232 DR. W. H. YOUNG ON THK GENERAL THEORY OF INTEGRATION. 



function, I call the new function the associated upper (lower) limiting function of the 

 given function. Using the term oscillation at a point for the excess of the maximum 

 over the minimum, we get a third associated function by taking as value at a point 

 the value of the oscillation there, this I call the associated oscillation function. 



Theorem 8. The associated upper (loioer) limiting function of any function with 

 respect to a fundamental set S is an upper (lower) semi-continuous function with 

 respect to S. 



For, complete the set S, and let H be the closed set so obtained. Form an 

 extended upper (lower) limiting function as in the proof of the preceding theorem. 

 The original discontinuous function may be supposed to have the same values in the 

 points of (H S) as this extended upper (lower) limiting function. A proof precisely 

 similar to that for the continuum* proves that the new upper (lower) limiting function 

 is upper (lower) semi-continuous. 



Now it is plain that, though the points of (H G) may have points of G for 

 limiting points, the upper (lower) limits of the values of the function in the neigh- 

 bourhood of points of G are the same for the old and new upper limiting functions. 

 Therefore the values of the extended upper (lower) limiting function at the points of 

 G are the maxima (minima) with respect to either H or G, so that the old upper 

 (lower) limiting function is upper (lower) semi-discontinuous with respect to S. 

 [Q.E.D.] 



Corollary. The associated oscillation function, being the sum of two upper semi- 

 continuous functions,] is itself an upper semi-continuous function. 



Theorem 9. If S' be a component of the fundamental set S, such that all the 

 limiting points of S' which belong to S are contained in S', then the upper (lower) limit 

 of a function in (S S') is the same as that of the associated upper (lower) limiting 

 function in (S S'). 



This follows from the fact that the maximum (minimum) at any point P of (S S') 

 is unaffected by the values of the function at the points of S', since P is not a limiting 

 point of S'. Whence the result is easily deduced. 



The fact that the value at a point of an upper (lower) semi-continuous function is 

 the maximum (minimum) at that point in the case of such a function, enables us to 

 substitute the word value instead of maximum (minimum) in the definition of 

 integration. 



The definition in this case takes the following simplified form : 



Divide the fundamental set S into a finite or countably infinite number of 

 measurable components, multiply the content of each component by the upper (lower) 

 limit of the values of an upper (loiver) semi-continuous function at points of that 

 component and sum all such products; then the Imver (upper) limit of all such 



* BAIRE, 'Ann. di Mat.' (3), vol. III., 1899. 



t This property is evidently unaffected by the substitution of a fundamental set instead of a segment 

 or region. 



