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1995 | 147 | 3 | 261-277
Tytuł artykułu

Properties of the class of measure separable compact spaces

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Abstrakty
EN
We investigate properties of the class of compact spaces on which every regular Borel measure is separable. This class will be referred to as MS. We discuss some closure properties of MS, and show that some simply defined compact spaces, such as compact ordered spaces or compact scattered spaces, are in MS. Most of the basic theory for regular measures is true just in ZFC. On the other hand, the existence of a compact ordered scattered space which carries a non-separable (non-regular) Borel measure is equivalent to the existence of a real-valued measurable cardinal ≤${\ninegot c}$. We show that not being in MS is preserved by all forcing extensions which do not collapse $ω_1$, while being in MS can be destroyed even by a ccc forcing.
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Twórcy
  • Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, U.S.A., kunen@cs.wisc.edu
Bibliografia
  • [1] I. Bandlow, On the origin of new compact spaces in forcing models, Math. Nachr. 139 (1988), 185-191.
  • [2] M. Džamonja and K. Kunen, Measures on compact HS spaces, Fund. Math. 143 (1993), 41-54.
  • [3] D. Fremlin, Consequences of Martin's Axiom, Cambridge Univ. Press, 1984.
  • [4] D. Fremlin, Real-valued measurable cardinals, in: Israel Math. Conf. Proceedings, Haim Judah (ed.), Vol. 6, 1993, 151-304.
  • [5] R. J. Gardner and W. F. Pfeffer, Borel measures, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (ed.), North-Holland, 1984, 961-1044.
  • [6] M. Gitik and S. Shelah, Forcings with ideals and simple forcing notions, Israel J. Math. 68 (1989), 129-160.
  • [7] M. Gitik and S. Shelah, More on simple forcing notions and forcings with ideals, Ann. Pure Appl. Logic 59 (1993), 219-238.
  • [8] P. Halmos, Measure Theory, Van Nostrand, 1950.
  • [9] R. Haydon, On dual $L^1$-spaces and injective bidual Banach spaces, Israel J. Math. 31 (1978), 142-152.
  • [10] J. Henry, Prolongement des mesures de Radon, Ann. Inst. Fourier 19 (1969), 237-247.
  • [11] K. Kunen, A compact L-space under CH, Topology Appl. 12 (1981), 283-287.
  • [12] K. Kunen and J. van Mill, Measures on Corson compact spaces, Fund. Math. this volume, 61-72.
  • [13] D. Maharam, On homogeneous measure algebras, Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 108-111.
  • [14] H. P. Rosenthal, On injective Banach spaces and the spaces $L^∞ (μ)$ for finite measures μ, Acta Math. 124 (1970), 205-248.
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Bibliografia
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bwmeta1.element.bwnjournal-article-fmv147i3p261bwm
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