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Displaying similar documents to “On some properties of Hamel bases and their applications to Marczewski measurable functions”

Continuous-, derivative-, and differentiable-restrictions of measurable functions

Jack Brown (1992)

Fundamenta Mathematicae

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We review the known facts and establish some new results concerning continuous-restrictions, derivative-restrictions, and differentiable-restrictions of Lebesgue measurable, universally measurable, and Marczewski measurable functions, as well as functions which have the Baire properties in the wide and restricted senses. We also discuss some known examples and present a number of new examples to show that the theorems are sharp.

Functions Equivalent to Borel Measurable Ones

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.

On the difference property of Borel measurable functions

Hiroshi Fujita, Tamás Mátrai (2010)

Fundamenta Mathematicae

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If an atomlessly measurable cardinal exists, then the class of Lebesgue measurable functions, the class of Borel functions, and the Baire classes of all orders have the difference property. This gives a consistent positive answer to Laczkovich's Problem 2 [Acta Math. Acad. Sci. Hungar. 35 (1980)]. We also give a complete positive answer to Laczkovich's Problem 3 concerning Borel functions with Baire-α differences.