PROPERTIES OF CLASSES OF FUNCTIONS 39 



(2) follows from (1) and § 23alli. (3) is a corollary of (2). 



(4) l^.A. 932''''^' 9 [all m"" '^^T"' ■ ^ ■ ^l'- 

 Proof. — Under the hypotheses on 3)? we have 



(a) ;u . : 3 : . 3 Mo ' e : 3 : 3 m, i K,,,,,^ . z> . An ^ eAno, 



(b) jx .m : 3 : 3 Mm,. » {K^m . 3 . Mm»i/> = 0) • (~-^j»» • => • Mump = Ajip), 



(c) Lm^.,,, = Ail (^1?; Mo), 



(5) ^13 . A^' . m'-"^"' s [all ;u"'""^>] *«■■'' . 3 . m-"''''''"-'. 



(6) "ip . A''', m'-"^'^' : 3 : 9[)?-V ~ . 9Jf^' . ~ . m"''. ~ . m""''- . ~ . [all m"" "'■"]^°'". 



(5) follows from (4) and § 7.2; § 3516; § 27.10. (6) follows from 

 (2, 3, 4, 5); §7.2; §3516; §27.10; §361; § 20a2. 



^1^ . A^' . W-^ . pi , A^'"'- .3.3 ^f' 9 t^,.,, = l{p=p,). t,„„ 

 (7) 



= (/J + Pi). 



Proof. — Considering the conditions (la), (16) of § 17 on the de- 

 velopmental system it is not difficult under the conditions of this 

 theorem to show that there exists a sequence {m„, ! of functions of 9)J 

 satisfying the following conditions: 



(a) e : 3 : 3 »?,. a m ^ m^ . 3 . A (Mm^,, — 1) ^ e, 



(6) M ■ : =) - 3 Mo 3 e : 3 : 3 vh 3 {m ^ m^ . p 4= Pi) ■ 3 . AM,,,M,n;. ^ eAfiop. 



It is not difficult to see that /j may be taken such that M;>, = 1, and 

 that there is available a scale function mq such that mop, = 1.* 



(c) Li Mm = spi (i^; Mo). 



(8) i^ . A^' . m"--^ . 3 . [aU m"" "'^1^°". 



From (6) and (8) we get 

 Theorem I. From 



it follows that 



* This readily follows on account of -Bf^ unless juop, = 0. In this case a mo, such that 

 Mop + 0, available as scale function in {b), may be secured by adding to mo a proper function 

 from the sequence {yuml. 



