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On some properties of Hamel bases and their applications to Marczewski measurable functions. (English) Zbl 1259.28005
Summary: We introduce new properties of Hamel bases. We show that it is consistent with ZFC that such Hamel bases exist. Under the assumption that there exists a Hamel basis with one of these properties, we construct a discontinuous and additive function that is Marczewski measurable. Moreover, we show that such a function can additionally have the intermediate value property (and even be an extendable function). Finally, we examine sums and limits of such functions.

28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
03E35 Consistency and independence results
26A51 Convexity of real functions in one variable, generalizations
Full Text: DOI
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