4 Proceedings of the Roijal Irish Aeailni///. 



On llie oilier hand, if /S" be a closed eomponeiit of *S' of coiileut >/- t, 

 theu 



|[ -I \<M{, + ,,.), 



J In }S'\ 



where ilia Ihe maximum of i/(.')i, and 



,L„- 



for all values of n ; 



l-l 



\<Mt„; 

 < M(t + 2e„), 

 <: Ah. 



Hence, if S^S^ &c., be a sequence of closed components of S, such 



that the content of Sr> I - e,-, where lim e„ = 0, then 



= lim , and similarly = lim 



-^^ISn 



The Integrals in a Measv^rahlc Set. 



4. Let ^ be a measurable set, and / its content. It is possible to enclose 

 ^ in a finite or enumerably infinite number of intervals of content < I + t, 

 where t is any assigned positive quantity. Consider then a sequence of sets 

 of intervals Z,, L^, . . . , &c., such that L,- contains E, and has content 

 < I + £,-. Also, take a sequence of closed components S,, S^ . . ■ ■ &c., such 

 that the content of Sr is > J - e',., where the sequences ei, jv . . . . &c., 

 and t'„ t'a . . . . &c., are positive, monotone, and have each the limit zero. 

 We have 



11 - I <M(.„ I t',„), 

 and 



j - < 31{i„^-i, + t',„), 



for all values of m, that is 



< il/(2f„ + 2t',„), 



< 23h„. 



