`Abstract and Applied AnalysisVolumeย 2011, Article IDย 637634, 7 pageshttp://dx.doi.org/10.1155/2011/637634`
Research Article

T. Kim,1ย D. V. Dolgy,2ย S. H. Lee,3ย and C. S. Ryoo4

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

Received 3 July 2011; Accepted 12 August 2011

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 [1โ8]).

It is known that the fermionic -adic -measure on is given by Kim as follows: (see [6, 9โ12]). 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, 9โ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 -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.

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

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