Opuscula Math. 44, no. 3 (2024), 323-339
https://doi.org/10.7494/OpMath.2024.44.3.323

 
Opuscula Mathematica

Finitely additive functions in measure theory and applications

Daniel Alpay
Palle Jorgensen

Abstract. In this paper, we consider, and make precise, a certain extension of the Radon-Nikodym derivative operator, to functions which are additive, but not necessarily sigma-additive, on a subset of a given sigma-algebra. We give applications to probability theory; in particular, to the study of \(\mu\)-Brownian motion, to stochastic calculus via generalized Itô-integrals, and their adjoints (in the form of generalized stochastic derivatives), to systems of transition probability operators indexed by families of measures \(\mu\), and to adjoints of composition operators.

Keywords: Hilbert space, reproducing kernels, probability space, Gaussian fields, transforms, covariance, Itô integration, Itô calculus, generalized Brownian motion.

Mathematics Subject Classification: 47B32, 60G20, 60G15, 60H05, 60J60, 46E22.

Full text (pdf)

  1. S. Albeverio, L. Nizhnik, A Schrödinger operator with a \(\delta'\)-interaction on a Cantor set and Krein-Feller operators, Math. Nachr. 279 (2006), no. 5-6, 467-476. https://doi.org/10.1002/mana.200310371
  2. D. Alpay, On linear combination of positive functions, associated reproducing kernel spaces and a non hermitian Schur algorithm, Arch. Math. (Basel) 58 (1992), 174-182. https://doi.org/10.1007/BF01191883
  3. D. Alpay, A theorem on reproducing kernel Hilbert spaces of pairs, Rocky Mountain J. Math. 22 (1992), 1243-1258. https://doi.org/10.1216/rmjm/1181072652
  4. D. Alpay, P. Jorgensen, New characterizations of reproducing kernel Hilbert spaces and applications to metric geometry, Opuscula Math. 41 (2021), no. 3, 283-300. https://doi.org/10.7494/OpMath.2021.41.3.283
  5. D. Alpay, P. Jorgensen, mu-Brownian motion, dualities, diffusions, transforms, and reproducing kernel spaces, J. Theoret. Probab. 35 (2022), no. 4, 2757-2783. https://doi.org/10.1007/s10959-021-01146-w
  6. D. Alpay, P. Jorgensen, D. Levanony, On the equivalence of probability spaces, J. Theoret. Probab. 30 (2017), no. 3, 813-841.
  7. N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337-404. https://doi.org/10.2307/1990404
  8. N. Aronszajn, Quadratic forms on vector spaces, Proc. Internat. Sympos. Linear Spaces, Jerusalem, 1960, 29-87.
  9. S.D. Chatterji, Les martingales et leurs applications analytiques, [in:] École d'Été de Probabilités: Processus Stochastiques (Saint Flour, 1971), Lecture Notes in Math., vol. 307, Springer Verlag, 1973, 27-164. https://doi.org/10.1007/BFb0059708
  10. H. Dym, H.P. McKean, Gaussian Processes, Function Theory and the Inverse Spectral Problem, Probability and Mathematical Statistics, vol. 31, Academic Press, New York-London, 1976.
  11. W. Feller, On boundaries defined by stochastic matrices, Proc. Sympos. Appl. Math. 7 (1957), 35-40.
  12. U. Freiberg, Refinement of the spectral asymptotics of generalized Krein Feller operators, Forum Math. 23 (2011), no. 2, 427-445. https://doi.org/10.1515/FORM.2011.017
  13. T. Hida, White noise analysis and applications in random fields, [in:] Dirichlet forms and stochastic processes (Beijing, 1993), de Gruyter, Berlin, 1995, 185-189.
  14. K. Itô, Stochastic Processes, Springer-Verlag, Berlin, 2004.
  15. P. Jorgensen, F. Tian, Reproducing kernels and choices of associated feature spaces, in the form of \(L^2\)-spaces, J. Math. Anal. Appl. 505 (2022), no. 2, Article no. 125535. https://doi.org/10.1016/j.jmaa.2021.125535
  16. L.A. Minorics, Spectral asymptotics for Krein-Feller operators with respect to \(V\)-variable Cantor measures, Forum Math. 32 (2020), no. 1, 121-138. https://doi.org/10.1515/forum-2018-0188
  17. E. Nelson, Topics in Dynamics I: Flows, Mathematical Notes, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1969.
  18. E. Nelson, Stochastic mechanics of particles and fields, [in:] Quantum Interaction, Lecture Notes in Comput. Sci., vol. 8369, Springer, Heidelberg, 2014, 1-5.
  19. J. Neveu, Processus aléatoires gaussiens, Number 34 in Séminaires de Mathématiques Supérieures, Les Presses de l'Université de Montréal, 1968.
  20. F. Riesz, Untersuchungen über Systeme integrierbarer Funkionen, Math. Ann. 69 (1910), 449-497. https://doi.org/10.1007/BF01457637
  21. L. Schwartz, Probabilités cylindriques et fonctions aléatoires, [in:] Séminaire Laurent Schwartz 1969-1970: Applications radonifiantes, Exp. No. 6, École Polytechnique, 1970, 1-8.
  • Daniel Alpay (corresponding author)
  • ORCID iD https://orcid.org/0000-0002-7612-3598
  • Schmid College of Science and Technology, Chapman University, One University Drive Orange, California 92866, USA
  • Communicated by Aurelian Gheondea.
  • Received: 2023-04-27.
  • Accepted: 2023-07-20.
  • Published online: 2024-02-15.
Opuscula Mathematica - cover

Cite this article as:
Daniel Alpay, Palle Jorgensen, Finitely additive functions in measure theory and applications, Opuscula Math. 44, no. 3 (2024), 323-339, https://doi.org/10.7494/OpMath.2024.44.3.323

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

We advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.