Cavdinal yumbeys of Aggregates of Functions, 325 



That a function should be integrable, it is necessary , and 

 sufficient that the aggreoate of its discontinuities should be a 

 content-less aggregate. Two functions which coincide except 

 at the points of such an aggregate have the same integral. 

 Thus the values of the function at these points are arbitrary 

 within Unite limits), and may thus independently take values 

 of an aggregate of cardinal number C without altering the 

 integral. 



Now. from the investigations of Harnack and others, we 

 know that content-less aggregates of cardinal number C exist. 

 Hence to every single definite integral belong an aggregate 

 of, so to speak, egidvalent integrable functions of cardinal 

 number 



Since every integral is a continuous function of its upper 

 limit, the cardinal number of the totality of integrable 

 functions is at most f ; and since also 



C. 0^ = 0^+1 =C^ 



we conclude that the cardinal number of the aggregate of all 

 integrable functions (of one real variable) is 



Further, we know that 



So this aggregate has a greater cardinal number than 

 Cantor t appeared to think. 



4. 



By application of the formula (1) w^e see at once that the 

 cardinal number of all functions (whether one-, finitely- 

 X-, or C-valued) of C variables, which are defined in a 

 continuum, is 



and 



* Cantor, Jahre-sber. d. d. M.- V. i. (1894). 

 t Math. Ann. xxi. p. 590 (1883). 



