The composition of operatorvalued measurable functions is measurable
HTML articles powered by AMS MathViewer
 by A. Badrikian, G. W. Johnson and Il Yoo PDF
 Proc. Amer. Math. Soc. 123 (1995), 18151820 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.References

A. Badrikian, Séminaire sur les fonctions aléatoires et les measures cylindriques, Lecture Notes in Math., vol. 139, Springer, Berlin, 1970.
 Donald L. Cohn, Measure theory, Birkhäuser, Boston, Mass., 1980. MR 578344, DOI 10.1007/9781489903990
 B. DeFacio, G. W. Johnson, and M. L. Lapidus, Feynman’s operational calculus as a generalized path integral, Stochastic processes, Springer, New York, 1993, pp. 51–60. MR 1427300 —, Feynman’s operational calculus and evolution equations (in preparation).
 N. Dinculeanu, Vector measures, Hochschulbücher für Mathematik, Band 64, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966. MR 0206189 —, Integration on locally compact spaces, Noordhoff, Leyden, the Netherlands, 1974.
 Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A WileyInterscience Publication. MR 1009162
 I. M. Gel’fand and N. Ya. Vilenkin, Generalized functions. Vol. 4: Applications of harmonic analysis, Academic Press, New YorkLondon, 1964. Translated by Amiel Feinstein. MR 0173945
 Einar Hille, Functional Analysis and SemiGroups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, New York, 1948. MR 0025077
 G. W. Johnson, The product of strong operator measurable functions is strong operator measurable, Proc. Amer. Math. Soc. 117 (1993), no. 4, 1097–1104. MR 1123654, DOI 10.1090/S0002993919931123654X
 G. W. Johnson and D. L. Skoug, The CameronStorvick function space integral: an $L(L_{p},L_{p’})$ theory, Nagoya Math. J. 60 (1976), 93–137. MR 407228, DOI 10.1017/S0027763000017189
 Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
 Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
 Laurent Schwartz, Radon measures on arbitrary topological spaces and cylindrical measures, Tata Institute of Fundamental Research Studies in Mathematics, No. 6, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1973. MR 0426084
 Maurice Sion, A theory of semigroup valued measures, Lecture Notes in Mathematics, Vol. 355, SpringerVerlag, BerlinNew York, 1973. MR 0450503, DOI 10.1007/BFb0060133
Additional Information
 © Copyright 1995 American Mathematical Society
 Journal: Proc. Amer. Math. Soc. 123 (1995), 18151820
 MSC: Primary 28B05; Secondary 46E40, 47A56, 47B99
 DOI: https://doi.org/10.1090/S00029939199512420729
 MathSciNet review: 1242072