DR. W. H. YOUNG ON THE GENERAL THEORY OF INTEGRATION. 249 



Now the blocks Px form a measurable set, by the usual argument, since they 

 consist of the part of a measurable set of blocks on a measurable component 8, as 

 base ; therefore the blocks PR at each point of S,, shifted up to the position of the 

 figure, being the difference of two measurable sets, form a measurable set, and the 

 content is PR . S,, that is, the same as it was before shifting. Since this is true for 

 each component $,-, it is true for the whole set of elongated blocks, that they form a 

 measurable set whose content is the same as it was before shifting. 



Now, by 27, we know that, before shifting, the content of this set was the upper 

 summation of the lengths of the given blocks, that is PQ, over the fundamental 

 set S, divided into Si, S s , ... Thus this upper summation, being the content of a 

 plane set containing the given set, is not less than the content of the given set (of 

 course in the shifted position). 



Similarly the corresponding lower summation is not greater than the same content. 

 But the lower limit of the upper summations is equal to the upper limit of the lower 

 summations, since either of them represents the content of the given blocks before 

 they were shifted up from the ar-axes. Thus the same content is itself neither less 

 nor greater than the content of the set in the shifted position, that is to say, the 

 content has l>een left unaltered by the shifting. [Q.E.D.] 



Corollary 1. In the shifted position the content is still the generalised integral of 

 the length of the blocks. 



Corollary 2. TJie sum of any finite number of summable functions is a summabfe 

 function, and its integral is the sum of their integrals, the fundamental set being any 

 measurable set of finite content. 



Corollary 3. Since the limit of a sequence of measurable sets is a measurable set, 

 it also follows that the sum of an absolutely convergent series of summable functions is 

 a summable function, and its generalised integral is the, sum of their integrals, 

 provided, as usual, the functions have finite upper and lower limits. 



30. By means of a theorem proved in my paper on " Upper and Lower Integra- 

 tion,"* we can extend the results of 29 still furth'er. The theorem quoted states 

 that, if X' be the content of the section of a closed plane set by the ordinate through 

 the point x of the as-axis, the content of the plane set is JX'rfx, and that, further, X 

 is an upper semi-continuous function of X. It follows, then, by 24, that the content 

 is the generalised integral of X'. 



The theorem now to be proved is as follows : 



Theorem 28. If X and X' be the outer and innei' measures of the content of the 

 ordinate section of a measuraMe set such that the set got by closing it is of finite 

 content, by the ordinate through the point x, X and X' are both summable functions, 

 and the generalised integral of either is the content of the measurable set. 



Let I be the content of the set, and e any assigned small positive quantity. Let us 



* ' Proc. Lend. Math. Soc.,' Ser. 2, vol. 2, Part I., p. 60. 

 VOL. CCIV. A. 2 K 



