DR. W. H. YOUNO ON THE GENERAL THEORY OF INTEGRATION. 229 



function, the upper (lower) integrals may be obtained in a similar way, so that, 

 in the case of an upper semi- continuous function the tentative definition discussed 

 in 6 of the upper integral can he retained, and the same is true of the lower 

 integral of a lower semi-continuous function. 



To prove this let us suppose the segment (A, B) = S divided up into a finite or 

 countably infinite number of sets of points EI, Ej, ..., so that the corresponding upper 



summation 



,/,+,/,+ ... 



differs from the minimum X by less than some assigned small positive quantity c. 



Now round every point P of EI describe a small interval, in which the maximum 

 of f differs from the value of f at P by a quantity less than some assigned small 

 positive quantity t'i. We thus get a definite set of intervals enclosing all the points 

 of EI ; their content is therefore not less than the content EI, but we can so construct 

 them that it is less than Ei + e'i, where <-\ is another small positive quantity as small 

 as we please. In each of these intervals the maximum of the function f is less 



Let us do likewise for each set E r , choosing the small quantities so that 



et+e,+ ... = e'l+e't-K.. = e. 



Applying the Heine- Borel theorem to all these intervals, which enclose every point 

 of EI, Ej, ..., that is of (A, B), we can determine a finite number of these intervals 

 enclosing every point of (A, B), and we can insure that the content of these intervals 

 differs from their sum by less than e.* To each of these intervals we can attach the 

 index of the first of the sets E^ , from which it was constructed ; the content 

 of those which have the index i will then be less than , + </, and the maximum in 

 each will be less than _/) + <",. Hence the upper summation over the non-overlapping 

 intervals, consisting partly of the simple parts of these intervals and partly of the 

 overlapping parts, is less than 



where M is a positive quantity greater than the numerically greatest values of f. 

 That is, this upper summation is less than X + e + 2M + S + <', which, since e is at our 

 disposal, proves the theorem. 



A similar proof can be given for the case of the lower integral of a lower semi- 

 continuous function, or we may deduce the corresponding theorem in this case from 

 the fact that a lower semi-continuous function becomes an upper semi-continuous 

 function when its sign is changed, and at the same time the lower integral becomes 

 the upper integral. 



Bearing in mind now that the upper integral of a function is equal to that of 



* Cp. " An extension of the Heine-Borel Theorem," ' Messenger of Mathematics,' New Series, No. 393, 

 January, 1904. 



