A note on the Lebesgue–Radon–Nikodym theorem with respect to weighted and twisted p-adic invariant integral on ℤp

Joo-Hee Jeong, Jin-Woo Park, Seog-Hoon Rim and Joung-Hee Jin
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 21, 2015, Number 1, Pages 10—17
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Authors and affiliations

Joo-Hee Jeong
Department of Mathematics Education, Kyungpook National University
Taegu 702-701, Republic of Korea

Jin-Woo Park
Department of Mathematics Education, Sehan University
YoungAm-gun, Chunnam, 526-702, Republic of Korea

Seog-Hoon Rim 
Department of Mathematics Education, Kyungpook National University
Taegu 702-701, Republic of Korea

Joung-Hee Jin 
Department of Mathematics Education, Kyungpook National University
Taegu 702-701, Republic of Korea

Abstract

In this paper we will give the Lebesgue–Radon–Nikodym theorem with respect to weighted and twisted p-adic q-measure on ℤp. In special case, if there is no twisted, then we can derive the same result as Jeong and Rim, 2012; If the case weight zero and no twist, then we derive the same result as Kim 2012.

Keywords

  • p-adic invariant integral
  • p-adic q-measure
  • Lebesgue–Radon–Nikodym theorem

AMS Classification

  • 11B68
  • 11S80

References

  1. Bayad, A. & T. Kim. (2011) Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials. Russ. J. Math. Phys., 18(2), 133–143.
  2. Calabuig, J. M., P. Gregori & E. A Sanchez Perez. (2008) Radon–Nikodym derivative for vector measures belonging to Kothe function space. J. Math. Anal. Appl. 348, 469–479.
  3. Choi, J., T. Kim & Y. H. Kim. (2011) A note on the q-analogues of Euler numbers and polynomials, Honam Mathematical Journal, 33(4), 529–534.
  4. George, K. (2008) On the Radon–Nikodym theorem, Amer. Math. Monthly, 115, 556–558.
  5. Jeong, J. & S. H. Rim. (2012) A note on the Lebesgue–Radon–Nikodym theorem with respect to weighted p-adic invariant integral on Zp, Abstract and Applied Analysis, 2012, Article ID 696720, 8 pages.
  6. Kim, T. (2002) q-Volkenborn integration. Russ. J. Math. Phys. 9(3), 288–299.
  7. Kim, T. (2012) Lebesgue–Radon–Nikodym theorem with respect to fermionic p-adic invariant measure on Zp, Russ. J. Math. Phys. 19, 193–196.
  8. Kim, T. (2007) Lebesgue–Radon–Nikodym theorem with respect to fermionic q-Volkenborn distribution on q. Appl. Math. Comp., 187, 266–271.
  9. Kim, T. (2011) A note on q-Bernstein polynomials. Russ. J. Math. Phys. 18(1), 73–82.
  10. Kim, T. (2008) Note on the Euler numbers and polynomials. Adv. Stud. Contemp. Math., 17, 131–156.
  11. Kim, T. (2009) Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Zp. Russ. J. Math. Phys. 16, 484–491.
  12. Kim, T. (2010) New approach to q-Euler polynimials of higher order. Russ. J. Math. Phys. 17, 218–225.
  13. Kim, T., J. Choi & H. Kim A note on the weighted Lebesgue–Radon–Nikodym theorem with respect to p-adic invariant integral on Zp. to appear in JAMI.
  14. Kim, T., D. V. Dolgy, S. H. Lee & C. S. Ryoo. (2011) Analogue of Lebesgue–Radon–Nikodym theorem with respect to p-adic q-measure on Zp. Abstract and Applied Analysis, 2011, Article ID637634, 6 pages.
  15. Kim, T., S. D. Kim, & D. W. Park. (2001) On Uniformly differntiabitity and q-Mahler expansion. Adv. Stud. Contemp. Math., 4, 35–41.
  16. Kim, Y. H., B. Lee & T. Kim. (2011) On the q-extension of the twisted generalized Euler numbers and polynomials attached to χ­. J. of Comp. Anal. and Appl., 13(7), 1208–1213.
  17. Jeong, J. & S. H. Rim. (2012) A note on the Lebesgue–Radon–Nikodym theorem with respect to weighted p-adic invariant integral on Zp. Abstract and Applied Analysis, 2012, Article ID 696720, 8 pages.

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Cite this paper

APA

Jeong, J.-H., Park, J.-W., Rim, S.-H., & Jin, J.-H. (2015). A note on the Lebesgue–Radon–Nikodym theorem with respect to weighted and twisted p-adic invariant integral on ℤp. Notes on Number Theory and Discrete Mathematics, 21(1), 10-17.

Chicago

Jeong, Joo-Hee, Jin-Woo Park, Seog-Hoon Rim and Joung-Hee Jin. “A Note on the Lebesgue–Radon–Nikodym Theorem with Respect to Weighted and Twisted p-adic Invariant Integral on ℤp.” Notes on Number Theory and Discrete Mathematics 21, no. 1 (2015): 10-17.

MLA

Jeong, Joo-Hee, Jin-Woo Park, Seog-Hoon Rim and Joung-Hee Jin. “A Note on the Lebesgue–Radon–Nikodym Theorem with Respect to Weighted and Twisted p-adic Invariant Integral on ℤp.” Notes on Number Theory and Discrete Mathematics 21.1 (2015): 10-17.. Print.

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