250 DR. W. II. VOlTffO Otf THE GENERAL THEORY OF INTEGRATION. 



take a closed component of the given set" of content greater than I e. Denoting by 

 X' the content of the ordinate section of this set, we have, by the theorem quoted, 

 I-e-<jXVx. 



Since X' is an upper semi-continuous function, it follows, by 24, that 



I e< the upper limit of the lower summations of X', 

 or, since X' is not greater than X', 



I <?< the upper limit of the lower summations of X'. 

 Since e may be as small as we please, 



I : the upper limit of the lower summations of X', 

 S the upper limit of the lower summations of X. 



Next let the content of the set got by closing the given set be denoted by S, and 

 that of the complementary set by J, so that 



I + J = S. 



Denoting by Y' and Z' the quantities for the complementary set and the whole 

 closed set corresponding to X' for the given set, we have, as before, 



J S the upper limit of the lower summations of Y'. 



Now (Z' Y') is the content of the difference of two closed sets, that is of an inner 

 limiting set,* containing the set X, therefore, 



Z'-Y'==Xo=:=X'. 

 Hence 



J ^ the upper limit of the lower summations of (Z' X"), 

 S the upper limit of (a lower summation of Z' minus an upper summation of X). 



But the upper limit of the lower summations of Z' is the generalised integral of Z', 

 that is, S ; therefore 



J S S the lower limit of the upper summations of X, 

 that is, 



I S the lower limit of the upper summations of X, 

 a fortiori, 



I i the lower limit of the upper summations of X'. 



Now ( 6) every upper summation of a function is greater than, or at least equal 

 to, any lower summation, so that a quantity cannot be less than the upper limit of the 



* ' Theory of Content,' p. 36. 



