Analogue of Lebesgue-Radon-Nikodym Theorem with respect to p-adic q-Measure on Zp

and Applied Analysis 3 By 2.2 , we get μf,−q ( a pZp ) 2 q 2 qp −qa ∫ Zp f ( a px ) dμ−qpn x . 2.3 Therefore, by 2.3 , we obtain the following theorem. Theorem 2.1. For f, g ∈ C Zp , one has μαf βg,−q αμf,−q βμg,−q, 2.4 where α, β are constants. From 2.2 and 2.4 , we note that ∣μf,−q ( a pZp )∣∣ ≤ M‖f‖∞, 2.5 where ‖f‖∞ supx∈Zp |f x | and M is some positive constant. Now, we recall the definition of the strongly fermionic p-adic q-measure on Zp. If μ−q is satisfied the following equation: ∣∣∣μ−q ( a pZp ) − μ−q ( a p Zp ∣∣ ≤ δn,q, 2.6 where δn,q → 0 and n → ∞ and δn,q is independent of a, then μ−q is called the weakly fermionic p-adic q-measure on Zp. If δn,q is replaced by Cp−νp 1−q n C is some constant , then μ−q is called strongly fermionic p-adic q-measure on Zp. Let P x ∈ Cp x q be an arbitrary q-polynomial with ∑ ai x i q. Then we see that μP,−q is strongly fermionic p-adic q-measure on Zp. Without a loss of generality, it is enough to prove the statement for P x x q . Let a be an integer with 0 ≤ a < p. Then we get μP,−q ( a pZp ) 2 q 2 qp −qa lim m→∞ 1 [ pm−n ] −qp pm−n ∑ i 0 [ a ip ]k q −1 q ni, q ni i ∑ l 0 ( i l ) [ p ]l q ( q − 1l. 2.7 By 2.7 , we easily get μP,−q ( a pZp ) ≡ 2 q 2 qp −qa a q ( mod [ p ] q ) ≡ 2 q 2 qp −qaP a ( mod [ p ] q ) . 2.8 4 Abstract and Applied Analysis Let x be an arbitrary in Zp with x ≡ xn mod p and x ≡ xn 1 mod p 1 , where xn and xn 1 are positive integers such that 0 ≤ xn < p and 0 ≤ xn 1 < p 1. Thus, by 2.8 , we have ∣∣∣μP,−q ( a pZp ) − μP,−q ( a p Zp ∣∣ ≤ Cp−νp 1−qp n , 2.9 where C is a positive some constant and n 0. Let fμP,−q a lim n→∞P,−q ( a pZp ) . 2.10 Then, 2.5 , 2.7 , and 2.8 , we get fμP,−q a 2 q 2 −qa a q 2 q 2 −qaP a . 2.11 Since fμP,−q x is continuous on Zp, it follows for all x ∈ Zp fμP,−q x 2 q 2 −qxP x . 2.12 Let g ∈ C Zp . By 2.10 , 2.11 , and 2.12 , we get ∫ Zp g x dμP,−q x lim m→∞ pn−1 ∑ i 0 g i μP,−q ( i pZp ) 2 q 2 pn−1 ∑ i 0 g i −qi i q ∫ Zp g x x qdμ−q x . 2.13 Therefore, by 2.13 , we obtain the following theorem. Theorem 2.2. Let P x ∈ Cp x q be an arbitrary q-polynomial with ∑ ai x i q. Then μP,−q is a strongly fermionic p-adic q-measure on Zp and for all x ∈ Zp fμP,−q −1 x 2 q 2 qP x . 2.14 Furthermore, for all g ∈ C Zp , one has ∫