Opuscula Math. 44, no. 3 (2024), 323-339

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.

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  • 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.
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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

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