We show that in a countably metacompact space, if a Baire measure admits a Borel extension, then it admits a regular Borel extension. We also prove that under the special axiom ♣ there is a Dowker space which is quasi-Mařík but not Mařík, answering a question of H. Ohta and K. Tamano, and under P(c), that there is a Mařík Dowker space, answering a question of W. Adamski. We answer further questions of H. Ohta and K. Tamano by showing that the union of a Mařík space and a compact space is Mařík, that under "c is real-valued measurable", a Baire subset of a Mařík space need not be Mařík, and finally, that the preimage of a Mařík space under an open perfect map is Mařík.
@article{bwmeta1.element.bwnjournal-article-fmv154i3p275bwm,
author = {J. Aldaz},
title = {Borel extensions of Baire measures},
journal = {Fundamenta Mathematicae},
volume = {154},
year = {1997},
pages = {275-293},
zbl = {0904.28013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv154i3p275bwm}
}
Aldaz, J. Borel extensions of Baire measures. Fundamenta Mathematicae, Tome 154 (1997) pp. 275-293. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv154i3p275bwm/
[00000] [Ad1] W. Adamski, On regular extensions of contents and measures, J. Math. Anal. Appl. 127 (1987), 211-225. | Zbl 0644.28002
[00001] [Ad2] W. Adamski, On the interplay between a topology and its associated Baire and Borel σ-algebra, Period. Math. Hungar. 21 (2) (1987), 85-93.
[00002] [Ad3] W. Adamski, τ-smooth Borel measures on topological spaces, Math. Nachr. 78 (1977), 97-107.
[00003] [Ba] W. Bade, Two properties of the Sorgenfrey plane, Pacific J. Math. 51 (1974), 349-354. | Zbl 0308.54031
[00004] [Be] M. G. Bell, On the combinatorial principle P(c), Fund. Math. 114 (1981), 149-157. | Zbl 0581.03038
[00005] [BB] K. P. S. Bhaskara Rao and M. Bhaskara Rao, Theory of Charges, Academic Press, 1983. | Zbl 0516.28001
[00006] [vD] E. K. van Douwen, Covering and separation properties of box products, in: Surveys in General Topology, G. M. Reed (ed.), Academic Press, 1980, 55-129. | Zbl 0453.54005
[00007] [Do] C. H. Dowker, On countably paracompact spaces, Canad. J. Math. 3 (1951), 219-224. | Zbl 0042.41007
[00008] [Eng] R. Engelking, General Topology, Heldermann, Berlin, 1989.
[00009] [F1] D. H. Fremlin, Consequences of Martin's Axiom, Cambridge Univ. Press, 1984. | Zbl 0551.03033
[00010] [F2] D. H. Fremlin, Real-valued-measurable cardinals, in: Set Theory of the Reals, H. Judah (ed.), Israel Math. Conf. Proc. 6, Bar-Ilan Univ., Ramat Gan, 1993, 151-304.
[00011] [F3] D. H. Fremlin, Uncountable powers of ℝ can be almost Lindelöf, Manuscripta Math. 22 (1977), 77-85. | Zbl 0396.54015
[00012] [Fro] Z. Frolík, Applications of complete families of continuous functions to the theory of Q-spaces, Czechoslovak Math. J. 11 (1961), 115-133.
[00013] [GJ] L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand, Princeton, N.J., 1989.
[00014] [Grz] E. Grzegorek, Solution of a problem of Banach on σ-fields without continuous measures, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), 7-10. | Zbl 0483.28003
[00015] [HRR] A. W. Hager, G. D. Reynolds and M. D. Rice, Borel-complete topological spaces, Fund. Math. 75 (1972), 135-143. | Zbl 0208.25101
[00016] [Ish] F. Ishikawa, On countably paracompact spaces, Proc. Japan Acad. 31 (1955), 686-687. | Zbl 0066.41001
[00017] [Ka] A. Kato, Union of realcompact spaces and Lindelöf spaces, Canad. J. Math. 31 (1979), 1247-1268. | Zbl 0453.54015
[00018] [Ki] R. B. Kirk, Locally compact, B-compact spaces, Indag. Math. 31 (1969), 333-344. | Zbl 0188.28003
[00019] [Kn] J. D. Knowles, Measures on topological spaces, Proc. London Math. Soc. (3) 17 (1967), 139-156.
[00020] [Ku] K. Kunen, Inaccessibility properties of cardinals, Ph.D. thesis, Stanford Univ., 1968.
[00021] [Lem] J. Lembcke, Konservative Abbildungen und Fortsetzung regulärer Masse, Z. Wahrsch. Verw. Gebiete 15 (1970), 57-96.
[00022] [Ma] J. Mařík, The Baire and Borel measure, Czechoslovak Math. J. 7 (1957), 248-253. | Zbl 0091.05501
[00023] [Mo] W. Moran, The additivity of measures on completely regular spaces, J. London Math. Soc. 43 (1968), 633-639. | Zbl 0159.07802
[00024] [Na] K. Nagami, Countable paracompactness of inverse limits and products, Fund. Math. 73 (1972), 261-270. | Zbl 0226.54005
[00025] [OT] H. Ohta and K. Tamano, Topological spaces whose Baire measure admits a regular Borel extension, Trans. Amer. Math. Soc. 317 (1990), 393-415. | Zbl 0691.54009
[00026] [Ost] A. J. Ostaszewski, On countably compact, perfectly normal spaces, J. London Math. Soc. (2) 14 (1976), 505-516. | Zbl 0348.54014
[00027] [Ox] J. C. Oxtoby, Homeomorphic measures in metric spaces, Proc. Amer. Math. Soc. 24 (1970), 419-423. | Zbl 0187.00902
[00028] [Pa] J. K. Pachl, Disintegration and compact measures, Math. Scand. 43 (1978), 157-168. | Zbl 0402.28006
[00029] [Ru1] M. E. Rudin, Dowker spaces, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (eds.), North-Holland, 1984, 761-780.
[00030] [Ru2] M. E. Rudin, A normal space X for which X × I is not normal, Fund. Math. 73 (1971), 179-186. | Zbl 0224.54019
[00031] [S] B. M. Scott, Some "almost-Dowker" spaces, Proc. Amer. Math. Soc. 68 (1978), 359-364. | Zbl 0378.54011
[00032] [SS] L. A. Steen and J. A. Seebach, Counterexamples in Topology, Springer, 1986. | Zbl 0211.54401
[00033] [St] A. H. Stone, Paracompactness and product spaces, Bull. Amer. Math. Soc. 58 (1948), 977-982. | Zbl 0032.31403
[00034] [Sz] P. J. Szeptycki, Dowker spaces, in: Topology Atlas 1, D. Shakhmatov and S. Watson (eds.), electronic publication, 1996, 45-47.
[00035] [T] F. Topsœ, On construction of measures, in: Proc. Conf. "Topology and Measure I" (Zinnowitz 1974), Part 2, J. Flachsmeyer, Z. Frolík and F. Terpe (eds.), Ernst-Moritz-Arndt Univ., Greifswald, 1978, 343-381.
[00036] [U] S. Ulam, Zur Masstheorie in der allgemeinen Mengenlehre, Fund. Math. 16 (1930), 140-150. | Zbl 56.0920.04
[00037] [W] W. Weiss, Versions of Martin's axiom, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (eds.), North-Holland, 1984, 827-886.
[00038] [Wh1] R. F. Wheeler, A survey of Baire measures and strict topologies, Exposition. Math. 77 (1983), 97-190. | Zbl 0522.28009
[00039] [Wh2] R. F. Wheeler, Extensions of a σ-additive measure to the projective cover, in: Lecture Notes in Math. 794, Springer, 1980, 81-104.