CoNHAN — Tlie Rieiiiuiui Intejral and Jleasurahle Sets. 



The Intcffrals in a- Set of Opm Intervals. 



2. If D denote the set of points of a finite or eniuneralily infinite nmuber 

 of non-overlapping open intervals («,-, b,), contained in a finite segment Z, 

 we may define the integrals in D thus 



= + + &c , 



and I = + + &c. 



L=J.rk 



These series are absolutely convergent. 



The Intefjrals in a Closed Set. 



3. Let iS be a closed set, and D the complementary set with respect to 

 the fundamental' segment (which may be regarded as open at both ends] ; 2) 

 consists of a set of non-overlapping open intervals, and we may define the 

 integrals in S thus 



/upper\ /upper \ /upper \ 



I integral in *S' = integral in Z - integral in D. 



\lower / \lower / \lower / 



The values of the integrals in S are obviously unaffected by removing 

 from Z a finite number of intervals which are aiso contained in D. Hence 

 if Z' denote any nimiber of open intervals containing S and I/, -the black 

 intervals of S with respect to Z', we infer that the 



/upperX /upper\ /upperX 



integral in »S' = integral in Z - J integral ni D . 



\lower / Vlower / \lower 7 



If / be the content of S, we can choose Z', so that its content is < / + f , 

 and consequently that of iy<t, where « is any assigned positive quantity. 



Hence, if Zi, Zj &c., be a sequence of sets of intervals containing )S^, 



and such that^ the content of Z,. is <I + er, where Km £„ = 0, then 



/upper \ , /upper \ . , . ^ 



integral in >S' = lini integral m Z„. 



Vlower/ K->M\lower/ 



Cor. The integrals in S are numerically less than M.I, where M is the 

 maximum of !/'(») I- ■ • 



[1*] 



