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.
@article{bwmeta1.element.doi-10_2478_s11533-012-0144-1,
author = {Fran\c cois Dorais and Rafa\l\ Filip\'ow and Tomasz Natkaniec},
title = {On some properties of Hamel bases and their applications to Marczewski measurable functions},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {487-508},
zbl = {1259.28005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0144-1}
}
François Dorais; Rafał Filipów; Tomasz Natkaniec. On some properties of Hamel bases and their applications to Marczewski measurable functions. Open Mathematics, Tome 11 (2013) pp. 487-508. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0144-1/
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