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Super-rigid families of strongly Blackwell spaces

Author: Manfred Droste
Journal: Proc. Amer. Math. Soc. 103 (1988), 803-808
MSC: Primary 28A20; Secondary 28A05, 54C05, 60A99
MathSciNet review: 947662
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Abstract: We construct a complete subfield $ F$ of $ P({\mathbf{R}})$, isomorphic to $ P({\mathbf{R}})$, of pairwise non-Borel-isomorphic rigid strong Blackwell subsets of $ {\mathbf{R}}$ such that there are only 'very few' measurable functions between any two members of $ F$. As a consequence, we obtain large chains and antichains of non-isomorphic rigid strong Blackwell subsets of $ {\mathbf{R}}$. Also, there is a collection of continuously many dense subsets of $ {\mathbf{R}}$ such that any two of them differ only by two elements, but none of them is a continuous image of any other.

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  • [1] K. P. S. Bhaskara Rao and B. V. Rao, Borel spaces, Dissertationes Math. (Rozprawy Mat.) 190 (1981), 1-63. MR 634451 (84c:28001)
  • [2] K. P. S. Bhaskara Rao and R. M. Shortt, Generalised Lusin sets with the Blackwell property, Fund. Math. 127 (1986), 9-39. MR 883149 (88j:54055)
  • [3] D. Blackwell, On the class of probability spaces, Proc. Third Berkeley Sympos. on Mathematical Statistics and Probability, vol. 2, Univ. of California Press, 1956, pp. 1-6. MR 0084882 (18:940d)
  • [4] M. Dugas and R. Göbel, Every cotorsion-free algebra is an endomorphism algebra, Math. Z. 181 (1982), 451-470. MR 682667 (84h:13008)
  • [5] D. Fremlin, On Blackwell algebras (unpublished manuscript, 1980).
  • [6] J. Hoffman-Jørgensen, The theory of analytic spaces, Various publication series, no. 10, Aarhus Univ., 1970. MR 0409748 (53:13500)
  • [7] J. Jasiński, On the combinatorial properties of Blackwell sets, Proc. Amer. Math. Soc. 93 (1985), 657-660. MR 776198 (86d:28002)
  • [8] K. Kuratowski, Topology, vol. I, Academic Press, New York; PWN, Warszawa, 1966. MR 0217751 (36:840)
  • [9] G. W. Mackey, Borel structures in groups and their duals, Trans. Amer. Math. Soc. 85 (1957), 134-165. MR 0089999 (19:752b)
  • [10] M. Orkin, A Blackwell space which is not analytic, Bull. Acad. Polon. Sci. 20 (1972), 437-438. MR 0316655 (47:5202)
  • [11] D. Ramachandran, Perfect measures, Macmillan-India, New Delhi, 1979.
  • [12] J. Rosenstein, Linear orderings, Academic Press, New York, 1982. MR 662564 (84m:06001)
  • [13] H. Sarbadhikari, Some contributions to descriptive set theory, Thesis, Indian Statistical Institute, Calcutta, 1977.
  • [14] R. M. Shortt, Borel-dense Blackwell spaces are strongly Blackwell, Colloq. Math. 53 (1987), 35-41. MR 890835 (88f:54073)
  • [15] -, Notions of independence for random variables, Probab. Math. Stat. (to appear). MR 928121 (89k:60006)
  • [16] -, A separation principle for Blackwell sets, Bull. Polon. Acad. Sci. Math. 34 (1986), 643-645. MR 890608 (88f:54074)
  • [17] W. Sierpiński, Sur les types d'ordre des ensembles linéaires, Fund. Math. 37 (1950), 253-264. MR 0041909 (13:19c)

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Keywords: Blackwell space, strong Blackwell space, rigid Borel space, separable space, measurable function
Article copyright: © Copyright 1988 American Mathematical Society

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