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



can be generalised. Tins theory has in fact taught us on the one hand that many of 

 the theorems hitherto stated for finite ntimben are true with or without modification 

 for a countably infinite number, and on the other hand that closed sets of points 

 possess many of the properties of intervals. We may, in accordance with these facts, 

 divide the segment (1) into an infinite number of non-overlapping intervals, in which 

 case, however, seeing that such a set of intervals always has points which are 

 external or semi-external to them, we must in general add a set of points to the 

 set of intervals, if the division of the segment is to be properly performed, that is, 

 if all the points of the segment are to be accounted for in our division ; or, more 

 generally, (2) into a finite or countably infinite number of sets of points. 



What would be the effect on the Riemann and Darboux definitions, if in those 

 definitions the word " finite " were replaced by " countably infinite," and the word 

 "interval" by "set of points"? A further question suggests itself: Are we at 

 liberty to replace the segment (, 6) itself by a closed set of points, and so define 

 integration with respect to any closed set of points ? 



Going one step further, recognising that the theory of the content of open sets 

 quite recently developed by M. LEBESGTTK* has enabled us to deal with all known open, 

 sets in much the same way as with closed sets as regards the very properties which 

 here come into consideration, we may attempt to replace both the segment and the 

 intervals of the segment by any kinds of measurable sets. 



Tn the Riemann and Darboux definitions it is tacitly assumed that the interval 

 (a, b) is finite, and that the function is throughout the interval finite and possesses 

 finite upper and lower limits. The discussion of the integration of a function which 

 is not necessarily finite, over an interval not necessarily finite in length, requires 

 separate consideration, and the definitions of such integrals, called improper integrals, 

 are of the nature of extensions of the definitions of ordinary integrals. Bearing in 

 mind the somewhat unsatisfactory and artificial character of such extensions, we may 

 hope finally that our discussion! may throw light on improper integrals also. 



In M. LEBESGUE'S valuable memoir, already referred to, a striking addition has been 

 made to the previously existing knowledge of the subjects dealt with. He has 

 shown that a more general definition than that of RIEMANN, available for all known 

 functions, one moreover coinciding with that of RIEMANN in the case of all functions 

 integrable in the Riemann sense, may be given ; a definition possessing, among 

 others, the remarkable property of permitting passage back from the derivatives of 

 a continuous function to the continuous function itself. 



Tn the present paper I attempt to discuss the whole matter, and take occasion in 

 the proper place to bring LEBESGUE'S work into connection with my own. Some 

 special cases of the results I obtain have been given by me in the paper presented to 



* "Integrate; Longueur; Aire," 'Ann. di Mat.,' 1902. Of. also a paper by the Author, "Open Sets 

 and the Theory of Content," ' Proc. Lond. Math. Soc.,' Ser. 2, vol. 2, Part I., p. 16. 



t In the instalment now presented to the Society I confine my attention however to proper integrajs. 



