446 



Dr. W. H. Young. 



[Apr. 23, 



general division of the fundamental segment into component sets of 

 points, whose content plays the part of the length of the intervals in 

 the original definitions, leads to summations which do not, in general, 

 have a definite limit even for integrable functions. The lower limit of 

 such generalised upper summations is shown to be not less than the 

 upper limit of such generalised lower summations ; but it is shown that 

 in general, only in case of upper continuous functions does the former 

 give us the upper integral, and in the case of lower semi-continuous 

 functions does the latter give us the lower integral. In general, 

 introducing the terms outer and inner measure of the integral for these 

 limits, the lower integral is less than the inner measure, which is less 

 than the outer measure, which is less than the upper integral. 



The property of semi-continuous functions just mentioned leads to a 

 new form of the definition of the upper (lower), integral in this case, 

 namely, as follows : — 



Divide the fundamental segment 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 

 (lower) semi-continuous function at points of that component 

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

 such summations for every conceivable mode of division is the 

 upper (lower) integral of the semi-continuous function. 



Introducing upper and lower limiting functions* we then have the 

 following theorem : — 



The upper (lower) integral of any function is the upper (lower) 

 integral of its associated upper (lower) semi-continuous function. 



This leads to a new definition of upper and lower integration, which 

 is as follows : — 



Divide the fundamental segment into any finite or countably infinite 

 number of measurable components, multiply the content of each 

 component by the upper (lower) limit of the maxima (minima) 

 of the function at points of that component and sum all such 

 products ; then the lower (upper), limit of all such summations 

 for every conceivable mode of division is the upper (lower) 

 integral of the function over the fundamental segment. 



This gives us also a definition of the integral in the case when it 

 exists, that is, when the upper and lower integrals are equal. 



This form of the definitions is at once extendable to the case when 

 the fundamental set S is any measurable set whatever, we merely have 

 to replace the word segment by set, or more precisely by measurable 



* "On Upper and Lower Integration," ' Lond. Math. Soc. Proc' 



