DR. W. II. YOlWO ON THE CI.M.KAI. TIIK'MJY OK INTEGRATION. 245 



content of a plane set containing the set E, and the latter the content of a plane set 

 contained in E, so that the common limit must he the content of E. In the case 

 when the function is sometimes positive and sometimes negative, the sum of the 

 positive terms of the first summation is the content of a set containing E|, and the 

 sum of the negative terms, taken with positive sign, is the content of a set contained 

 in EH ; similarly the second summation is the content of a component of E t minus the 

 content of a set containing E a ; thus the common limit must be the content of E, 

 minus that of E Thus the two definitions are identical.* 



The second of LEBKSGUE'S definitions enables us without difficulty to identify the 

 Lebesgue integral with the generalised integral which I defined in 24. In fact, 

 comparing LEBESGUE'S notation with that used by myself in 21, it is clear that 



Whence the former of the two given expressions is equal to 

 that is, 

 which, as a is decreased indefinitely, approaches the limit 



that is, the generalised integral of 24. 



Thus the Lt'besgue integral is the name as the generalised integral of 24, the 

 fundamental set being a finite segment. 



26. Contrasting the definition of 24 (8 being now a finite segment) with the 

 geometrical definition of LEBESGUE, we see that they stand to one another in the 

 same relation as the ordinary definition of integration of, say, a continuous function 

 to its definition as a certain area. Just as, however, the mathematical conception of 

 area is more complex than and indeed depends on that of length, so does the theory 

 of the content of a plane set of points depend naturally on that of the content of a 

 linear set. Just as the determination of area requires the application of the processes 

 explained in the first definition of integration of continuous functions, so with the 

 content of a plane set. Thus the comparative simplicity of the geometrical definition 

 is only apparent. With regard to LEBESGUE'S analytical definition, I have pointed 

 out that it is equivalent to what seems to me the much more convenient form in 

 which I have expressed it as an ordinary integral ( 24). 



If we know I as a function of k, which may be the case, all Lebesgue integration 

 and all upper and lower integration reduce to ordinary integration. It may, however, 

 happen that I is not readily found as a function of k. The definition of 24 seems 

 then the most fundamental, and is, in many respects, very convenient in the theory. 



* LEBESUUE, Inc. fit., p. 252. 



