Bimeasurable functions
R. Purves (1966)
Fundamenta Mathematicae
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R. Purves (1966)
Fundamenta Mathematicae
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Fisher, David, Morris, Dave Witte, Whyte, Kevin (2004)
The New York Journal of Mathematics [electronic only]
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Steven Shreve (1981)
Fundamenta Mathematicae
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Andrzej Komisarski, Henryk Michalewski, Paweł Milewski (2010)
Bulletin of the Polish Academy of Sciences. Mathematics
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Let X and Y be two Polish spaces. Functions f,g: X → Y are called equivalent if there exists a bijection φ from X onto itself such that g∘φ = f. Using a theorem of J. Saint Raymond we characterize functions equivalent to Borel measurable ones. This characterization answers a question asked by M. Morayne and C. Ryll-Nardzewski.
B. Rao (1970)
Fundamenta Mathematicae
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Ashok Maitra (1970)
Fundamenta Mathematicae
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Greg Hjorth, Alexander S. Kechris (2001)
Fundamenta Mathematicae
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Let E₀ be the Vitali equivalence relation and E₃ the product of countably many copies of E₀. Two new dichotomy theorems for Borel equivalence relations are proved. First, for any Borel equivalence relation E that is (Borel) reducible to E₃, either E is reducible to E₀ or else E₃ is reducible to E. Second, if E is a Borel equivalence relation induced by a Borel action of a closed subgroup of the infinite symmetric group that admits an invariant metric, then either E is reducible...
Gundersen, Gary G., Steinbart, Enid M., Wang, Shupei (1998)
Annales Academiae Scientiarum Fennicae. Mathematica
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Petr Holický (2004)
Bulletin of the Polish Academy of Sciences. Mathematics
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We prove an abstract version of the Kuratowski extension theorem for Borel measurable maps of a given class. It enables us to deduce and improve its nonseparable version due to Hansell. We also study the ranges of not necessarily injective Borel bimeasurable maps f and show that some control on the relative classes of preimages and images of Borel sets under f enables one to get a bound on the absolute class of the range of f. This seems to be of some interest even within separable spaces. ...
K. Musiał (1973)
Colloquium Mathematicae
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Michael Rice, George Reynolds (1980)
Fundamenta Mathematicae
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Ashok Maitra, Victor Pestien, S. Ramakrishnan (1990)
Fundamenta Mathematicae
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Dana Scott (1964)
Fundamenta Mathematicae
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Alessandro Andretta, Donald A. Martin (2003)
Fundamenta Mathematicae
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Two sets of reals are Borel equivalent if one is the Borel pre-image of the other, and a Borel-Wadge degree is a collection of pairwise Borel equivalent subsets of ℝ. In this note we investigate the structure of Borel-Wadge degrees under the assumption of the Axiom of Determinacy.
Verónica Becher, Pablo Ariel Heiber, Theodore A. Slaman (2014)
Fundamenta Mathematicae
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We show that the set of absolutely normal numbers is Π⁰₃-complete in the Borel hierarchy of subsets of real numbers. Similarly, the set of absolutely normal numbers is Π⁰₃-complete in the effective Borel hierarchy.
Benjamin D. Miller (2007)
Fundamenta Mathematicae
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Answering a question of Kłopotowski, Nadkarni, Sarbadhikari, and Srivastava, we characterize the Borel sets S ⊆ X × Y with the property that every Borel function f: S → ℂ is of the form f(x,y) = u(x) + v(y), where u: X → ℂ and v: Y → ℂ are Borel.