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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The composition of operator-valued measurable functions is measurable
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by A. Badrikian, G. W. Johnson and Il Yoo PDF
Proc. Amer. Math. Soc. 123 (1995), 1815-1820 Request permission

Abstract:

Given separable Frechet spaces, E, F, and G, let $\mathcal {L}(E,F),\mathcal {L}(F,G)$, and $\mathcal {L}(E,G)$ denote the space of continuous linear operators from E to F , F to G, and E to G, respectively. We topologize these spaces of operators by any one of a family of topologies including the topology of pointwise convergence and the topology of compact convergence. We will show that if $(X,\mathcal {F})$ is any measurable space and both $A:X \to \mathcal {L}(E,F)$ and $B:X \to \mathcal {L}(F,G)$ are Borelian, then the operator composition $BA:X \to \mathcal {L}(E,G)$ is also Borelian. Further, we will give several consequences of this result.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 1815-1820
  • MSC: Primary 28B05; Secondary 46E40, 47A56, 47B99
  • DOI: https://doi.org/10.1090/S0002-9939-1995-1242072-9
  • MathSciNet review: 1242072