LOCALIZED TRANSFUNCTIONS
Jason Bentley1, Piotr Mikusiński2
1,2 Department of Mathematics
University of Central Florida
Orlando - 32816, USA

Abstract

Generalized functions, called transfunctions, are defined as maps between spaces of measures on measurable spaces $(X,\Sigma_X)$ and $(Y, \Sigma_Y)$. Measurable functions $f: (X,\Sigma_X) \to (Y,\Sigma_Y)$ can be identified with transfunctions via the push forward operator $f_\char93 (\mu) (B) = \mu (f^{-1}(B))$. In this paper we introduce the notion of localization of transfunctions that gives an insight into which transfunctions arise from continuous functions or measurable functions or are close to such functions. We also introduce the notion of a graph of a transfunction and describe what it tells us about the transfunction. In our investigation of transfunctions, we are motivated by applications that include Monge-Kantorovich transportation problems and population dynamics.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 31
Issue: 6
Year: 2018

DOI: 10.12732/ijam.v31i6.1

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