DR. W. H. YOUNG ON THE GENERAL THEORY OF INTEGRATION. 241 



that is, not less than 



,-K' 



and therefore not less than !'// + KS.* 



it 



(K- 

 Idk. In precisely 



the same way it may be shown that the lower integral of a lower semi-continuous 

 function can l>e expressed as an ordinary integral. For variety I give the following 

 proof. 



Let f be a lower semi-continuous function, and /'= f; then f is an upper semi- 

 continuous function ; and if K be not greater than the lower limit of f, and K' not 

 less than the upper limit of f, then K' is not greater than the lower limit of f, and 

 -K not less than the upper limit of/'. Hence, by what has just been proved, 



l/dx = -\fdx = -(-K')S-p Jrftf, 

 where J is the content of the set of points of S at which f has a value S M. 



IK' 

 Jltk, 



where J is the content of the set of points of S at which f has a value S k. 

 Summing up, we have the following theorems : 



Theorem. Tfie upper integral of an upper semi-continuous function with respect to 



(K' 

 Idle, where I is the content of that component of S at 

 K 



every point of ivhich the function has a value greater than or equal to k. 



Theorem 21. The lower integral of a lower semi- continuous function with respect 



fK' 

 Jdk, where J is the content of that component o/S at 



every point of which the function has a value less than or equal to k. 



In both these theorems K and K' are quantities which are respectively less than or 

 equal to the lower limit, and greater than or equal to the upper limit of the function 

 for points of S. Hence also 



Theorem 22. The upper integral of any function with respect to a measurable set 



(K' 

 Idk, where I is the content of that component of S at every point of 



which the maximum of the function is greater than or equal to k. 



ft.' 

 Jdk, where J is the content of that component of S 



at every point of which the minimum of the function is less than or equal to k. 



* This argument shows that in the case of an upper semi-continuous function the upper integral is equal 

 not only to the lower limit of the upper summations, but also to the upper limit of the lower summations 

 the latter is not to be confounded with the lower integral. Similarly for a lower semi-continuous function 



tli limits give the lower integral. 



VOL. CC1V. A. 2 I 



