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VI. On the General Theory of Integration. 

 tty W. FT. YODN<J, Sc.D., St. P<>t<>rs f'nUrf/i; 



Communicated !/ Dr. E. W. Honsox, F.R.S. 

 Ilcceived April 23, Rend May 19, 1904. 



Introductory.* 



RIEMANN was the first to consider the theory of integration of non-continuous 

 functions. As is well known, his definition of the integral of a function between 

 the limits a and 6 is as follows : Divide the segment (ft, It) into any finite number 

 of intervals, each less, say, than a positive quantity, or norm d ; take the product of 

 each such interval by the value of the function at any point of that interval, and 

 form the sum of all these products ; if this sum has a limit, when (/ is indefinitely 

 diminished which is independent of the mode of division into intervals, and of the 

 choice of the points in those intervals at which the values of the function are 

 considered, this limit is called the integral of the function from a to 6. 



The most convenient mode, however, of defining a lliemann (that is an ordinary) 

 integral of a function, is due to DARBOUX ; it is Iwvsed on the introduction of upper 

 and Imver integral* (inte"grale par exces, par deTaut : oberes, unteres Integral). The 

 definitions of these are as follows : -It may be shown that, if the interval (a, b) be 

 divided as befoi'e, ami the sum of the products taken as )>efore, but with this 

 difference, that instead of the value of the function at an arbitrary point of the 

 part, the upper (lower) limit of the values of the function in the part be taken and 

 multiplied by the length of the corresponding part, these summations have, whatever 

 be the type of function, each of them a definite limit, independent of the mode of 

 division and the mode in which d approaches the value zero. This limit is called the 

 upper (lower) integral of the function. In the special case in which these two limits 

 agree, the common value is called the integral of the function. 



The progress of the modern theory of sets of points (Thdorie des ensembles ; 

 Mengenlehre), due, as is well known, chiefly to G. CANTOR, though taking its origin 

 in RIEMANN'S paper ' Ueber die Darstellbarkeit einer Funktion durch eine trigono- 

 metrische Reihe,' naturally leads us to put the question how far these definitions 



* An abridged statement of the contents of this memoir will l>e found in the " Abstract " published in 

 the ' Proceedings of the Royal Society,' vol. 73, pp. 445-449. 



(377.) 22.2.05 



