PROPERTIES OF CLASSES OF FUNCTIONS 9 



in case there exists a function a of 2 such that the sequence {p.n\ 

 converges to B uniformly as to a, that is in case 



30-9 Lm™ = ^ {^■,<y')- 



n 



4. Definitions* — In §4a are stated the simple relations holding 

 between various cases of relatively uniform convergence, the sequences 

 being the same and the scale functions different. 



A function a is said to be dominated by\ a function t in case 

 Act ^ At, that is, in case AcTp ^ Arp for every p. 



A class ® = [a] of functions is said to be dominated by a class 

 2; = [t] of functions in case for each function a- there exists a function 

 T depending on a such that a is dominated by t. 



4a. Propositions. % — (1) Uniformity (of convergence) as to a implies 

 uniformity as to every function dominating a. 



(2) Uniformity as to a and uniformity as to aa {a =# 0) are equivalent. 



(3) Uniformity as to implies that the functions of the sequence 

 are ultimately equal and hence implies uniformity as to every func- 

 tion a. 



(4) If Aa is uniformly bounded from oo, viz., 



3 6 9.4(7 ^ e, 



then uniformity as to <r implies uniformity (i. e., ordinary uniform 

 convergence). Similarly if Aa is uniformly bounded from 0, viz., 



3696 ^ A<T, 



then uniformity implies uniformity as to a. 



5. The extension of a class SCJ of functions as to a function a or as 

 to a class © of functions.^ — 9??^, the class W extended as to the function <t, 

 is the class of all functions 6 of the form: 



n 



viz., of all limit functions of sequences !;u„| of the class 9}J converging 

 on ''^^ uniformly as to the function cr. In symbols: 



m^ = [a\le = JLnn (^l?;<r)]. 



*I. G. A., §§9;18. 



t This property dominated by is a relation between the two functions a and t. In 

 practice it is thought of as a property, in uotationiBir, of the function a. Similarly in 

 case the class © is dominated by the class X, the class © is said to have the property BiS. 



tl. G. A.,§9a. 



§1. G. A., §14 



