Lebesque-Radon-Nikodym theorem with respect to p-adic invariant measure on Zp

TL;DR

This paper establishes a p-adic analogue of the Lebesgue-Radon-Nikodym theorem for fermionic invariant measures on Zp, expanding measure theory into the p-adic setting.

Contribution

It introduces a new version of the Lebesgue-Radon-Nikodym theorem tailored for fermionic p-adic invariant measures on Zp, a novel extension in p-adic measure theory.

Findings

Derived the p-adic Lebesgue-Radon-Nikodym theorem for fermionic measures

Extended classical measure theory into the p-adic context

Provided foundational results for p-adic measure analysis

Abstract

In this paper we derive the analogue of Lebesque-Radon Nikody theorem with respect to fermionic p-adic invariant measures on Zp

Equations

$$[x{]}_q=[x:q]=\frac{1-q^x}{1-q},[x{]}_{-q}=\frac{1-(-q{)}^x}{1+q},$$

$${\mu }_q(a+p^N{\mathbb{Z}}_p)=\frac{q^a}{[p^N{]}_q},\text{(see [4-8])}.$$

$$I_q(f)={\int }_{{\mathbb{Z}}_p}f(x)d{\mu }_q(x)=\underset{N\to \infty }{\lim }\frac{1}{[p^N{]}_q}\sum _{x=0}^{p^N-1}f(x)q^x.$$

$${\mu }_{-q}(a+d{p}^N{\mathbb{Z}}_p)=\frac{(-q{)}^a}{[d{p}^N{]}_{-q}},$$

$$I_{-q}(f)={\int }_{{\mathbb{Z}}_p}f(x)d{\mu }_{-q}(x)=\underset{N\to \infty }{\lim }\frac{1}{[p^N{]}_{-q}}\sum _{x=0}^{p^N-1}f(x)(-q{)}^x,$$

$$I_{-1}(f)=\underset{q\to 1}{\lim }{\int }_{{\mathbb{Z}}_p}f(x)d{\mu }_{-q}(x)={\int }_{{\mathbb{Z}}_p}f(x)d{\mu }_{-1}(x),\text{ (see [6])}.$$

$${△}_{1}f(m,x)=\frac{f(x+m)-f(x)}{m},$$

$$|{△}_{1}f(m,x)|\le M.$$

$${\mu }_{f,-1}(a+p^n{\mathbb{Z}}_p)={\int }_{a+p^n{\mathbb{Z}}_p}f(x)d{\mu }_{-1}(x),$$

$${\mu }_{f,-1}(a+p^n{\mathbb{Z}}_p)$$

$${\mu }_{f,-1}(a+p^n{\mathbb{Z}}_p)=(-1{)}^a{\int }_{{\mathbb{Z}}_p}f(a+p^{n}x)d{\mu }_{-1}(x).$$

$${\mu }_{\alpha f+\beta g,-1}=\alpha {\mu }_{f,-1}+\beta {\mu }_{g,-1},$$

$$|{\mu }_{f,-1}(a+p^n{\mathbb{Z}}_p)|\le \Vert f{\Vert }_{\infty }.$$

$$|{\mu }_{-1}(a+p^n{\mathbb{Z}}_p)-{\mu }_{-1}(a+p^{n+1}{\mathbb{Z}}_p)|\le {\delta }_n,$$

$${\mu }_{P,-1}(a+p^n{\mathbb{Z}}_p)=\underset{m\to \infty }{\lim }(-1{)}^a\sum _{i=0}^{p^{m-n}-1}(a+i{p}^n{)}^k(-1{)}^i.$$

$$(a+i{p}^n{)}^k=\sum _{l=0}^k{a}^{k-l}(\genfrac{}{}{0ex}{}{k}{l})(i{p}^n{)}^l=a^k+\left(\genfrac{}{}{0ex}{}{k}{1}\right)a^{k-1}{p}^{n}i+\cdots +p^{n^k}{i}^k.$$

$${\mu }_{P,-1}(a+p^n{\mathbb{Z}}_p)$$

$$=\underset{m\to \infty }{\lim }(-1{)}^a\{a^k+k{a}^{k-1}{p}^n\sum _{i=0}^{p^{m-n}-1}i(-1{)}^i+\cdots +p^{n^k}\sum _{i=0}^{p^{m-n}-1}{i}^k(-1{)}^i\}.$$

$$E_n={\int }_{{\mathbb{Z}}_p}{x}^{n}d{\mu }_{-1}(x)=\underset{m\to \infty }{\lim }\sum _{i=0}^{p^m-1}{i}^n(-1{)}^i,$$

$${\mu }_{P,-1}(a+p^n{\mathbb{Z}}_p)$$

$$|{\mu }_{P,-1}(x+p^n{\mathbb{Z}}_p)-{\mu }_{P,-1}(x+p^{n+1}{\mathbb{Z}}_p)|\le C{p}^{-n},$$

$$f_{{\mu }_{P,-1}}(a)=\underset{n\to \infty }{\lim }{\mu }_{P,-1}(a+p^n{\mathbb{Z}}_p).$$

$$f_{{\mu }_{P,-1}}(a)$$

$$f_{{\mu }_{P,-1}}(x)=(-1{)}^{x}P(x).$$

$${\int }_{{\mathbb{Z}}_p}g(x)d{\mu }_{P,-1}(x)$$

$$f_{{\mu }_{P,-1}}=(-1{)}^{x}P(x).$$

$${\int }_{{\mathbb{Z}}_p}g(x)d{\mu }_{P,-1}(x)={\int }_{{\mathbb{Z}}_p}g(x)P(x)d{\mu }_{-1}(x),$$

$$||f-f_m|{|}_{\infty }\le \underset{n\ge m}{\sup }n|a_n|.$$

$$|{\mu }_{f,-1}(a+p^n{\mathbb{Z}}_p)-{\mu }_{f,-1}(a+p^{n+1}{\mathbb{Z}}_p)|$$

$$\le \max \{|{\mu }_{f_{m,-1}}(a+p^n{\mathbb{Z}}_p)-{\mu }_{f_{m,-1}}(a+p^{n+1}{\mathbb{Z}}_p)|,$$

$$|{\mu }_{f-f_{m,-1}}(a+p^n{\mathbb{Z}}_p)-{\mu }_{f-f_{m,-1}}(a+p^{n+1}{\mathbb{Z}}_p)|\}.$$

$$|{\mu }_{f-f_m,-1}(a+p^n{\mathbb{Z}}_p)|\le ||f-f_m|{|}_{\infty }\le C_1{p}^{-n}.$$

Full text

Lebesque-Radon-Nikodym theorem with respect to fermionic $p$-adic invariant measure on $\mathbb{Z}_{p}$

Taekyun Kim

Taekyun Kim. Division of General Education-Mathematics

Kwangwoon University, Seoul 139-701, Republic of Korea

tkkim@kw.ac.kr

Abstract   In this paper we derive the analogue of the Lebesque-Radon-Nikodym theorem with respect to fermionic $p$-adic invariant measure on $\mathbb{Z}_{p}$.

2010 Mathematics Subject Classification : 11S80, 48B22, 28B99

Key words and phrases : Lebesque-Radon-Nikodym theorem, fermionic $p$-adic invariant integral

1. Introduction

Let $p$ be a fixed odd prime number. Throughout this paper, the symbol $\mathbb{Z}$, $\mathbb{Z}_{p}$, $\mathbb{Q}_{p}$, and $\mathbb{C}_{p}$ denote the ring of rational integers, the ring of $p$-adic integers, the field of $p$-adic rational numbers, and the completion of algebraic closure of $\mathbb{Q}_{p}$, respectively. Let $\nu_{p}$ be the normalized exponential valuation of $\mathbb{C}_{p}$ with $|p|_{p}=p^{-\nu_{p}(p)}=p^{-1}$. In this paper, we assume that $q\in\mathbb{C}_{p}$ with $|q-1|<p^{-\frac{1}{p-1}}$ as an indeterminate. We use the notations

[TABLE]

for all $x\in\mathbb{Z}_{p}$. For any positive integer $N$, let $a+p^{N}\mathbb{Z}_{p}=\{x\in X|\,x\equiv a\pmod{p^{N}}\}$, where $a\in\mathbb{Z}$ lies in $0\leq a<p^{N}$. The $p$-adic $q$-invariant distribution $\mu_{q}$ on $\mathbb{Z}_{p}$ is defined by

[TABLE]

We say that $f$ is a uniformly differentiable function at a point $a\in\mathbb{Z}_{p}$ and denote this property by $f\in UD(\mathbb{Z}_{p})$, if the difference quotients $F_{f}(x,y)=\frac{f(x)-f(y)}{x-y}$ have a limit $l=f^{\prime}(a)$ as $(x,y)\rightarrow(a,a)$. For $f\in UD(\mathbb{Z}_{p})$, the $p$-adic invariant $q$-integral is defined as

[TABLE]

The fermionic $p$-adic $q$-measures on $\mathbb{Z}_{p}$ are defined as

[TABLE]

and the fermionic $p$-adic invariant $q$-integral on $\mathbb{Z}_{p}$ is defined as

[TABLE]

for $f\in UD(\mathbb{Z}_{p})$. The fermionic $p$-adic invariant integral on $\mathbb{Z}_{p}$ is defined as

[TABLE]

Let $C(\mathbb{Z}_{p},\mathbb{C}_{p})$ be the space of continuous function on $\mathbb{Z}_{p}$ with values in $\mathbb{C}_{p}$, provided with norm $||f||_{\infty}=\underset{x\in\mathbb{Z}_{p}}{\sup}|f(x)|$. The difference quotient $\bigtriangleup_{1}f$ of $f$ is the function of two variables given by

[TABLE]

for all $x\in\mathbb{Z}_{p}$ and $m\in\mathbb{Z}_{p}$ with $m\neq 0$. A function $f:\mathbb{Z}_{p}\longrightarrow\mathbb{C}_{p}$ is said to be a Lipschitz function if there exists a constant $M>0$ such that for all $m\in\mathbb{Z}_{p}\setminus\{0\}$ and $x\in\mathbb{Z}_{p}$,

[TABLE]

Here $M$ is called the Lipschitz constant of $f$. The $\mathbb{C}_{p}$-linear space consisting of all Lipschitz function is denoted by Lip$(\mathbb{Z}_{p},\mathbb{C}_{p})$. This space is a Banach space with respect to the norm $||f||_{1}=||f||_{\infty}\vee||\bigtriangleup_{1}f||_{\infty}$, (see [8]). 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 measures, the Riesz representation theorem for functionals on Hilbert spaces, or martingale theory, ( see [9, 10, 11, 12]). Recently, several authors have studied related to Radon-Nikodym theorems ( see [1, 2, 3]). In previous paper [7], author have studied the analogue of the Lebesque-Radon-Nikodym theorem with respect to $p$-adic $q$-invariant distribution on $\mathbb{Z}_{p}$. The purpose of this paper is to derive the Lebesque-Radon-Nikodym’s type theorem with respect to fermionic $p$-adic invariant measures on $\mathbb{Z}_{p}$.

2. Analogue of the

Lebesque-Radon-Nikodym theorem with respect to fermionic $p$-adic invariant integral on $\mathbb{Z}_{p}$

Let $f\in UD(\mathbb{Z}_{p})$. For any positive integers $a$ and $n$ with $a<p^{n}$, define

[TABLE]

where the integral is the fermionic $p$-adic invariant integral on $\mathbb{Z}_{p}$.

It is easy to see that

[TABLE]

Thus it follows that

[TABLE]

By (3), we have that for $f,\,g\in UD(\mathbb{Z}_{p},\mathbb{C}_{p})$,

[TABLE]

where $\alpha,\,\beta$ are constants. We also see that

[TABLE]

Here we recall the definition of the strongly fermionic $p$-adic invariant measure on $\mathbb{Z}_{p}$. If $\mu_{-1}$ is satisfied the following equation

[TABLE]

where $\delta_{n}\rightarrow 0$ as $n\rightarrow\infty$ and $\delta_{n}$ is independent of $a$, then $\mu_{-1}$ is called the weakly fermionic invariant measure on $\mathbb{Z}_{p}$. If $\delta_{n}$ is replaced by $cp^{-n}$ with a constant $c$, then $\mu_{-1}$ is called strongly fermionic $p$-adic invariant measure on $\mathbb{Z}_{p}$.

Let $P(x)\in\mathbb{C}_{p}[[x]]$ be an arbitrary polynomial. Now we claim that $\mu_{P,-1}$ is a strongly fermionic $p$-adic invariant measure on $\mathbb{Z}_{p}$. Without a loss of generality, it is enough to prove the statement for $P(x)=x^{k}$.

Let $a$ be an integer with $0\leq a<p^{n}$. Then we see that

[TABLE]

It is easy to see that

[TABLE]

By (6) and (7), we have

[TABLE]

From the definition of the Euler numbers $E_{n}$, we note that

[TABLE]

(see [6]). By (8) and (9), we obtain that

[TABLE]

Let $x$ be an arbitrary in $\mathbb{Z}_{p}$. Let $x\equiv x_{n}\,\,(\text{mod}\,p^{n})$ and $x\equiv x_{n+1}\,\,(\text{mod}\,p^{n+1})$, where $x_{n}$ and $x_{n+1}$ are positive integers such that $0\leq x_{n}<p^{n}$ and $0\leq x_{n+1}<p^{n+1}$. Then we have

[TABLE]

where $C$ is some constant and $n\gg 0$.

Let

[TABLE]

Then we have from (8) and (9) that

[TABLE]

Since $f_{\mu_{P,-1}}(x)$ is continuous, it follows that for all $x\in\mathbb{Z}_{p}$,

[TABLE]

For $g\in UD(\mathbb{Z}_{p},\mathbb{C}_{p})$, we see that

[TABLE]

Therefore we have the following theorem.

Theorem 1**.**

Let $P(x)\in\mathbb{C}_{p}[[x]]$ be an arbitrary polynomial. Then $\mu_{P,-1}$ is a strongly fermionic $p$-adic invariant measure and for all $x\in\mathbb{Z}_{p}$,

[TABLE]

Furthermore, for any $g\in UD(\mathbb{Z}_{p},\mathbb{C}_{p})$,

[TABLE]

where the second integral is fermionic $p$-adic invariant integral on $\mathbb{Z}_{p}$.

Let $f(x)=\sum_{n=0}^{\infty}a_{n}\binom{x}{n}$ be the Mahler expansion of uniformly differential function of $f$, where $\binom{x}{n}$ is the binomial coefficient. Then we note that $\lim_{n\rightarrow\infty}n|a_{n}|=0.$ Let $f_{m}(x)=\sum_{i=0}^{m}a_{i}\binom{x}{i}\in\mathbb{C}_{p}[[x]].$ Then

[TABLE]

Writing $f=f_{m}+f-f_{m}$, we easily see that

[TABLE]

By Theorem 1, we note that for some constant $C_{1}$,

[TABLE]

Also, for $m\gg 0$, it follows that $||f||_{\infty}=||f_{m}||_{\infty}$, and so

[TABLE]

where $C_{2}$ is some positive constant.

From (14), we note that

[TABLE]

If we fix $\epsilon>0$, and fix $m$ such that $||f-f_{m}||_{\infty}\leq\epsilon$, then for $n\gg 0$, we have

[TABLE]

Hence we see that

[TABLE]

Let $m$ be the sufficiently large number such that $||f-f_{m}||_{\infty}\leq p^{-n}$. Then we have

[TABLE]

Let $g\in UD(\mathbb{Z}_{p},\mathbb{C}_{p})$. Then

[TABLE]

Let $f$ be the function from $UD(\mathbb{Z}_{p},\mathbb{C}_{p})$ to Lip$(\mathbb{Z}_{p},\mathbb{C}_{p})$. We know that $\mu_{-1}$ is a strongly $p$-adic invariant measure on $\mathbb{Z}_{p}$, and

[TABLE]

for any positive integer $n$ and some constant $C_{4}$.

If $\mu_{1,-1}$ is the associated strongly fermionic $p$-adic invariant measures on $\mathbb{Z}_{p}$, then we have

[TABLE]

for $n\gg 0$. Thus, we note that for sufficiently large $n$,

[TABLE]

where $K$ is a constant. Therefore $\mu_{-1}-\mu_{1,-1}$ is a measure on $\mathbb{Z}_{p}$. Hence we obtain the following theorem.

Theorem 2**.**

Let $\mu_{-1}$ be a strongly fermionic $p$-adic invariant on $\mathbb{Z}_{p}$, and assume that the fermionic Radon-Nikodym derivative $f_{\mu_{-1}}$ on $\mathbb{Z}_{p}$ is uniformly differentiable function. Suppose that $\mu_{1,-1}$ is the strongly fermionic $p$-adic invariant measure associated to $f_{\mu_{-1}}$. Then there exists a measure $\mu_{2,-1}$ on $\mathbb{Z}_{p}$ such that

[TABLE]

Remark 1**.**

Theorem 2 seems to be the $p$-adic analogue of Lebesque decomposition with respect to strongly fermionic $p$-adic invariant measure on $\mathbb{Z}_{p}$, ( see [9, 10]).