# Analogue of Lebesgue-Radon-Nikodym Theorem with respect top-adic q-Measure on ℤp

Authors: T. Kim; D. V. Dolgy; S. H. Lee; C. S. Ryoo
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Recently, Lebesgue-Radon-Nikodym theorem with respect to fermionicp-adic invariant measure on ℤp was studied in Kim. In this paper we will give the analogue
of the Lebesgue-Radon-Nikodym theorem with respect to p-adic q-measure on ℤp. In special case, q=1, we can derive the same results in Kim.


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## Body

## 1. Introduction

Letp be a fixed odd prime number.  Throughout this paper, the symbols ℤp, ℚp, and ℂp denote the ring of p-adic integers, the field of p-adic rational numbers, and the p-adic completion of the algebraic closure of ℚp, respectively. Let νp be the normalized exponential valuation of ℂp with |p|=p-νp(p)=1/p and νp(0)=∞.When one speaks ofq-extension, q can be regarded as an indeterminate, a complex q∈ℂ, or a p-adic number q∈ℂp. In this paper we assume that q∈ℂp with |1-q|<1, and we use the notations of q-numbers as follows: (1.1)[x]q=[x:q]=1-qx1-q,[x]-q=1-(-q)x1+q.
For any positive integer N, let (1.2)a+pNZp={x∈Zp∣x≡a(mod⁡pN)},
where a∈ℤ satisfies the condition 0≤a<pN (see [1–8]).It is known that the fermionicp-adic q-measure on ℤp is given by Kim as follows: (1.3)μ-q(a+pnZp)=(-q)a[pN]-q=1+q1+qpN(-q)a,
(see [6, 9–12]). Let C(ℤp) be the space of continuous functions on ℤp. From (1.3), the fermionic p-adic q-integral on ℤp is defined by Kim as follows: (1.4)I-q(f)=∫Zpf(x)dμ-q(x)=lim⁡N→∞1[pN]-q∑x=0pN-1f(x)(-q)x,
where f∈C(ℤp) (see [1, 6, 9–12]).Various proofs of the Radon-Nikodym theorem can be found in many books on measure theory, analysis, or probability theory. Usually they use the Hahn decomposition theorem for signed measure, the Riesz representation theorem for functionals on Hilbert space, or a martingale theory (see [13, 14]). In the previous paper [3], the author has studied the analogue of the Lebesgue-Radon-Nikodym theorem with respect to fermionic p-adic invariant measure on ℤp. The purpose of this paper is to derive the analogue of the Lebesgue-Radon-Nikodym theorem with respect to p-adic q-measure on ℤp in the sense of fermionic.


## 2. Lebesgue-Radon-Nikodym's Type Theorem with respect top-Adic q-Measure on ℤp

For any positive integera and n with a<pn and f∈C(ℤp), let us define
(2.1)μf,-q(a+pnZp)=∫a+pnZpf(x)dμ-q(x),
where the integral is the fermionic p-adic q-integral on ℤp.From (1.3), (1.4), and (2.1), we note that(2.2)μf,-q(a+pnZp)=lim⁡m→∞1[pm+n]-q∑x=0pm-1f(a+pnx)(-q)a+pnx=lim⁡m→∞1[pm]-q∑x=0pm-n-1f(a+pnx)(-q)aqpnx(-1)x=[2]q[2]qpn(-q)alim⁡m→∞1[pm-n]-qpn∑x=0pm-n-1f(a+pnx)(-qpn)x=[2]q[2]qpn(-q)a∫Zpf(a+pnx)dμ-qpn(x).
By (2.2), we get
(2.3)μf,-q(a+pnZp)=[2]q[2]qpn(-q)a∫Zpf(a+pnx)dμ-qpn(x).
Therefore, by (2.3), we obtain the following theorem.Theorem 2.1.
Forf,g∈C(ℤp), one has
(2.4)μαf+βg,-q=αμf,-q+βμg,-q,
where α,β are constants.From (2.2) and (2.4), we note that (2.5)|μf,-q(a+pnZp)|≤M∥f∥∞,
where ∥f∥∞=supx∈ℤp|f(x)| and M is some positive constant.Now, we recall the definition of the strongly fermionicp-adic q-measure on ℤp. If μ-q is satisfied the following equation: (2.6)|μ-q(a+pnZp)-μ-q(a+pn+1Zp)|≤δn,q,
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.Ifδn,q is replaced by Cp-νp(1-qn) (C is some constant), then μ-q is called strongly fermionic p-adic q-measure on ℤp.LetP(x)∈Cp[[x]]q be an arbitrary q-polynomial with ∑ai[x]qi. 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]qk.Leta be an integer with 0≤a<pn. Then we get (2.7)μP,-q(a+pnZp)=[2]q[2]qpn(-q)alim⁡m→∞1[pm-n]-qpn∑i=0pm-n[a+ipn]qk(-1)iqpni,qpni=∑l=0i(il)[pn]ql(q-1)l.
By (2.7), we easily get (2.8)μP,-q(a+pnZp)≡[2]q[2]qpn(-q)a[a]qk(mod⁡[pn]q)≡[2]q[2]qpn(-q)aP(a)(mod⁡[pn]q).
Let x be an arbitrary in ℤp with x≡xn(modpn) and x≡xn+1(modpn+1), where xn and xn+1 are positive integers such that 0≤xn<pn and 0≤xn+1<pn+1. Thus, by (2.8), we have (2.9)|μP,-q(a+pnZp)-μP,-q(a+pn+1Zp)|≤Cp-νp(1-qpn),
where C is a positive some constant and n≫0.Let(2.10)fμP,-q(a)=lim⁡n→∞μP,-q(a+pnZp).
Then, (2.5), (2.7), and (2.8), we get (2.11)fμP,-q(a)=[2]q2(-q)a[a]qk=[2]q2(-q)aP(a).
Since fμP,-q(x) is continuous on ℤp, it follows for all x∈ℤp(2.12)fμP,-q(x)=[2]q2(-q)xP(x).
Let g∈C(ℤp). By (2.10), (2.11), and (2.12), we get (2.13)∫Zpg(x)dμP,-q(x)=lim⁡m→∞∑i=0pn-1g(i)μP,-q(i+pnZp)=[2]q2∑i=0pn-1g(i)(-q)i[i]qk=∫Zpg(x)[x]qkdμ-q(x).
Therefore, by (2.13), we obtain the following theorem.Theorem 2.2.
LetP(x)∈Cp[[x]]q be an arbitrary  q-polynomial with  ∑ai[x]qi. Then  μP,-q is a strongly fermionic  p-adic  q-measure on  ℤp and for all  x∈ℤp(2.14)fμP,-q=(-1)x[2]q2qxP(x).
Furthermore, for all g∈C(ℤp), one has
(2.15)∫Zpg(x)dμP,-q(x)=∫Zpg(x)P(x)dμ-q(x),
where the second integral is fermionic p-adic q-integral on ℤp.Letf(x)=∑n=0∞an,q(xn)q be the q-Mahler expansion of continuous function on ℤp, where (2.16)(xn)q=[x]q[x-1]q⋯[x-n+1]q[n]q!
(see [4]). Then we note that limm→∞|an,q|=0.Let(2.17)fm(x)=∑i=0mai,q(xi)q∈Cp[[x]]q.
Then (2.18)‖f-fm‖∞≤sup⁡m≤n|an,q|.
Writing f=fm+f-fm, we easily get (2.19)|μf,-q(a+pnZp)-μf,-q(a+pn+1Zp)|≤max⁡{|μfm,-q(a+pnZp)-μfm,-q(a+pn+1Zp)|,|μf-fm,-q(a+pnZp)-μf-fm,-q(a+pn+1Zp)|}.
From Theorem 2.2, we note that (2.20)|μf-fm,-q(a+pnZp)|≤∥f-fm∥∞≤C1p-νp(1-qpn),
where C1 is some positive constant.Form≫0, we have ∥f∥∞=∥fm∥∞.So(2.21)|μfm,-q(a+pnZp)-μfm,-q(a+pn+1Zp)|≤C2p-νp(1-qpn),
where C2 is also some positive constant.By (2.20) and (2.21), we see that (2.22)|f(a)-μf,-q(a+pnZp)|≤max⁡{|f(a)-fm(a)|,|fm(a)-μfm,-q(a+pnZp)|,|μf-fm,-q(a+pnZp)|}≤max⁡{|f(a)-fm(a)|,|fm(a)-μfm,-q(a+pnZp)|,≤‖f-fm‖∞}.
If we fix ϵ>0 and fix m such that ∥f-fm∥≤ϵ, then, for n≫0, we have
(2.23)|f(a)-μf,-q(a+pnZp)|≤ϵ.
Hence, we have (2.24)fμf,-q(a)=lim⁡n→∞μf,-q(a+pnZp)=[2]q2(-q)af(a).
Let m be the sufficiently large number such that ∥f-fm∥∞≤p-n.Then we get(2.25)μf,-q(a+pnZp)=μfm,-q(a+pnZp)+μf-fm,-q(a+pnZp)=μfm,-q(a+pnZp)=(-1)aqa[2]q2f(a)(mod⁡[pn]q).
For all g∈C(ℤp), we have (2.26)∫Zpg(x)dμf,-q(x)=∫Zpf(x)g(x)dμ-q(x).
Assume that f is the function from C(ℤp,Cp) to Lip(ℤp,Cp). By the definition of μ-q, we easily see that μ-q is a strongly p-adic q-measure on ℤp and for n≫0(2.27)|fμ-q(a)-μ-q(a+pnZp)|≤C3p-νp(1-qpn),
where C3 is some positive constant.Ifμ1,-q is associated strongly fermionic p-adic q-measure on ℤp, then we have (2.28)|μ1,-q(a+pnZp)-fμ-q(a)|≤C4p-νp(1-qpn),
where n≫0 and C4 is some positive constant.From (2.28), we get (2.29)|μ-q(a+pnZp)-μ1,-q(a+pnZp)|≤|μ-q(a+pnZp)-fμ-q(a)|+|fμ-q(a)-μ1,-q(a+pnZp)|≤K,
where K is some positive constant.Therefore,μ-q-μ1,-q is a q-measure on ℤp. Hence, we obtain the following theorem.Theorem 2.3.
Letμ-q be a strongly fermionic  p-adic  q-measure on  ℤp, and assume that the fermionic Radon-Nikodym derivative  fμ-q on  ℤp is continuous function on  ℤp. Suppose that  μ1,-q is the strongly fermionic  p-adic  q-measure associated to  fμ-q. Then there exists a  q-measure  μ2,-q on  ℤp such that
(2.30)μ-q=μ1,-q+μ2,-q.



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*Source: 637634-2011-10-11.xml*