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



taken over the, same fundamental set S is equal to the integral of the sum of the 

 functions taken over the set. This follows from Theorem \ 3, which asserts the same 

 for upper (lower) integrals of upper (lower) semi-continuous functions. The proof is of 

 precisely the same nature as the preceding. 



21. We now proceed to obtain a formula for the upper (lower) integral of an 

 upper (lower) semi-continuous function over any measurable set of points, in terms of 

 an ordinary integral. These formulae of course at the same time give us the upper 

 (lower) integral of any function, by reason of 11. 



Let K' be any quantity not less than the greatest, and K not greater than the 

 least, value of an upper semi-continuous function f defined for a fundamental set S. 



Divide (K, K') into n parts, and consider the sets of points G,, G 2 , ... G B , where G r 



rrt _ TT 



denotes all those points of S for which / is S K' -- r, and G B is S, so that 



n 



By Theorem 7, GI, G 2 , ... are measurable sets ; let I,, I 2 , ... be their contents. Then 

 by the definition at the end of 11, the upper integral, being not greater than any 

 upper summation, is not greater than 



n 

 however great n may be, and is therefore 



K/ - K (I. + I 2 +-+In- 1 + In) + KI n , 



since I is a monotone function of k and therefore an integrable one. 



Here S is the content of the fundamental set, and I that of the set of points for 

 which the values of the function are greater than or equal to k. But it was shown, 

 in 6, that for any function whatever the greatest value of the lower summations is 

 certainly not greater than the least value of the upper summations, therefore the 

 upper integral is not less than all the lower summations. Hence the upper integral 

 is certainly not less than 



- n *-^ 

 n 



