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THE FATOU COMPLETION OF A FRÉCHET FUNCTION SPACE AND APPLICATIONS

Part of: Set functions, measures and integrals with values in abstract spaces Connections with other structures, applications Linear function spaces and their duals Topological linear spaces and related structures

Published online by Cambridge University Press:  08 December 2009

R. DEL CAMPO  and
W. J. RICKER
R. DEL CAMPO
Affiliation:
Dpto. Matemática Aplicada I, EUITA, Ctra. de Utrera Km. 1, E–41013 Sevilla, Spain (email: [email protected])
W. J. RICKER*
Affiliation:
Math.-Geogr. Fakultät, Katholische Universität Eichstätt-Ingolstadt, D–85072 Eichstätt, Germany (email: [email protected])
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Abstract

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Given a metrizable locally convex-solid Riesz space of measurable functions we provide a procedure to construct a minimal Fréchet (function) lattice containing it, called its Fatou completion. As an application, we obtain that the Fatou completion of the space L1(ν) of integrable functions with respect to a Fréchet-space-valued measure ν is the space L1w(ν) of scalarly ν-integrable functions. Further consequences are also given.

Keywords

Fréchet space (lattice)vector measureFatou propertyLebesgue topologyscalarly integrable function

MSC classification

Secondary: 28B05: Vector-valued set functions, measures and integrals 46A40: Ordered topological linear spaces, vector lattices 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 54H12: Topological lattices, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 1 , February 2010 , pp. 49 - 60
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1]Albanese, A. A., ‘Primary products of Banach spaces’, Arch. Math. 66 (1996), 397–405.CrossRefGoogle Scholar
[2]Albanese, A. A., ‘On subspaces of the spaces L ploc and of their strong duals’, Math. Nachr. 197 (1999), 5–18.CrossRefGoogle Scholar
[3]Aliprantis, C. D. and Burkinshaw, O., Locally Solid Riesz Spaces (Academic Press, New York, 1978).Google Scholar
[4]Bonet, J., Okada, S. and Ricker, W. J., ‘The canonical spectral measure and Köthe function spaces’, Quaest. Math. 29 (2006), 91–116.CrossRefGoogle Scholar
[5]Castillo, J. M. F., Díaz, J.C. and Motos, J., ‘On the Fréchet space L p−’, Manuscripta Math. 96 (1988), 219–230.CrossRefGoogle Scholar
[6]Curbera, G. P., ‘Operators into L 1 of a vector measure and applications to Banach lattices’, Math. Ann. 293 (1992), 317–330.CrossRefGoogle Scholar
[7]Curbera, G. P. and Ricker, W. J., ‘Banach lattices with the Fatou property and optimal domains of kernel operators’, Indag. Math. (N.S.) 17 (2006), 187–204.CrossRefGoogle Scholar
[8]del Campo, R. and Ricker, W. J., ‘The space of scalarly integrable functions for a Fréchet space-valued measure’, J. Math. Anal. Appl. 354 (2009), 641–647.CrossRefGoogle Scholar
[9]Drewnowski, L. and Labuda, I., ‘Copies of c 0 and ℓ ∞ in topological Riesz spaces’, Trans. Amer. Math. Soc. 350 (1998), 3555–3570.CrossRefGoogle Scholar
[10]Fremlin, D. H., Topological Riesz Spaces and Measure Theory (Cambridge University Press, Cambridge, 1974).CrossRefGoogle Scholar
[11]Kalton, N. J., ‘Exhaustive operators and vector measures’, Proc. Edinb. Math. Soc. 19 (1974), 291–300.CrossRefGoogle Scholar
[12]Kluvánek, I. and Knowles, G., Vector Measures and Control Systems (North-Holland, Amsterdam, 1976).Google Scholar
[13]Lewis, D. R., ‘On integrability and summability in vector spaces’, Illinois J. Math. 16 (1972), 294–307.CrossRefGoogle Scholar
[14]Luxemburg, W. A. J. and Zaanen, A. C., Riesz Spaces I (North-Holland, Amsterdam, 1971).Google Scholar
[15]Meise, R. G. and Vogt, D., Introduction to Functional Analysis (Clarendon Press, Oxford, 1997).CrossRefGoogle Scholar
[16]Metafune, G. and Moscatelli, V. B., ‘On the space ℓ p+=⋂ q>pℓ q’, Math. Nachr. 147 (1990), 7–12.CrossRefGoogle Scholar
[17]Okada, S. and Ricker, W. J., ‘Vector measures and integration in non-complete spaces’, Arch. Math. 63 (1994), 344–353.CrossRefGoogle Scholar
[18]Reiher, K., ‘Weighted inductive and projective limits of normed Köthe function spaces’, Results Math. 13 (1988), 147–161.CrossRefGoogle Scholar
[19]Ronglu, L. and Qingying, B., ‘Locally convex spaces containing no copy of c 0’, J. Math. Anal. Appl. 172 (1993), 205–211.Google Scholar
[20]Stefansson, G. F., ‘L 1 of a vector measure’, Le Matematiche 48 (1993), 219–234.Google Scholar
[21]Wnuk, W., ‘Locally solid Riesz spaces not containing c 0’, Bull. Pol. Acad. Sci. Math. 36 (1988), 51–56.Google Scholar
[22]Zaanen, A. C., Integration, 2nd rev. edn (North-Holland, Interscience, Amsterdam, New York, 1967).Google Scholar
[23]Zaanen, A. C., Riesz Spaces II (North-Holland, Amsterdam, 1983).Google Scholar