The composition of operator-valued measurable functions is measurable

Badrikian, A.; Johnson, G. W.; Yoo, Il

doi:10.1090/S0002-9939-1995-1242072-9

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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