the discussion of the inner 

 and outer measures of the integral of any function, and in the case 

 when they are equal of the generalised integral of a function, which is, 

 in this generalised sense, integrable. In particular it is shown that 

 such functions are none other than the functions which Lebesgue has 

 named summable, and the generalised integral is shown to be identical 

 with the Lebesgue integral in the case when S is a finite segment; a 

 geometrical interpretation of the integral, similar to that used by 

 Lebesgue, is given in the general case. 



Contrasting the first definition given of the generalised integral 

 with the geometrical definition, it is seen that they stand to one 

 another in the same relation as the ordinary definition of integration, 

 say of a continuous function, to its definition as a certain area. Just 

 as, however, the mathematical concept of area is more complex than, 

 and, indeed, depends on that of length, so does the theory of the 

 content of a plane set of points depend naturally on that of a linear 

 set. Just as the determination of area requires the application of the 

 processes explained in the first definition of integration of continuous 

 functions, so with the content of a plane set. Thus the comparative 

 simplicity of the geometrical definition is only apparent. 



Sis 



ISk, 



J K 



