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THE ROYAL IRISH ACADEMY 



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I. 



THE EIEMANX INTEGRAL AND IMEASUEABLE SETS. 

 By M. J. C0N1?AN. 



Bead JuxE 24. Published August 16, 191-2. 

 lilfroducflO/l. 



The geiieral theory of mtegration in any measurable set has been discussed 

 by W. H. Young.' He discards the Darboux-Eiemann definition, and ulti- 

 mately defines a generalized integral not essentially different from that of 

 Lebesgue.^ 



In this paper the Darboux-Eiemann definition is taken as the starting- 

 point and the extension to any measurable set made as follows : — 



Commencing with the integral for a single interval, the integral is defined 

 in succession for («) a set of open intervals, (b) a closed set regarded as the 

 set complementary to a set of open intervals, (c) a measurable set by 

 considering the integrals in the closed components of the set. 



In applying these ideas to multiple integrals I found it convenient to 

 make a slight modification in the form of the Darboux-Eiemami definitions. 

 The region of integration is divided in the usual manner into a finite number 

 of elementary parts, and the upper and lower limits of the function for the 

 internal points of each elementary part are selected, and the usual 

 summations formed ; or, to express it in another form, the functional 

 values at the boundary-points of the elementary parts are not considered 

 in estimating the upper and lower limits. The values of the upper and 

 lower integrals remain the same as before. 



> Phil. Trans, of the Eoyal Soc, Sei. A, vol. ceiv. 



-"Integr.il, Longiieuv, Aire" — Annali di Matheinatica, Ser. iii, vol. vii, 1902. 

 B.I i. I'ROC, VOL. XXX.. SEOT. A, [1] 



