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Measurability in \(C({2^\kappa })\) } and Kunen cardinals

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Abstract

A cardinal κ is called a Kunen cardinal if the σ-algebra on κ × κ generated by all products A×B, where A,B ⊂ κ, coincides with the power set of κ×κ. For any cardinal κ, let \(C({2^\kappa })\) be the Banach space of all continuous real-valued functions on the Cantor cube \(C({2^\kappa })\) . We prove that κ is a Kunen cardinal if and only if the Baire σ-algebra on \(C({2^\kappa })\) for the pointwise convergence topology coincides with the Borel σ-algebra on \(C({2^\kappa })\) for the norm topology. Some other links between Kunen cardinals and measurability in Banach spaces are also given.

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Authors and Affiliations

  1. Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30100, Espinardo (Murcia), Spain

    A. Avilés

  2. Instytut Matematyczny, Uniwersytet Wrocławski, Pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland

    G. Plebanek

  3. Departamento de Matemática Aplicada, Facultad de Informática, Universidad de Murcia, 30100, Espinardo (Murcia), Spain

    J. Rodríguez

Authors
  1. A. Avilés
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  2. G. Plebanek
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  3. J. Rodríguez
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Correspondence to A. Avilés.

Additional information

A. Avilés and J. Rodríguez were supported by MEC and FEDER (Project MTM2008-05396) and Fundación Séneca (Project 08848/PI/08). A. Avilés was supported by Ramon y Cajal contract (RYC-2008-02051) and an FP7-PEOPLEERG-2008 action.

G. Plebanek was partially supported by MNiSW Grant N N201 418939 (2010–2013)

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Avilés, A., Plebanek, G. & Rodríguez, J. Measurability in \(C({2^\kappa })\) } and Kunen cardinals. Isr. J. Math. 195, 1–30 (2013). https://doi.org/10.1007/s11856-012-0122-0

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  • DOI: https://doi.org/10.1007/s11856-012-0122-0

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