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Functional operators and properties of set functions

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Abstract

Let X be the F-space of the functions x(t) defined on the measurable space (T, Σ, μ) with values in B-space Y. We consider the operators f mapping X to the B-space Z. X, Y, and Z are considered over the scalar field R. To each operator f is associated the family Φf of vector-valued functions\(\varphi _x (e):\Sigma \to Z,\varphi _x (e) = f(x\dot \chi _e ), e \in \Sigma\). The characteristics of these families are given for various classes of operators. The relationship of convergence and continuation of the operators f with convergence and continuation of the corresponding families Φf is considered. Riesz' theorem on integral representation of linear functionals is generalized.

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Authors
  1. G. Ya. Areshkin
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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 159, pp. 113–118, 1987.

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Areshkin, G.Y. Functional operators and properties of set functions. J Math Sci 47, 2903–2907 (1989). https://doi.org/10.1007/BF01305219

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  • DOI: https://doi.org/10.1007/BF01305219

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