Abstract
It is an open problem if any separable compact space K whose function space C(K) with the cylindrical σ-algebra is a standard measurable space, embeds in the space of the first Baire class functions on the Cantor set, with the pointwise topology. We prove that this is true for separable linearly ordered compacta.
Other problems discussed in this note concern the Borel structures in C(K) generated by the norm, weak or pointwise topology in C(K). We give an example of a compact space K such that the weak and the pointwise topology generate different Borel structures in C(K).
Resumen
Es un problema abierto saber si cada espacio compacto separable K cuyo espacio de funciones C(K) con la σ-álgebra cilíndrica es un espacio medible estándar, se sumerge en el espacio de las funciones de la primera clase de Baire definidas en el conjunto de Cantor, con la topología de convergencia puntual. Aquí probamos que lo anterior es cierto para compactos separables linealmente ordenados. Discutimos en esta nota otros problemas relativos a las estructuras de Borel en C(K) generadas por la norma, topología débil o topología de convergencia puntual en C(K). Damos un ejemplo de un espacio compacto K tal que la topología débil y la topología de convergencia puntual generan estructuras de Borel diferentes en C(K).
Similar content being viewed by others
References
Albiac, F. and Kalton, N. J., (2006). Topics in Banach Space Theory, Springer.
Bessaga, C. and Pelczyński, A., (1975). Selected Topics in Infinite-Dimensional Topology, Polish Scientific Publishers, Warsaw.
Burke, D. K. and Pol, R., (2003). On Borel sets in function spaces with the weak topology, J. Lond. Math. Soc. (2), 68, 3, 725–738. DOI: 10.1112/S0024610703004812
Dodos, P., (2006). Codings of separable compact subsets of the first Baire class, Ann. Pure Appl. Logic, 142, 425–441. DOI: 10.1016/j.apal.2006.05.006
Dow, A. and Hart, K. P., (2000). The measure algebra does not always embed, Fund. Math., 163, 163–176.
Edgar, G. A., (1977). Measurability in a Banach space, Indiana Univ. Math. J., 26, 4, 663–677.
Edgar, G. A., (1979). Measurability in a Banach space. II, Indiana Univ. Math. J., 28, 4, 559–579.
Fremlin, D. H., (1980). Borel sets in non-separable Banach spaces, Hokkaido Math. J., 9, 179–183.
Fremlin, D. H., (2003). Measure Theory, Vol. 4: Topological Measure Theory, Torres Fremlin.
Godefroy, G., (1980). Compacts de Rosenthal, Pacific J. Math., 91, 293–306.
Halmos, P. R., (1963). Lectures on Boolean Algebras, Van Nostrand.
Haydon, R.; Moltó, A. and Orihuela, J., (2007). Spaces of functions with countably many discontinuities, Israel J. Math., 158, 19–39. DOI: 10.1007/s11856-007-0002-1
Kechris, A. S., (1995). Classical Descriptive Set Theory, Springer-Verlag, New York.
Kenderov, P. S.; Kortezov, I. S. and Moors, W. B., (2006). Norm continuity of weakly continuous mappings into Banach spaces, Topology Appl., 153, 2745–2759. DOI: 10.1016/j.topol.2005.11.007
Kuratowski, K., (1966). Topology I, Academic Press and PWN, New York and London.
Marciszewski, W., (1989). On a classification of pointwise compact sets of the first Baire class functions, Fund. Math., 133, 195–209.
Marciszewski, W., (2008). Modifications of the double arrow space and related Banach spaces C(K), Studia Math., 184, 249–262. DOI: 10.4064/sm184-3-4
Marciszewski, W. and Pol, R., (2009). On Banach spaces whose norm-open sets are F σ -sets in the weak topology, J. Math. Anal. Appl., 350, 708–722. DOI: 10.1016/j.jmaa.2008.06.007
Marciszewski, W. and Pol, R., (2010). On Borel almost disjoint families, preprint.
Orihuela, J., (2007). Topological open problems in the geometry of Banach spaces, Extracta Math., 22, 197–213.
Ostaszewski, A. J., (1974). A characterization of compact, separable, ordered spaces, J. Lond. Math. Soc. (2), 7, 758–760. DOI: 10.1112/jlms/s2-7.4.758
Oxtoby, J. C., (1960/1961). Spaces that admit a category measure, J. Reine Angew. Math., 205, 156–170. DOI: 10.1515/crll.1960.205.156
Pol, R., (1986). Note on compact sets of first Baire class functions, Proc. Amer. Math. Soc., 96, 152–154. DOI: 10.1090/S0002-9939-1986-0813828-3. DOI: 10.2307/2045670
Raja, M., (1999). Kadec norms and Borel sets in a Banach space, Studia Math., 136, 1–16.
Sakai, S., (1971). C*-algebras and W*-algebras, Springer.
Talagrand, M., (1978). Comparaison des boreliens d’un espace de Banach pour les topologies fortes et faibles, Indiana Univ. Math. J., 27, 1001–1004.
Todorcevic, S., (2005). Representing trees as relatively compact subsets of the first Baire class, Bull. Cl. Sci. Math. Nat. Sci. Math., 30, 29–45. DOI: 10.2298/BMAT0530029T
Vakhania, N. N.; Tarieladze, V. I. and Chobanyan, S. A., (1991). Probability Distributions on Banach Spaces, Kluwer.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Manuel Valdivia on the occasion of his 80th birthday
Submitted by Fernando Bombal
Rights and permissions
About this article
Cite this article
Marciszewski, W., Pol, R. On some problems concerning Borel structures in function spaces. RACSAM 104, 327–335 (2010). https://doi.org/10.5052/RACSAM.2010.20
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.5052/RACSAM.2010.20