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

Abstract

Recently, Lebesgue-Radon-Nikodym theorem with respect to fermionic 𝑝-adic invariant measure on ℤ_𝑝 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 ℤ_𝑝. In special case, 𝑞=1, we can derive the same results in Kim.

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper, the symbols ℤ_𝑝, ℚ_𝑝, and ℂ_𝑝 denote the ring of 𝑝-adic integers, the field of 𝑝-adic rational numbers, and the 𝑝-adic completion of the algebraic closure of ℚ_𝑝, respectively. Let 𝜈_𝑝 be the normalized exponential valuation of ℂ_𝑝 with |𝑝|=𝑝^{−𝜈𝑝(𝑝)}=1/𝑝 and 𝜈_𝑝(0)=∞.

When one speaks of 𝑞-extension, 𝑞 can be regarded as an indeterminate, a complex 𝑞∈ℂ, or a 𝑝-adic number 𝑞∈ℂ_𝑝. In this paper we assume that 𝑞∈ℂ_𝑝 with |1−𝑞|<1, and we use the notations of 𝑞-numbers as follows: [𝑥]_𝑞=𝑥∶𝑞=1−𝑞^𝑥1−𝑞,[𝑥]_−𝑞=1−−𝑞^𝑥.1+𝑞 (1.1) For any positive integer 𝑁, let 𝑎+𝑝^𝑁ℤ_𝑝=𝑥∈ℤ_𝑝∣𝑥≡𝑎mod𝑝^𝑁, (1.2) where 𝑎∈ℤ satisfies the condition 0≤𝑎<𝑝^𝑁 (see [1–8]).

It is known that the fermionic 𝑝-adic 𝑞-measure on ℤ_𝑝 is given by Kim as follows: 𝜇_−𝑞𝑎+𝑝^𝑛ℤ_𝑝=−𝑞^𝑎𝑝^𝑁_−𝑞=1+𝑞1+𝑞^𝑝𝑁−𝑞^𝑎, (1.3) (see [6, 9–12]). Let 𝐶(ℤ_𝑝) be the space of continuous functions on ℤ_𝑝. From (1.3), the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 is defined by Kim as follows: 𝐼_−𝑞𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁𝑝_−𝑞^𝑁−1_𝑥=0𝑓(𝑥)−𝑞^𝑥, (1.4) where 𝑓∈𝐶(ℤ_𝑝) (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 ℤ_𝑝. The purpose of this paper is to derive the analogue of the Lebesgue-Radon-Nikodym theorem with respect to 𝑝-adic 𝑞-measure on ℤ_𝑝 in the sense of fermionic.

2. Lebesgue-Radon-Nikodym's Type Theorem with respect to 𝑝-Adic 𝑞-Measure on ℤ_𝑝

For any positive integer 𝑎 and 𝑛 with 𝑎<𝑝^𝑛 and 𝑓∈𝐶(ℤ_𝑝), let us define 𝜇_𝑓,−𝑞𝑎+𝑝^𝑛ℤ_𝑝=_{𝑎+𝑝𝑛ℤ𝑝}𝑓(𝑥)𝑑𝜇_−𝑞(𝑥), (2.1) where the integral is the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝.

From (1.3), (1.4), and (2.1), we note that𝜇_𝑓,−𝑞𝑎+𝑝^𝑛ℤ_𝑝=lim_𝑚→∞1𝑝^{𝑚+𝑛𝑝}_−𝑞^𝑚−1_𝑥=0𝑓𝑎+𝑝^𝑛𝑥−𝑞^{𝑎+𝑝𝑛𝑥}=lim_𝑚→∞1𝑝^𝑚𝑝_−𝑞^{𝑚−𝑛−1}_𝑥=0𝑓𝑎+𝑝^𝑛𝑥−𝑞^𝑎𝑞^𝑝𝑛𝑥(−1)^𝑥=[2]_𝑞[2]_𝑞𝑝^𝑛−𝑞^𝑎lim_𝑚→∞1𝑝^𝑚−𝑛_−𝑞𝑝^{𝑛𝑝𝑚−𝑛−1}_𝑥=0𝑓𝑎+𝑝^𝑛𝑥−𝑞^𝑝𝑛𝑥=[2]_𝑞[2]_𝑞𝑝^𝑛−𝑞^𝑎_ℤ𝑝𝑓𝑎+𝑝^𝑛𝑥𝑑𝜇_−𝑞𝑝^𝑛(𝑥). (2.2) By (2.2), we get 𝜇_𝑓,−𝑞𝑎+𝑝^𝑛ℤ_𝑝=[2]_𝑞[2]_𝑞𝑝^𝑛−𝑞^𝑎_ℤ𝑝𝑓𝑎+𝑝^𝑛𝑥𝑑𝜇_−𝑞𝑝^𝑛(𝑥). (2.3) Therefore, by (2.3), we obtain the following theorem.

Theorem 2.1. For 𝑓,𝑔∈𝐶(ℤ_𝑝), one has 𝜇_{𝛼𝑓+𝛽𝑔,−𝑞}=𝛼𝜇_𝑓,−𝑞+𝛽𝜇_𝑔,−𝑞, (2.4) where 𝛼,𝛽 are constants.

From (2.2) and (2.4), we note that 𝜇_𝑓,−𝑞𝑎+𝑝^𝑛ℤ_𝑝≤𝑀‖𝑓‖_∞, (2.5) where ‖𝑓‖_∞=sup_{𝑥∈ℤ𝑝}|𝑓(𝑥)| and 𝑀 is some positive constant.

Now, we recall the definition of the strongly fermionic 𝑝-adic 𝑞-measure on ℤ_𝑝. If 𝜇_−𝑞 is satisfied the following equation: 𝜇_−𝑞𝑎+𝑝^𝑛ℤ_𝑝−𝜇_−𝑞𝑎+𝑝^𝑛+1ℤ_𝑝≤𝛿_𝑛,𝑞, (2.6) where 𝛿_𝑛,𝑞→0 and 𝑛→∞ and 𝛿_𝑛,𝑞 is independent of 𝑎, then 𝜇_−𝑞 is called the weakly fermionic 𝑝-adic 𝑞-measure on ℤ_𝑝.

If 𝛿_𝑛,𝑞 is replaced by 𝐶𝑝^{−𝜈𝑝(1−𝑞𝑛)} (𝐶 is some constant), then 𝜇_−𝑞 is called strongly fermionic 𝑝-adic 𝑞-measure on ℤ_𝑝.

Let 𝑃(𝑥)∈𝐶_𝑝[[𝑥]]_𝑞 be an arbitrary 𝑞-polynomial with ∑𝑎_𝑖[𝑥]^𝑖_𝑞. Then we see that 𝜇_𝑃,−𝑞 is strongly fermionic 𝑝-adic 𝑞-measure on ℤ_𝑝. Without a loss of generality, it is enough to prove the statement for 𝑃(𝑥)=[𝑥]^𝑘_𝑞.

Let 𝑎 be an integer with 0≤𝑎<𝑝^𝑛. Then we get 𝜇_𝑃,−𝑞𝑎+𝑝^𝑛ℤ_𝑝=[2]_𝑞[2]_𝑞𝑝^𝑛−𝑞^𝑎lim_𝑚→∞1𝑝^𝑚−𝑛_−𝑞𝑝^{𝑛𝑝𝑚−𝑛}_𝑖=0𝑎+𝑖𝑝^𝑛𝑘_𝑞(−1)^𝑖𝑞^𝑝𝑛𝑖,𝑞^𝑝𝑛𝑖=^𝑖_𝑙=0𝑖𝑙𝑝^𝑛𝑙_𝑞𝑞−1^𝑙. (2.7) By (2.7), we easily get 𝜇_𝑃,−𝑞𝑎+𝑝^𝑛ℤ_𝑝≡[2]_𝑞[2]_𝑞𝑝^𝑛−𝑞^𝑎[𝑎]^𝑘_𝑞mod𝑝^𝑛_𝑞≡[2]_𝑞[2]_𝑞𝑝^𝑛−𝑞^𝑎𝑃(𝑎)mod𝑝^𝑛_𝑞. (2.8) Let 𝑥 be an arbitrary in ℤ_𝑝 with 𝑥≡𝑥_𝑛(mod𝑝^𝑛) and 𝑥≡𝑥_𝑛+1(mod𝑝^𝑛+1), where 𝑥_𝑛 and 𝑥_𝑛+1 are positive integers such that 0≤𝑥_𝑛<𝑝^𝑛 and 0≤𝑥_𝑛+1<𝑝^𝑛+1. Thus, by (2.8), we have 𝜇_𝑃,−𝑞𝑎+𝑝^𝑛ℤ_𝑝−𝜇_𝑃,−𝑞𝑎+𝑝^𝑛+1ℤ_𝑝≤𝐶𝑝^{−𝜈𝑝(1−𝑞𝑝𝑛)}, (2.9) where 𝐶 is a positive some constant and 𝑛≫0.

Let 𝑓_{𝜇𝑃,−𝑞}(𝑎)=lim_𝑛→∞𝜇_𝑃,−𝑞𝑎+𝑝^𝑛ℤ_𝑝. (2.10) Then, (2.5), (2.7), and (2.8), we get 𝑓_{𝜇𝑃,−𝑞}(𝑎)=[2]_𝑞2−𝑞^𝑎[𝑎]^𝑘_𝑞=[2]_𝑞2−𝑞^𝑎𝑃(𝑎). (2.11) Since 𝑓_{𝜇𝑃,−𝑞}(𝑥) is continuous on ℤ_𝑝, it follows for all 𝑥∈ℤ_𝑝𝑓_{𝜇𝑃,−𝑞}(𝑥)=[2]_𝑞2−𝑞^𝑥𝑃(𝑥). (2.12) Let 𝑔∈𝐶(ℤ_𝑝). By (2.10), (2.11), and (2.12), we get _ℤ𝑝𝑔(𝑥)𝑑𝜇_𝑃,−𝑞(𝑥)=lim^𝑝_𝑚→∞^𝑛−1_𝑖=0𝑔(𝑖)𝜇_𝑃,−𝑞𝑖+𝑝^𝑛ℤ_𝑝=[2]_𝑞2^𝑝𝑛−1_𝑖=0𝑔(𝑖)−𝑞^𝑖[𝑖]^𝑘_𝑞=_ℤ𝑝𝑔(𝑥)[𝑥]^𝑘_𝑞𝑑𝜇_−𝑞(𝑥). (2.13) Therefore, by (2.13), we obtain the following theorem.

Theorem 2.2. Let 𝑃(𝑥)∈𝐶_𝑝[[𝑥]]_𝑞 be an arbitrary 𝑞-polynomial with ∑𝑎_𝑖[𝑥]^𝑖_𝑞. Then 𝜇_𝑃,−𝑞 is a strongly fermionic 𝑝-adic 𝑞-measure on ℤ_𝑝 and for all 𝑥∈ℤ_𝑝𝑓_{𝜇𝑃,−𝑞}=(−1)^𝑥[2]_𝑞2𝑞^𝑥𝑃(𝑥). (2.14) Furthermore, for all 𝑔∈𝐶(ℤ_𝑝), one has _ℤ𝑝𝑔(𝑥)𝑑𝜇_𝑃,−𝑞(𝑥)=_ℤ𝑝𝑔(𝑥)𝑃(𝑥)𝑑𝜇_−𝑞(𝑥), (2.15) where the second integral is fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝.

Let 𝑓(𝑥)=∑^∞_𝑛=0𝑎_𝑛,𝑞(^𝑥_𝑛)_𝑞 be the 𝑞-Mahler expansion of continuous function on ℤ_𝑝, where 𝑥𝑛_𝑞=[𝑥]_𝑞[𝑥−1]_𝑞⋯[𝑥−𝑛+1]_𝑞[𝑛]_𝑞! (2.16) (see [4]). Then we note that lim_𝑚→∞|𝑎_𝑛,𝑞|=0.

Let 𝑓_𝑚(𝑥)=^𝑚_𝑖=0𝑎_𝑖,𝑞𝑥𝑖_𝑞∈𝐶_𝑝[[𝑥]]_𝑞. (2.17) Then 𝑓−𝑓_𝑚∞≤sup_𝑚≤𝑛𝑎_𝑛,𝑞. (2.18) Writing 𝑓=𝑓_𝑚+𝑓−𝑓_𝑚, we easily get 𝜇_𝑓,−𝑞𝑎+𝑝^𝑛ℤ_𝑝−𝜇_𝑓,−𝑞𝑎+𝑝^𝑛+1ℤ_𝑝≤max𝜇_{𝑓𝑚,−𝑞}𝑎+𝑝^𝑛ℤ_𝑝−𝜇_{𝑓𝑚,−𝑞}𝑎+𝑝^𝑛+1ℤ_𝑝,𝜇_{𝑓−𝑓𝑚,−𝑞}𝑎+𝑝^𝑛ℤ_𝑝−𝜇_{𝑓−𝑓𝑚,−𝑞}𝑎+𝑝^𝑛+1ℤ_𝑝. (2.19) From Theorem 2.2, we note that 𝜇_{𝑓−𝑓𝑚,−𝑞}𝑎+𝑝^𝑛ℤ_𝑝≤‖𝑓−𝑓_𝑚‖_∞≤𝐶₁𝑝^{−𝜈𝑝(1−𝑞𝑝𝑛)}, (2.20) where 𝐶₁ is some positive constant.

For 𝑚≫0, we have ‖𝑓‖_∞=‖𝑓_𝑚‖_∞.

So 𝜇_{𝑓𝑚,−𝑞}𝑎+𝑝^𝑛ℤ_𝑝−𝜇_{𝑓𝑚,−𝑞}𝑎+𝑝^𝑛+1ℤ_𝑝≤𝐶₂𝑝^{−𝜈𝑝(1−𝑞𝑝𝑛)}, (2.21) where 𝐶₂ is also some positive constant.

By (2.20) and (2.21), we see that 𝑓(𝑎)−𝜇_𝑓,−𝑞𝑎+𝑝^𝑛ℤ_𝑝≤max𝑓(𝑎)−𝑓_𝑚(𝑎),𝑓_𝑚(𝑎)−𝜇_{𝑓𝑚,−𝑞}𝑎+𝑝^𝑛ℤ_𝑝,𝜇_{𝑓−𝑓𝑚,−𝑞}𝑎+𝑝^𝑛ℤ_𝑝≤max𝑓(𝑎)−𝑓_𝑚(𝑎),𝑓_𝑚(𝑎)−𝜇_{𝑓𝑚,−𝑞}𝑎+𝑝^𝑛ℤ_𝑝,≤𝑓−𝑓_𝑚∞. (2.22) If we fix 𝜖>0 and fix 𝑚 such that ‖𝑓−𝑓_𝑚‖≤𝜖, then, for 𝑛≫0, we have 𝑓(𝑎)−𝜇_𝑓,−𝑞𝑎+𝑝^𝑛ℤ_𝑝≤𝜖. (2.23) Hence, we have 𝑓_{𝜇𝑓,−𝑞}(𝑎)=lim_𝑛→∞𝜇_𝑓,−𝑞𝑎+𝑝^𝑛ℤ_𝑝=[2]_𝑞2−𝑞^𝑎𝑓(𝑎). (2.24) Let 𝑚 be the sufficiently large number such that ‖𝑓−𝑓_𝑚‖_∞≤𝑝^−𝑛.

Then we get 𝜇_𝑓,−𝑞𝑎+𝑝^𝑛ℤ_𝑝=𝜇_{𝑓𝑚,−𝑞}𝑎+𝑝^𝑛ℤ_𝑝+𝜇_{𝑓−𝑓𝑚,−𝑞}𝑎+𝑝^𝑛ℤ_𝑝=𝜇_{𝑓𝑚,−𝑞}𝑎+𝑝^𝑛ℤ_𝑝=(−1)^𝑎𝑞^𝑎[2]_𝑞2𝑓(𝑎)mod𝑝^𝑛_𝑞. (2.25) For all 𝑔∈𝐶(ℤ_𝑝), we have _ℤ𝑝𝑔(𝑥)𝑑𝜇_𝑓,−𝑞(𝑥)=_ℤ𝑝𝑓(𝑥)𝑔(𝑥)𝑑𝜇_−𝑞(𝑥). (2.26) Assume that 𝑓 is the function from 𝐶(ℤ_𝑝,𝐶_𝑝) to 𝐿𝑖𝑝(ℤ_𝑝,𝐶_𝑝). By the definition of 𝜇_−𝑞, we easily see that 𝜇_−𝑞 is a strongly 𝑝-adic 𝑞-measure on ℤ_𝑝 and for 𝑛≫0𝑓_𝜇−𝑞(𝑎)−𝜇_−𝑞𝑎+𝑝^𝑛ℤ_𝑝≤𝐶₃𝑝^{−𝜈𝑝(1−𝑞𝑝𝑛)}, (2.27) where 𝐶₃ is some positive constant.

If 𝜇_1,−𝑞 is associated strongly fermionic 𝑝-adic 𝑞-measure on ℤ_𝑝, then we have 𝜇_1,−𝑞𝑎+𝑝^𝑛ℤ_𝑝−𝑓_𝜇−𝑞(𝑎)≤𝐶₄𝑝^{−𝜈𝑝(1−𝑞𝑝𝑛)}, (2.28) where 𝑛≫0 and 𝐶₄ is some positive constant.

From (2.28), we get 𝜇_−𝑞𝑎+𝑝^𝑛ℤ_𝑝−𝜇_1,−𝑞𝑎+𝑝^𝑛ℤ_𝑝≤𝜇_−𝑞𝑎+𝑝^𝑛ℤ_𝑝−𝑓_𝜇−𝑞(𝑎)+𝑓_𝜇−𝑞(𝑎)−𝜇_1,−𝑞𝑎+𝑝^𝑛ℤ_𝑝≤𝐾, (2.29) where 𝐾 is some positive constant.

Therefore, 𝜇_−𝑞−𝜇_1,−𝑞 is a 𝑞-measure on ℤ_𝑝. Hence, we obtain the following theorem.

Theorem 2.3. Let 𝜇_−𝑞 be a strongly fermionic 𝑝-adic 𝑞-measure on ℤ_𝑝, and assume that the fermionic Radon-Nikodym derivative 𝑓_𝜇−𝑞 on ℤ_𝑝 is continuous function on ℤ_𝑝. Suppose that 𝜇_1,−𝑞 is the strongly fermionic 𝑝-adic 𝑞-measure associated to 𝑓_𝜇−𝑞. Then there exists a 𝑞-measure 𝜇_2,−𝑞 on ℤ_𝑝 such that 𝜇_−𝑞=𝜇_1,−𝑞+𝜇_2,−𝑞. (2.30)