12 pitcher: interrelations of 



Theorem II. If a class ©^' is the scale class of each of a finite 

 number of instances of relative uniform convergence, then the corresponding 

 finite sequence {(r„) of scale functions may he replaced by a single scale 

 function a effective for each of the instances. 



8. Five properties of general reference. — The properties A; L; C; D; 

 Di defined in the previous sections are properties of general reference* 

 and are possessed by many classes of functions of recognized importance 

 in Analysis. In particular the classes,! 



m'; W"; 50?"'; W"; W-; SOJ'^, 



possess these properties. 



9. The linear and ^-extension of a class Tl of functions.X — The 

 linear extension of the class 90'? of functions on "i^ to 31, denoted by 



is defined to be the class of all functions 



Mi 



zLdiiJ-i, 



where n is any positive integer. 



The ^-extension of the class 90?, denoted by 



is by definition the class of functions 90?/, extended as to the class 

 m. That is, 



9a. Propositions. — In the following propositions 91 denotes the 

 classes Ml, M^ derived from 9}J, and 3l9Jt = [all an]. 



(1)§ m""''; 9?*°^*. 



* See preface. 



t § 2; I. G. A., §§ 16; 23. 



1 1. G. A., §§27; 43. The property L is a property extensionally attainable. In an 

 excursus (I. G. A., §§ 28-42), Moore discusses such properties in the theory of classes in 

 general. 



§(1) The class 'SI of functions belongs to the class 9i of functions and the class 3i 

 belongs to the class 9Ji,. 



(2) For any class 9J1 the linear extension of 'SI is linear. 



(3) If a class 9)J is Unear the linear extension of 5)i is the same as 3Ji and the *-extension 

 of 3)1 is the same as the extension of -D! as to itself. 



(4) If a class 3)1 is linear and closed then the «-extension of 9Ji is the same as ilt. 



(5) If 9J1 has the property Di then 9i has the property Di and is dominated by the 

 class 2I3Jt of functions, and the class 9J!« is linear and has the property Di. 



