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The composition of operator-valued measurable functions is measurable

Authors: A. Badrikian, G. W. Johnson and Il Yoo
Journal: Proc. Amer. Math. Soc. 123 (1995), 1815-1820
MSC: Primary 28B05; Secondary 46E40, 47A56, 47B99
MathSciNet review: 1242072
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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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Keywords: Lusin space, Souslin space, operator-valued measurable function, measurability of operator composition, strong operator measurability, Frechet space, Banach space, topology of simple convergence, topology of compact convergence
Article copyright: © Copyright 1995 American Mathematical Society

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