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# A note on the Analogue of Lebesgue-Radon-Nikodym theorem with respect to weighted p-adic q-measure on ${\mathbb{Z}}_{p}$

Journal of Inequalities and Applications20132013:15

https://doi.org/10.1186/1029-242X-2013-15

• Received: 2 September 2012
• Accepted: 13 December 2012
• Published:

## Abstract

We give the analogue of the Lebesgue-Radon-Nikodym theorem with respect to a weighted p-adic q-measure on ${\mathbb{Z}}_{p}$. In a special case, when the weight ${q}^{x}$ is 1, we can derive the same result as Kim et al. (Abstr. Appl. Anal. 2011:637634, 2011). And if $q=1$, we have the same result as Kim (Russ. J. Math. Phys. 19:193-196, 2012).

MSC:11B68, 11S80.

## 1 Introduction

Let p be a fixed odd prime number. Throughout this paper, the symbols ${\mathbb{Z}}_{p}$, ${\mathbb{Q}}_{p}$, and ${\mathbb{C}}_{p}$ denote the ring of p-adic integers, the field of p-adic rational numbers, and the p-adic completion of the algebraic closure of ${\mathbb{Q}}_{p}$, respectively. Let ${\nu }_{p}$ be the normalized exponential valuation of ${\mathbb{C}}_{p}$ with $|p|={p}^{-{\nu }_{p}\left(p\right)}=\frac{1}{p}$ and ${\nu }_{p}\left(0\right)=\mathrm{\infty }$.

When one speaks of q-extension, q can be regarded as an indeterminate, a complex $q\in \mathbb{C}$, or a p-adic number $q\in {\mathbb{C}}_{p}$. In this paper, we assume that $q\in {\mathbb{C}}_{p}$ with $|1-q|<1$, and we use the notations of q-numbers as follows:
${\left[x\right]}_{q}=\left[x:q\right]=\frac{1-{q}^{x}}{1-q}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\left[x\right]}_{-q}=\frac{1-{\left(-q\right)}^{x}}{1+q}.$
(1.1)
For any positive integer N, let
$a+{p}^{N}{\mathbb{Z}}_{p}=\left\{x\in {\mathbb{Z}}_{p}|x\equiv a\phantom{\rule{1em}{0ex}}\left(mod{p}^{N}\right)\right\},$
(1.2)

where $a\in \mathbb{Z}$ satisfies the condition $0\le a<{p}^{N}$ (see ).

It is known that the fermionic p-adic q-measure on ${\mathbb{Z}}_{p}$ is given by Kim as follows:
${\mu }_{-q}\left(a+{p}^{N}{\mathbb{Z}}_{p}\right)=\frac{{\left(-q\right)}^{a}}{{\left[{p}^{N}\right]}_{-q}}=\frac{1+q}{1+{q}^{{p}^{N}}}{\left(-q\right)}^{a},\phantom{\rule{1em}{0ex}}\text{(see [1, 6, 9–12])}.$
(1.3)
Let $C\left({\mathbb{Z}}_{p}\right)$ be the space of continuous functions on ${\mathbb{Z}}_{p}$. From (1.3), the fermionic p-adic q-integral on ${\mathbb{Z}}_{p}$ is defined by Kim as follows:
${I}_{-q}\left(f\right)={\int }_{{\mathbb{Z}}_{p}}f\left(x\right)\phantom{\rule{0.2em}{0ex}}d{\mu }_{-q}\left(x\right)=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{{\left[{p}^{N}\right]}_{-q}}\sum _{x=0}^{{p}^{N}-1}f\left(x\right){\left(-q\right)}^{x},$
(1.4)

where $f\in C\left({\mathbb{Z}}_{p}\right)$ (see [1, 6, 912]).

Let us assume $q\in {\mathbb{C}}_{p}$ with $|q-1|<1$. By (1.4), we get
${\int }_{{\mathbb{Z}}_{p}}{q}^{-x}{e}^{{\left[x\right]}_{q}t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{-q}\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}{E}_{n,q}\frac{{t}^{n}}{n!},$
(1.5)
(see [7, 8, 13]) where ${E}_{n,q}$ are q-Euler numbers. The q-Euler polynomials, ${E}_{n,q}\left(x\right)$, are also defined by
${\int }_{{\mathbb{Z}}_{p}}{q}^{-y}{e}^{{\left[x+y\right]}_{q}t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{-y}\left(t\right)=\sum _{n=0}^{\mathrm{\infty }}{E}_{n,q}\left(x\right)\frac{{t}^{n}}{n!}.$
(1.6)
By (1.5) and (1.6), we get
${E}_{n,q}\left(x\right)=\sum _{l=0}^{n}\left(\begin{array}{c}n\\ l\end{array}\right){x}^{n-l}{E}_{l,q}={\left(x+{E}_{q}\right)}^{n},$
with the usual convention of replacing ${\left({E}_{q}\right)}^{n}$ by ${E}_{n,q}$ (see [1, 2, 7, 8, 13]),
${E}_{n,q}={\int }_{{\mathbb{Z}}_{p}}{q}^{-x}{\left[x\right]}_{q}^{n}\phantom{\rule{0.2em}{0ex}}d{\mu }_{-q}\left(x\right)=\frac{{\left[\iota \right]}_{q}}{{\left(1-q\right)}^{n}}\sum _{l=0}^{n}{\left(-1\right)}^{l}\left(\begin{array}{c}n\\ l\end{array}\right)\frac{1}{{\left[l\right]}_{q}}.$

We will give the analogue of the Lebesgue-Radon-Nikodym theorem with respect to a weighted p-adic q-measure on ${\mathbb{Z}}_{p}$. In a special case, when the weight ${q}^{x}$ is 1, we can derive the same result as Kim et al. . And if $q=1$, we have the same result as Kim .

## 2 Lebesgue-Radon-Nikody-type theorem with respect to a weighted p-adic q-measure on ${\mathbb{Z}}_{p}$

For any positive integer a and n, with $a<{p}^{n}$ and $f\in C\left({\mathbb{Z}}_{p}\right)$, let us define
${\stackrel{˜}{\mu }}_{f,-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right)={\int }_{a+{p}^{n}{\mathbb{Z}}_{p}}{q}^{-x}f\left(x\right)\phantom{\rule{0.2em}{0ex}}d{\mu }_{-q}\left(x\right),$
(2.1)

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

By (2.2), we get
${\stackrel{˜}{\mu }}_{f,-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right)=\frac{{\left[2\right]}_{q}}{{\left[2\right]}_{{q}^{{p}^{n}}}}{\left(-1\right)}^{a}{\int }_{{\mathbb{Z}}_{p}}{q}^{{p}^{n}x}f\left(a+{p}^{n}x\right)\phantom{\rule{0.2em}{0ex}}d{\mu }_{-{q}^{{p}^{n}}}\left(x\right).$
(2.3)

Therefore, by (2.3), we obtain the following theorem.

Theorem 1 For $f,g\in C\left({\mathbb{Z}}_{p}\right)$, we have
${\stackrel{˜}{\mu }}_{\alpha f+\beta g,-q}=\alpha {\stackrel{˜}{\mu }}_{f,-q}+\beta {\stackrel{˜}{\mu }}_{g,-q},$
(2.4)

where α, β are constants.

From (2.2) and (2.4), we note that
$|{\stackrel{˜}{\mu }}_{f,-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right)|\le M{\parallel {f}_{q}\parallel }_{\mathrm{\infty }},$
(2.5)

where ${\parallel {f}_{q}\parallel }_{\mathrm{\infty }}={sup}_{x\in {\mathbb{Z}}_{p}}|{q}^{-x}f\left(x\right)|$ and M is some positive constant.

Now, we recall the definition of the strongly fermionic p-adic q-measure on ${\mathbb{Z}}_{p}$. If ${\mu }_{-q}$ satisfies the following equation:
$|{\mu }_{-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right)-{\mu }_{-q}\left(a+{p}^{n+1}{\mathbb{Z}}_{p}\right)|\le {\delta }_{n,q},$
(2.6)

where ${\delta }_{n,q}\to 0$ and $n\to \mathrm{\infty }$ and ${\delta }_{n,q}$ is independent of a, then ${\mu }_{-q}$ is called a weakly fermionic p-adic q-measure on ${\mathbb{Z}}_{p}$.

If ${\delta }_{n,q}$ is replaced by $C{p}^{-{\nu }_{p}\left(1-{q}^{n}\right)}$ (C is some constant), then ${\mu }_{-q}$ is called a strongly fermionic p-adic q-measure on ${\mathbb{Z}}_{p}$.

Let $P\left(x\right)\in {\mathbb{C}}_{p}\left[{\left[x\right]}_{q}\right]$ be an arbitrary q-polynomial with $\sum {a}_{i}{\left[x\right]}_{q}^{i}$. Then we see that ${\mu }_{P,-q}$ is a strongly fermionic p-adic q-measure on ${\mathbb{Z}}_{p}$. Without loss of generality, it is enough to prove the statement for $P\left(x\right)={\left[x\right]}_{q}^{k}$.

Let a be an integer with $0\le a<{p}^{n}$. Then we get
${\stackrel{˜}{\mu }}_{P,-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right)=\frac{{\left[2\right]}_{q}}{{\left[2\right]}_{{q}^{{p}^{n}}}}{\left(-1\right)}^{a}\underset{m\to \mathrm{\infty }}{lim}\frac{1}{{\left[{p}^{m-n}\right]}_{-{q}^{{p}^{n}}}}\sum _{i=0}^{{p}^{m-n}-1}{\left[a+i{p}^{n}\right]}_{q}^{k}{\left(-1\right)}^{i}{q}^{{p}^{n}i},$
(2.7)
and
${q}^{{p}^{n}i}=\sum _{l=0}^{i}\left(\genfrac{}{}{0}{}{i}{l}\right){\left[{p}^{n}\right]}_{q}^{l}{\left(q-1\right)}^{l}.$
By (2.7), we easily get
$\begin{array}{rl}{\stackrel{˜}{\mu }}_{P,-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right)& \equiv \frac{{\left[2\right]}_{q}}{{\left[2\right]}_{{q}^{{p}^{n}}}}{\left(-1\right)}^{a}{\left[a\right]}_{q}^{k}\phantom{\rule{1em}{0ex}}\left(mod{\left[{p}^{n}\right]}_{q}\right)\\ \equiv \frac{{\left[2\right]}_{q}}{{\left[2\right]}_{{q}^{{p}^{n}}}}{\left(-1\right)}^{a}P\left(a\right)\phantom{\rule{1em}{0ex}}\left(mod{\left[{p}^{n}\right]}_{q}\right).\end{array}$
(2.8)
Let x be arbitrary in ${\mathbb{Z}}_{p}$ with $x\equiv {x}_{n}\left(mod{p}^{n}\right)$ and $x\equiv {x}_{n+1}\left(mod{p}^{n+1}\right)$, where ${x}_{n}$ and ${x}_{n+1}$ are positive integers such that $0\le {x}_{n}<{p}^{n}$ and $0\le {x}_{n+1}<{p}^{n+1}$. Thus, by (2.8), we have
$|{\stackrel{˜}{\mu }}_{P,-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right)-{\stackrel{˜}{\mu }}_{P,-q}\left(a+{p}^{n+1}{\mathbb{Z}}_{p}\right)|\le C{p}^{-{\nu }_{p}\left(1-{q}^{{p}^{n}}\right)},$
(2.9)

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

Let
${f}_{{\stackrel{˜}{\mu }}_{P,-q}}\left(a\right)=\underset{n\to \mathrm{\infty }}{lim}{\stackrel{˜}{\mu }}_{P,-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right).$
(2.10)
Then by (2.5), (2.7), and (2.8), we get
${f}_{{\stackrel{˜}{\mu }}_{P,-q}}\left(a\right)=\frac{{\left[2\right]}_{q}}{2}{\left(-1\right)}^{a}{\left[a\right]}_{q}^{k}=\frac{{\left[2\right]}_{q}}{2}{\left(-1\right)}^{a}P\left(a\right).$
(2.11)
Since ${f}_{{\stackrel{˜}{\mu }}_{P,-q}}\left(x\right)$ is continuous on ${\mathbb{Z}}_{p}$, it follows, for all $x\in {\mathbb{Z}}_{p}$,
${f}_{{\stackrel{˜}{\mu }}_{P,-q}}\left(x\right)=\frac{{\left[2\right]}_{q}}{2}{\left(-1\right)}^{x}P\left(x\right).$
(2.12)
Let $g\in C\left({\mathbb{Z}}_{p}\right)$. By (2.10), (2.11), and (2.12), we get
$\begin{array}{rcl}{\int }_{{\mathbb{Z}}_{p}}g\left(x\right)\phantom{\rule{0.2em}{0ex}}d{\stackrel{˜}{\mu }}_{P,-q}\left(x\right)& =& \underset{n\to \mathrm{\infty }}{lim}\sum _{i=0}^{{p}^{n}-1}g\left(i\right){\stackrel{˜}{\mu }}_{P,-q}\left(i+{p}^{n}{\mathbb{Z}}_{p}\right)\\ =& \frac{{\left[2\right]}_{q}}{2}\underset{n\to \mathrm{\infty }}{lim}\sum _{i=0}^{{p}^{n}-1}g\left(i\right){\left(-q\right)}^{i}{\left[i\right]}_{q}^{k}\\ =& {\int }_{{\mathbb{Z}}_{p}}{q}^{-x}g\left(x\right){\left[x\right]}_{q}^{k}\phantom{\rule{0.2em}{0ex}}d{\mu }_{-q}\left(x\right).\end{array}$
(2.13)

Therefore, by (2.13), we obtain the following theorem.

Theorem 2 Let $P\left(x\right)\in {\mathbb{C}}_{p}\left[{\left[x\right]}_{q}\right]$ be an arbitrary q-polynomial with $\sum {a}_{i}{\left[x\right]}_{q}^{i}$. Then ${\stackrel{˜}{\mu }}_{P,-q}$ is a strongly fermionic weighted p-adic q-measure on ${\mathbb{Z}}_{p}$, and for all $x\in {\mathbb{Z}}_{p}$,
${f}_{{\stackrel{˜}{\mu }}_{P,-q}}={\left(-1\right)}^{x}\frac{{\left[2\right]}_{q}}{2}P\left(x\right).$
(2.14)
Furthermore, for any $g\in C\left({\mathbb{Z}}_{p}\right)$, we have
${\int }_{{\mathbb{Z}}_{p}}g\left(x\right)\phantom{\rule{0.2em}{0ex}}d{\stackrel{˜}{\mu }}_{P,-q}\left(x\right)={\int }_{{\mathbb{Z}}_{p}}{q}^{-x}g\left(x\right)P\left(x\right)\phantom{\rule{0.2em}{0ex}}d{\mu }_{-q}\left(x\right),$
(2.15)

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

Let $f\left(x\right)={\sum }_{n=0}^{\mathrm{\infty }}{a}_{n,q}{\left(\genfrac{}{}{0}{}{x}{n}\right)}_{q}$ be the q-Mahler expansion of a continuous function on ${\mathbb{Z}}_{p}$, where
${\left(\genfrac{}{}{0}{}{x}{n}\right)}_{q}=\frac{{\left[x\right]}_{q}{\left[x-1\right]}_{q}\cdots {\left[x-n+1\right]}_{q}}{{\left[n\right]}_{q}!}\phantom{\rule{1em}{0ex}}\left(\text{see }\right).$
(2.16)

Then we note that ${lim}_{n\to \mathrm{\infty }}|{a}_{n,q}|=0$.

Let
${f}_{m}\left(x\right)=\sum _{i=0}^{m}{a}_{i,q}{\left(\genfrac{}{}{0}{}{x}{i}\right)}_{q}\in {\mathbb{C}}_{p}\left[{\left[x\right]}_{q}\right].$
(2.17)
Then
${\parallel f-{f}_{m}\parallel }_{\mathrm{\infty }}\le \underset{m\le n}{sup}|{a}_{n,q}|.$
(2.18)
Writing $f={f}_{m}+f-{f}_{m}$, we easily get
From Theorem 2, we note that
$|{\stackrel{˜}{\mu }}_{f-{f}_{m},-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right)|\le {\parallel f-{f}_{m}\parallel }_{\mathrm{\infty }}\le {C}_{1}{p}^{-{\nu }_{p}\left(1-{q}^{{p}^{n}}\right)},$
(2.20)

where ${C}_{1}$ is some positive constant.

For $m\gg 0$, we have ${\parallel f\parallel }_{\mathrm{\infty }}={\parallel {f}_{m}\parallel }_{\mathrm{\infty }}$.

So,
$|{\stackrel{˜}{\mu }}_{{f}_{m},-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right)-{\stackrel{˜}{\mu }}_{{f}_{m},-q}\left(a+{p}^{n+1}{\mathbb{Z}}_{p}\right)|\le {C}_{2}{p}^{-{\nu }_{p}\left(1-{q}^{{p}^{n}}\right)},$
(2.21)

where ${C}_{2}$ is also some positive constant.

If we fix $ϵ>0$ and fix m such that $\parallel f-{f}_{m}\parallel \le ϵ$, then for $n\gg 0$, we have
$|f\left(a\right)-{\stackrel{˜}{\mu }}_{f,-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right)|\le ϵ.$
(2.23)
Hence, we have
${f}_{{\mu }_{f,-q}}\left(a\right)=\underset{n\to \mathrm{\infty }}{lim}{\stackrel{˜}{\mu }}_{f,-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right)=\frac{{\left[2\right]}_{q}}{2}{\left(-1\right)}^{a}f\left(a\right).$
(2.24)

Let m be a sufficiently large number such that ${\parallel f-{f}_{m}\parallel }_{\mathrm{\infty }}\le {p}^{-n}$.

Then we get
$\begin{array}{rcl}{\stackrel{˜}{\mu }}_{f,-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right)& =& {\stackrel{˜}{\mu }}_{{f}_{m},-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right)+{\stackrel{˜}{\mu }}_{f-{f}_{m},-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right)\\ =& {\stackrel{˜}{\mu }}_{{f}_{m},-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right)\\ =& {\left(-1\right)}^{a}\frac{{\left[2\right]}_{q}}{{\left[2\right]}_{{q}^{{p}^{n}}}}f\left(a\right)\phantom{\rule{1em}{0ex}}\left(mod{\left[{p}^{n}\right]}_{q}\right).\end{array}$
(2.25)
For any $g\in C\left({\mathbb{Z}}_{p}\right)$, we have
${\int }_{{\mathbb{Z}}_{p}}g\left(x\right)\phantom{\rule{0.2em}{0ex}}d{\stackrel{˜}{\mu }}_{f,-q}\left(x\right)={\int }_{{\mathbb{Z}}_{p}}{q}^{-x}f\left(x\right)g\left(x\right)\phantom{\rule{0.2em}{0ex}}d{\mu }_{-q}\left(x\right).$
(2.26)
Assume that f is the function from $C\left({\mathbb{Z}}_{p},{\mathbb{C}}_{p}\right)$ to $Lip\left({\mathbb{Z}}_{p},{\mathbb{C}}_{p}\right)$. By the definition of ${\stackrel{˜}{\mu }}_{-q}$, we easily see that ${\stackrel{˜}{\mu }}_{-q}$ is a strongly p-adic q-measure on ${\mathbb{Z}}_{p}$, and for $n\gg 0$,
$|{f}_{{\stackrel{˜}{\mu }}_{-q}}\left(a\right)-{\stackrel{˜}{\mu }}_{-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right)|\le {C}_{3}{p}^{-{\nu }_{p}\left(1-{q}^{{p}^{n}}\right)},$
(2.27)

where ${C}_{3}$ is some positive constant.

If ${\stackrel{˜}{\mu }}_{1,-q}$ is associated with strongly fermionic weighted p-adic q-measure on ${\mathbb{Z}}_{p}$, then we have
$|{\stackrel{˜}{\mu }}_{1,-q}\left(a+{p}^{n}{\mathbb{Z}}_{p}\right)-{f}_{{\stackrel{˜}{\mu }}_{-q}}\left(a\right)|\le {C}_{4}{p}^{-{\nu }_{p}\left(1-{q}^{{p}^{n}}\right)},$
(2.28)

where $n\gg 0$ and ${C}_{4}$ is some positive constant.

where K is some positive constant.

Therefore, ${\stackrel{˜}{\mu }}_{-q}-{\stackrel{˜}{\mu }}_{1,-q}$ is a q-measure on ${\mathbb{Z}}_{p}$. Hence, we obtain the following theorem.

Theorem 3 Let ${\stackrel{˜}{\mu }}_{-q}$ be a strongly fermionic weighted p-adic q-measure on ${\mathbb{Z}}_{p}$, and assume that the fermionic weighted Radon-Nikodym derivative ${f}_{{\stackrel{˜}{\mu }}_{-q}}$ on ${\mathbb{Z}}_{p}$ is a continuous function on ${\mathbb{Z}}_{p}$. Suppose that ${\stackrel{˜}{\mu }}_{1,-q}$ is the strongly fermionic weighted p-adic q-measure associated to ${f}_{{\stackrel{˜}{\mu }}_{-q}}$. Then there exists a q-measure ${\stackrel{˜}{\mu }}_{2,-q}$ on ${\mathbb{Z}}_{p}$ such that
${\stackrel{˜}{\mu }}_{-q}={\stackrel{˜}{\mu }}_{1,-q}+{\stackrel{˜}{\mu }}_{2,-q}.$
(2.30)

## Declarations

### Acknowledgements

This research was supported by the Kyungpook National University Research Fund 2012. The authors would like to thank professor T. Kim who suggested this problem to us.

## Authors’ Affiliations

(1)
Department of Medical Information Technology, Daegu Haany University, Kyungsan, 712-715, Republic of Korea
(2)
Department of Mathematics Education, Kyungpook National University, Daegu, 702-701, Republic of Korea

## References 