Abstract and Applied Analysis

Volumeย 2011, Article IDย 637634, 7 pages

http://dx.doi.org/10.1155/2011/637634

## Analogue of Lebesgue-Radon-Nikodym Theorem with respect to -adic -Measure on

^{1}Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea^{2}Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea^{3}Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea^{4}Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea

Received 3 July 2011; Accepted 12 August 2011

Academic Editor: H. B.ย Thompson

Copyright ยฉ 2011 T. Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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

#### 1. Introduction

Let be a fixed odd prime number. Throughout this paper, the symbols , , and denote the ring of -adic integers, the field of -adic rational numbers, and the -adic completion of the algebraic closure of , respectively. Let be the normalized exponential valuation of with and .

When one speaks of -extension, can be regarded as an indeterminate, a complex , or a -adic number . In this paper we assume that with , and we use the notations of -numbers as follows: For any positive integer , let where satisfies the condition (see [1โ8]).

It is known that the fermionic -adic -measure on is given by Kim as follows: (see [6, 9โ12]). Let be the space of continuous functions on . From (1.3), the fermionic -adic -integral on is defined by Kim as follows: where (see [1, 6, 9โ12]).

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 -adic invariant measure on . The purpose of this paper is to derive the analogue of the Lebesgue-Radon-Nikodym theorem with respect to -adic -measure on in the sense of fermionic.

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

For any positive integer and with and , let us define where the integral is the fermionic -adic -integral on .

From (1.3), (1.4), and (2.1), we note that By (2.2), we get Therefore, by (2.3), we obtain the following theorem.

Theorem 2.1. *For , one has
**
where are constants.*

From (2.2) and (2.4), we note that where and is some positive constant.

Now, we recall the definition of the strongly fermionic -adic -measure on . If is satisfied the following equation: where and and is independent of , then is called the weakly fermionic -adic -measure on .

If is replaced by ( is some constant), then is called strongly fermionic -adic -measure on .

Let be an arbitrary -polynomial with . Then we see that is strongly fermionic -adic -measure on . Without a loss of generality, it is enough to prove the statement for .

Let be an integer with . Then we get By (2.7), we easily get Let be an arbitrary in with and , where and are positive integers such that and . Thus, by (2.8), we have where is a positive some constant and .

Let Then, (2.5), (2.7), and (2.8), we get Since is continuous on , it follows for all Let . By (2.10), (2.11), and (2.12), we get Therefore, by (2.13), we obtain the following theorem.

Theorem 2.2. *Let be an arbitrary -polynomial with . Then is a strongly fermionic -adic -measure on and for all **
Furthermore, for all , one has
**
where the second integral is fermionic -adic -integral on .*

Let be the -Mahler expansion of continuous function on , where (see [4]). Then we note that .

Let Then Writing , we easily get From Theorem 2.2, we note that where is some positive constant.

For , we have .

So where is also some positive constant.

By (2.20) and (2.21), we see that If we fix and fix such that , then, for , we have Hence, we have Let be the sufficiently large number such that .

Then we get For all , we have Assume that is the function from to . By the definition of , we easily see that is a strongly -adic -measure on and for where is some positive constant.

If is associated strongly fermionic -adic -measure on , then we have where and is some positive constant.

From (2.28), we get where is some positive constant.

Therefore, is a -measure on . Hence, we obtain the following theorem.

Theorem 2.3. *Let be a strongly fermionic -adic -measure on , and assume that the fermionic Radon-Nikodym derivative on is continuous function on . Suppose that is the strongly fermionic -adic -measure associated to . Then there exists a -measure on such that
*

#### References

- A. Bayad and T. Kim, โIdentities involving values of Bernstein,
*q*-Bernoulli, and*q*-Euler polynomials,โ*Russian Journal of Mathematical Physics*, vol. 18, no. 2, pp. 133โ143, 2011. View at Publisher ยท View at Google Scholar - M. Cenkci, V. Kurt, S. H. Rim, and Y. Simsek, โOn $(i,q)$ Bernoulli and Euler numbers,โ
*Applied Mathematics Letters*, vol. 21, no. 7, pp. 706โ711, 2008. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - T. Kim, โLebesgue-Radon-Nikodym theorem with respect to fermionic
*p*-adic invariant measure on${\mathbb{Z}}_{p}$,โ to appear in*Russian Journal of Mathematical Physics*. - T. Kim, S. D. Kim, and D.-W. Park, โOn uniform differentiability and
*q*-Mahler expansions,โ*Advanced Studies in Contemporary Mathematics*, vol. 4, no. 1, pp. 35โ41, 2001. View at Google Scholar - S.-H. Rim, K. H. Park, and E. J. Moon, โOn Genocchi numbers and polynomials,โ
*Abstract and Applied Analysis*, vol. 2008, Article ID 898471, 7 pages, 2008. View at Google Scholar ยท View at Zentralblatt MATH - S.-H. Rim, J.-H. Jin, E.-J. Moon, and S.-J. Lee, โSome identities on the
*q*-Genocchi polynomials of higher-order and*q*-Stirling numbers by the fermionic*p*-adic integral on ${\mathbb{Z}}_{p}$,โ*International Journal of Mathematics and Mathematical Sciences*, vol. 2010, Article ID 860280, 14 pages, 2010. View at Google Scholar - S.-H. Rim, J.-H. Jin, E.-J. Moon, and S.-J. Lee, โOn multiple interpolation functions of the
*q*-Genocchi polynomials,โ*Journal of Inequalities and Applications*, vol. 2010, Article ID 351419, 13 pages, 2010. View at Google Scholar - Y. Simsek, โSpecial functions related to Dedekind-type DC-sums and their applications,โ
*Russian Journal of Mathematical Physics*, vol. 17, no. 4, pp. 495โ508, 2010. View at Publisher ยท View at Google Scholar - T. Kim, โSome identities on the
*q*-Euler polynomials of higher order and*q*-Stirling numbers by the fermionic*p*-adic integral on ${\mathbb{Z}}_{p}$,โ*Russian Journal of Mathematical Physics*, vol. 16, no. 4, pp. 484โ491, 2009. View at Publisher ยท View at Google Scholar ยท View at MathSciNet - T. Kim, โNew approach to
*q*-Euler polynomials of higher order,โ*Russian Journal of Mathematical Physics*, vol. 17, no. 2, pp. 218โ225, 2010. View at Publisher ยท View at Google Scholar - T. Kim, โSymmetry of power sum polynomials and multivariate fermionic
*p*-adic invariant integral on ${\mathbb{Z}}_{p}$,โ*Russian Journal of Mathematical Physics*, vol. 16, no. 1, pp. 93โ96, 2009. View at Publisher ยท View at Google Scholar ยท View at MathSciNet - T. Kim, โOn the multiple
*q*-Genocchi and Euler numbers,โ*Russian Journal of Mathematical Physics*, vol. 15, no. 4, pp. 481โ486, 2008. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - W. Rudin,
*Real and Complex Analysis*, McGraw-Hill, New York, NY, USA, 3rd edition, 1987. - H. L. Royden,
*Real Analysis*, Prentice-Hall, Upper Saddle River, NJ, USA, 1998.